2026 paper questions

$F\propto 1/r^2$. With the distance doubled, $F^{\prime }=G\frac{m^2}{(2r{)}^2}=\frac{1}{4}G\frac{m^2}{r^2}=\frac{F}{4}$. Option C ($F/2$) would be true only if the force were inversely proportional to the distance, not to its square.

At $h=R_e$ the distance from the centre is $2R_e$, so $g_h=g{\left(\frac{R_e}{R_e+h}\right)}^2=g{\left(\frac{R_e}{2R_e}\right)}^2=\frac{g}{4}$. Option D ($g/9$) is the value at $h=2R_e$, where the distance from the centre is $3R_e$.

How bright a star appears is its radiant flux intensity, $F=L/4\pi d^2$. With the same $L$, a smaller $F$ can only come from a larger distance $d$. Options B, C and D would each change the luminosity itself ($L=4\pi r^2\sigma T^4$), but the question says the luminosities are equal.

The formula list printed on the paper gives $v=v_0\sin \theta$. Since $\sin \theta$ has its largest value, $1$, at $\theta =\pi /2$, the velocity is maximum ($v=v_0$) there. This is the moment the body passes through the mean position; at $\theta =0$ it is at an extreme and $v=0$.

$\text{T.E.}=\frac{1}{2}m{\omega }^2{x}_0^2$, so the total energy is proportional to $m{\omega }^2{x}_0^2$. The other options do not have this form (energy grows with the square of both $\omega$ and the amplitude).

More lines per unit length means a smaller grating element $d=1/N$. From $\sin \theta =m\lambda /d$ every order moves to a larger angle, so the maxima are more widely spaced. With more lines sharing the beam the maxima are also narrower and brighter, that is sharper. Options C and D each get only one of the two changes right.

Compare with $V=V_0\sin \omega t$: the peak value is the number in front of the sine, $V_0=16\mathrm{V}$. The number $2\times 10^3$ is the angular frequency $\omega$ in $\mathrm{rad}{\mathrm{s}}^{-1}$, and $10^3$ is the reactance $X_C=1/(\omega C)=1000\Omega$, not a voltage.

$T=2\pi \sqrt{l/g}$, so $T\propto \sqrt{l}$. Four times the length doubles the period: $2\times 2.5=5\mathrm{s}$.

A bridge rectifier uses four diodes to convert AC into (pulsating) DC by full-wave rectification. A transformer changes the voltage, a capacitor stores charge, and an inverter changes DC into AC.

In a resistor $I=V/R$ at every instant, so the voltage and current pass through zero and reach their peaks together: they are in phase, $\Phi =0$. A phase difference of $\pi /2$ belongs to a pure inductor or a pure capacitor.

Brighter light means more photons per second, and each photon frees one electron, so more electrons are emitted per second (larger current). The kinetic energy and stopping potential depend on frequency, not intensity, and the work function is a property of the metal.

de Broglie's relation is $\lambda =h/p$ with momentum $p=mv$, so $\lambda =h/mv$. The other options put wrong numerical factors in front.

In the Sun, hydrogen nuclei fuse to form helium, releasing energy: this is fusion. Uranium decay and radioactive decay are spontaneous emissions from unstable nuclei, and an atomic bomb works by uncontrolled fission.

$F=L/4\pi d^2$, so $F\propto 1/d^2$. Doubling $d$ multiplies $F$ by $(1/2{)}^2=1/4$. 'Half' would be true only if $F$ were proportional to $1/d$.

Using $F_c=2mv\omega \sin \theta$ with $\omega =7.3\times 10^{-5}\mathrm{rad}{\mathrm{s}}^{-1}$ (printed on the paper): $F_c=2(2\times 10^5)(22.2)(7.3\times 10^{-5})\sin 25^{\circ }=648\times 0.423=274\mathrm{N}$.

These are the numbers of the book's Example 24.2. The other options have the same digits with the wrong power of ten.

A receding source's waves are stretched (Doppler effect), so its spectral lines move to longer wavelengths, towards the red end of the spectrum: a red shift. A blue shift is seen when a source approaches; 'green shift' and 'yellow light' are not Doppler effects.

Red light has the longest wavelength in the visible band (about $750\mathrm{nm}$). Since $f=c/\lambda$, it has the lowest frequency. Violet has the highest.

Effect (1 mark). The orbital velocity decreases: a satellite in a higher orbit moves more slowly.

Mathematical explanation (2 marks). For a circular orbit of radius $r$ (from the Earth's centre), gravity provides the centripetal force:

A higher orbit has a larger $r$, so $v_o$ is smaller. If $r$ becomes 4 times larger, $v_o$ becomes half. Example: the ISS at $400\mathrm{km}$ moves at $7.7\mathrm{km}{\mathrm{s}}^{-1}$, but a geostationary satellite ($r=4.23\times 10^7\mathrm{m}$) moves at only $3.1\mathrm{km}{\mathrm{s}}^{-1}$.

Marks: formula (1), substitution in SI units (1), answer with unit (1). This is Assignment 17.2 of the book.

The pressure increases (for a gas kept in a fixed volume).

When the temperature rises, the average translational kinetic energy of the molecules, $\frac{3}{2}kT$, increases, so the molecules move faster ($v_{\mathrm{rms}}\propto \sqrt{T}$). Faster molecules raise the pressure in two ways:

1. each collision with a wall gives a larger change of momentum, $2m{v}_x$ (the molecules hit harder);
2. the molecules reach the walls more often, because the time between collisions with the same wall, $2l/v_x$, is shorter.

So more momentum is delivered to each unit area of the wall every second, and the pressure rises. Mathematically:

At constant volume $P\propto ⟨v^2⟩\propto T$: doubling the absolute temperature doubles the pressure, and doubling the rms speed makes the pressure four times larger. (If the gas is free to expand at constant pressure, its volume increases instead.)

Constructive interference (1.5 marks)

When two coherent light waves meet in phase, crest on crest and trough on trough, their displacements add and the resultant wave has a larger amplitude. The light there is brighter: a bright fringe. This happens when the path difference is a whole number of wavelengths:

Destructive interference (1.5 marks)

When two coherent light waves meet in anti-phase, crest on trough, they cancel and the resultant amplitude is small or zero. The point is dark: a dark fringe. This happens when the path difference is an odd number of half wavelengths:

In Young's double slit the path difference to a point at angle $\theta$ is $d\sin \theta$, so bright fringes need $d\sin \theta =m\lambda$ and dark fringes need $d\sin \theta =\left(m+\frac{1}{2}\right)\lambda$.

The frequency response of a forced oscillator is the way its amplitude changes with the driving frequency. The amplitude is largest (resonance) when the driving frequency equals the natural frequency $f_0$. The sharpness of resonance is how quickly the amplitude falls on either side of $f_0$. The factors are:

• Natural frequency of the oscillator. It decides where the peak lies. It is set by the oscillator itself, for example $f_0=\frac{1}{2\pi }\sqrt{k/m}$ for a spring or $\frac{1}{2\pi }\sqrt{g/l}$ for a pendulum.
• Damping (the main factor for sharpness). With small damping little energy is lost per cycle, so near $f_0$ energy builds up to a large amplitude and the peak is tall and narrow: a sharp resonance. With heavy damping energy is lost quickly, so the peak is low and broad: a flat resonance.
• Strength of the driving force. A stronger driving force raises the whole curve but does not move the peak.

The output of a bridge rectifier is pulsating DC. A capacitor connected in parallel with the load smooths it into a nearly constant DC; it acts as a filter.

1. As the rectified voltage rises to its peak, the capacitor charges to the peak voltage V_0.
2. When the rectified voltage falls, the diodes switch off and the capacitor discharges slowly through the load, supplying the current.
3. The next pulse recharges it before its voltage has fallen much, so the gaps between the peaks are filled in.

The ripple left in the output is small when the time constant RC is much longer than the time between pulses: a larger capacitance or a larger load resistance gives better smoothing, while a small load resistance makes the capacitor discharge faster.

Self-inductance (1 mark)

When the current in a coil changes, an emf \varepsilon_L=-L\,\Delta I/\Delta t is induced in the coil itself (self-induction). The constant L is the self-inductance: the ratio of the emf induced in the coil to the time rate of change of current through it.

SI unit and factors (2 marks)

SI unit: the henry (H), 1\,\mathrm{H}=1\,\mathrm{V\,s\,A^{-1}}. A coil has a self-inductance of 1 H if an emf of 1 V is induced in it when its current changes at 1\,\mathrm{A\,s^{-1}}.

For a long coil L=\dfrac{\mu_r\mu_0N^2A}{l}, so L depends on:

• the number of turns N (L\propto N^2);
• the area of cross-section A (L\propto A);
• the length l of the coil for the same turns (L\propto1/l);
• the core material, its relative permeability \mu_r (an iron core increases L).

From kinetic theory, the pressure of an ideal gas of N molecules, each of mass m, in volume V is P=\dfrac13\,\dfrac{Nm}{V}\langle v^{2}\rangle.

The absolute temperature of an ideal gas is directly proportional to the average translational kinetic energy of its molecules. Temperature is therefore a measure of the average kinetic energy of molecular motion; at 0\,\mathrm{K} this energy would be zero.

Capacitive reactance is inversely proportional to frequency:

When the frequency is doubled (f'=2f) with the same capacitor:

So doubling the frequency halves the capacitive reactance, and for the same voltage the current through the capacitor doubles. Reason: the capacitor is charged and discharged twice as often, so twice as much charge flows per second.

Marks: one for each of three explained points.

Wind patterns (the trade winds, westerlies and polar easterlies, and local winds) are an essential part of the climate system because they move heat, moisture and energy around the globe. Over a 30-year period their regular patterns form part of the climate of a region.

1. Heat transport: winds carry heat from the warm tropics towards the cold poles and reduce the temperature difference between them. They also drive the surface ocean currents, which carry warm water towards the poles.
2. Moisture transport and rainfall: winds carry water vapour from the oceans over the land and so decide where and when rain falls; for example, the monsoon winds bring most of Pakistan's rain.
3. Dispersion and daily weather: winds spread pollutants (dust, smoke, industrial emissions) and pollen over large areas, and land and sea breezes caused by differential heating change the daily weather of coastal regions.

Changes in wind patterns over long periods therefore change regional and even global climate.

Principle. Different tissues absorb X-rays by different amounts. Bone (calcium, high atomic number and density) absorbs X-rays strongly; soft tissue such as fat and muscle absorbs much less; air in the lungs hardly absorbs at all.

Image. A beam of X-rays is passed through the body onto a photographic plate or detector. Where many X-rays arrive (behind soft tissue) the image is dark; behind bone few arrive, so bones cast a white shadow. The result is an image in shades of grey.

Uses. It shows fractures, foreign bodies (a swallowed coin), teeth and chest problems. For soft tissues a contrast medium (barium meal, iodine dye) is used to increase contrast, or a CT scan gives a detailed image.

Pair production (1 mark)

Pair production is the creation of a particle and its antiparticle from energy. When a high-energy gamma-ray photon passes near a heavy nucleus, it is completely absorbed and an electron-positron pair is created. Charge, energy and momentum are conserved; energy is converted into mass.

Minimum energy (2 marks)

Climate inertia is the tendency of the Earth's climate to resist, or respond slowly to, changes in factors such as greenhouse gas concentrations and solar radiation. It is of two types (1.5 marks each):

1. Short-term inertia: the oceans and land surfaces absorb and release heat relatively slowly, so the temperature cannot change instantly. This smooths the daily and seasonal temperature changes: the hottest part of the day comes in the afternoon, a few hours after the Sun is highest, and places near the sea have smaller day and night temperature differences than deserts.

2. Long-term inertia: changes in greenhouse gas concentrations, such as \mathrm{CO_2}, take decades to centuries to affect the climate fully, because the deep ocean warms slowly and the carbon cycle and ice sheets respond slowly. Even if emissions were stabilised today, the Earth would keep warming for many years because of the heat already stored in the oceans.

Piezoelectric effect (1 mark)

The piezoelectric effect is the ability of certain materials (quartz, lead zirconate titanate) to generate an electric charge in response to applied mechanical stress. Squeezing the crystal separates its positive and negative charge centres, so a p.d. appears across its faces. The reverse also happens: an applied electric field changes the shape of the crystal (inverse piezoelectric effect).

Two applications (2 marks)

1. Production and detection of ultrasound: in medical ultrasound scanners an alternating p.d. makes the crystal vibrate and emit ultrasound, and the returning echoes make it produce electrical signals.
2. Generation of high voltage: a sharp blow on a piezoelectric crystal gives a spark in gas lighters and stove igniters.

Other applications: quartz crystals keep time in watches and computers (frequency generation), microbalances, and scanning probe microscopes.

The resolution of a microscope (its ability to show two close points separately) is limited by diffraction: it cannot show details much smaller than the wavelength of the radiation used. So the shorter the wavelength, the higher the resolution.

An electron microscope uses fast electrons, which behave as waves with de Broglie wavelength

where V is the accelerating potential difference. For V=10\,\mathrm{kV}, \lambda\approx1.2\times10^{-11}\,\mathrm{m} (0.12\,\text{\AA}), whereas visible light has \lambda\approx4000 to 7000\,\text{\AA}. The electron wavelength is thousands of times shorter, so the electron microscope resolves details down to about 0.1 to 0.2\,\mathrm{nm}, compared with about 200\,\mathrm{nm} for an optical microscope. Increasing V makes \lambda even shorter and the resolution even higher.

Control rods (1.5 marks)

Control rods are made of a material that absorbs neutrons strongly, such as boron or cadmium. They can be moved into or out of the reactor core. Pushing them in absorbs more neutrons, so fewer fissions occur; pulling them out allows more fissions. They are set so that only one neutron from each fission goes on to cause another fission, which keeps the chain reaction steady and the power output constant. Dropped fully in, they stop the reaction in an emergency.

Moderator (1.5 marks)

The moderator (water or graphite) slows down the fast neutrons produced by fission (energies of a few MeV) to thermal neutrons of about 0.025\,\mathrm{eV}. Slow neutrons are far more likely to be captured by uranium-235 and cause fission, so the moderator keeps the chain reaction going. The neutrons lose energy by colliding many times with the light nuclei (hydrogen or carbon) of the moderator.

Waveform (2 marks)

For a pure inductor, if I=I_0\sin\omega t then V=V_0\sin(\omega t+\pi/2) with V_0=\omega LI_0. The voltage reaches its peak a quarter cycle (90^\circ) before the current.

Phasor diagram (1 mark)

The current phasor is drawn along the horizontal and the voltage phasor at 90^\circ anticlockwise from it: the voltage leads the current by 90^\circ (the current lags).

The equatorial region receives more solar energy per square metre than the polar regions because:

1. Angle of the Sun's rays: at the equator sunlight falls almost vertically (direct sunlight), so a beam covers a small area and the energy per unit area is high. Near the poles the same beam falls at a low angle (oblique sunlight) and spreads over a much larger area, so the energy per unit area is small. On level ground I = I_0\cos\theta, where \theta is the latitude.
2. Longer path through the atmosphere: oblique rays near the poles pass through more air, so more of their energy is absorbed, scattered and reflected before it reaches the ground.
3. Ice and daylight: polar ice and snow reflect most of the sunlight that arrives (high albedo), while oceans and forests near the equator absorb it; the poles also have months of darkness in winter.

Idea (1 mark): red shift leads to the idea that the universe is expanding: galaxies, with their stars, are moving away from us and from each other.

Why (2 marks): red shift means the spectral lines of the star or galaxy are at longer wavelengths than their laboratory values. By the Doppler effect, this happens when the source moves away from the observer, with speed given by

Almost all galaxies, in every direction, show red shift, and the farther they are the greater the shift (v=H_0d). The simple explanation is that the space between the galaxies is stretching, like the rubber of an inflating balloon carrying its dots apart. So the universe is expanding; running this backwards leads to the Big Bang.

Alternative method: find the mass of one molecule, then use v_{\mathrm{rms}}=\sqrt{3kT/m}.

This is Example 16.2 of the book. Marks: formula 1, substitution 1, answer with unit 1.

(a) Mass defect (1.5 marks)

The mass of a nucleus is always less than the sum of the masses of its separate protons and neutrons. This difference is the mass defect: \Delta m=\left[Zm_p+(A-Z)m_n\right]-m_{\text{nucleus}}. For deuterium, \Delta m=(1.007276+1.008665)-2.013553=0.002388\,\mathrm{u}. The missing mass has been converted into the energy released when the nucleons bound together.

(b) Binding energy (1.5 marks)

The binding energy is the minimum energy required to break a nucleus into its separate nucleons; the same energy is released when the nucleus forms. It is the energy equivalent of the mass defect: E_b=\Delta mc^{2}, or E_b(\mathrm{MeV})=\Delta m(\mathrm{u})\times931.5. For deuterium, E_b=0.002388\times931.5=2.22\,\mathrm{MeV}. The greater the binding energy per nucleon, the more stable the nucleus.

The field strength at a distance r from the centre of a planet of mass M is

Factor 1: mass of the planet (1.5 marks). g\propto M. A more massive planet has a stronger field; for example Jupiter, with 316 times the Earth's mass, has g\approx24.8\,\mathrm{m\,s^{-2}} at its surface.

Factor 2: distance from the planet's centre (1.5 marks). g\propto1/r^{2}. At the surface r equals the planet's radius R, so for the same mass a smaller planet has a stronger surface field. Above the surface, g_h=GM/(R+h)^{2} falls with height; at h=R it is only g/4.

g does not depend on the mass of the body placed in the field.

Light damping (1 mark). The damping force is small, so the body keeps oscillating while its amplitude decreases gradually with time. Example: a swing, or a pendulum swinging in air.

Heavy damping (1 mark). The damping force is very large, so the body does not complete even one oscillation; it moves back slowly to its equilibrium position and takes a long time to come to rest. Example: a mass on a spring in thick oil.

Graph (1 mark).

Grating element (2 marks)

A diffraction grating is a plate with a very large number of equally spaced, parallel slits. The grating element (grating constant) d is the distance between the centres of two adjacent slits. It equals the width of one transparent slit a plus the width of one opaque line b, so d=a+b. It is a few micrometres, a little larger than the wavelength of light, and its SI unit is the metre.

Formula (1 mark)

For example, a grating with 5000 lines per cm has d=\dfrac{1\times10^{-2}\,\mathrm{m}}{5000}=2.0\times10^{-6}\,\mathrm{m}. The grating element appears in the grating equation d\sin\theta=m\lambda.

Half-wave rectification (1.5 marks)

One diode passes only the positive half-cycles; the output is zero during every negative half-cycle. One pulse per input cycle.

Full-wave rectification (1.5 marks)

Four diodes (bridge) pass both half-cycles, and the negative halves are turned upward, so the current through the load flows one way all the time. Two pulses per input cycle.

The full-wave output has no gaps, twice the pulse frequency and twice the average value (2V_0/\pi against V_0/\pi).

Effect (2 marks)

Increasing the slit separation d decreases the fringe spacing: the bright and dark fringes come closer together. With the slits farther apart, the path difference d\sin\theta grows faster as we move across the screen, so it reaches one more wavelength after a shorter distance. If d is made too large, the fringes are so close that they cannot be seen separately.

Mathematical relation (1 mark)

For example, doubling d halves the fringe spacing: with \lambda=600\,\mathrm{nm} and L=1\,\mathrm{m}, d=0.25\,\mathrm{mm} gives \Delta y=2.4\,\mathrm{mm}, while d=0.50\,\mathrm{mm} gives \Delta y=1.2\,\mathrm{mm}.

Gravitational potential (2 marks)

The gravitational potential \varphi (or V) at a point is the work done per unit mass in bringing a small test mass from infinity to that point. It equals the gravitational potential energy per unit mass, \varphi=U/m. It is a scalar with SI unit \mathrm{J\,kg^{-1}}. Its zero is at infinity, so it is negative at every finite distance from the Earth, and it increases (becomes less negative) with distance.

Derivation (4 marks)

Here M is the mass of the Earth (M=M_E). The force changes with distance, so the path is split into small steps and the work in each step is added.

Sketch (1 mark)

The labelled diagram of the path (above) is the sketch that goes with the derivation. The graph of \varphi against r shows the result:

Simple harmonic motion (1 mark)

SHM is the oscillatory motion in which the acceleration of a particle is directly proportional to its displacement from the mean position and is always directed towards the mean position:

Diagram (1 mark)

A block of mass m on a smooth horizontal surface is attached to a spring of spring constant k whose other end is fixed. At x=0 the spring is unstretched. When the block is displaced by x and released, the spring exerts a restoring force towards x=0, and the block oscillates about this mean position.

Mass-spring system executes SHM (3 marks)

Frequency and time period (2 marks)

Since the acceleration is proportional to the displacement and directed towards the mean position, the mass-spring system executes SHM with T=2\pi\sqrt{m/k} and f=\frac{1}{2\pi}\sqrt{k/m}, independent of the amplitude.

Working of Young's double-slit experiment (4 marks)

Young's experiment (1801) produces interference of light from two coherent sources and shows that light behaves as a wave.

1. Monochromatic light (for example from a sodium lamp or a laser) falls on a narrow single slit S_0. Light spreading out from S_0 by diffraction reaches two narrow, parallel slits S_1 and S_2, placed close together (separation d, about 0.5\,\mathrm{mm}) and at equal distances from S_0.
2. S_1 and S_2 receive light from the same wavefront, so they act as two coherent sources vibrating in phase.
3. Each slit diffracts the light, and the two sets of waves overlap and interfere in the region beyond the slits. A screen is placed at a distance L (about 1\,\mathrm{m}) from the slits.
4. At points where the path difference S_2P-S_1P is 0,\ \lambda,\ 2\lambda,\dots the waves arrive in phase and interfere constructively, giving bright fringes.
5. At points where the path difference is \tfrac12\lambda,\ \tfrac32\lambda,\dots the waves arrive in anti-phase and interfere destructively, giving dark fringes.
6. So equally spaced, alternate bright and dark fringes parallel to the slits appear on the screen. The central fringe at O is bright because the path difference there is zero. With white light the central fringe is white and the others are coloured.

Diagram (1 mark)

Derivation of fringe spacing (2 marks)

Electric potential (1 mark)

Electric potential at a point is the work done in bringing a unit positive charge from infinity to that point, keeping the charge in equilibrium. It is the potential energy per unit charge, V=\dfrac{W}{q_0}=\dfrac{U}{q_0}. It is a scalar, measured in volts (1\,\mathrm{V}=1\,\mathrm{J\,C^{-1}}).

Derivation (5 marks)

The field of a point charge is not uniform, so the path from infinity is divided into small steps in which the field is almost constant. The work in every step is found and the steps are added.

Sketch (1 mark)

The potential falls off as 1/r with distance from the charge:

The current lags the voltage (2 marks)

In an RL series circuit the current lags behind the applied voltage by an angle \varphi, where 0<\varphi<90^\circ. The same current flows through R and L. The voltage across R is in phase with the current, but the voltage across L leads the current by 90^\circ, because the back emf of the coil opposes every change in current. The applied voltage is the phasor sum of the two, so it is ahead of the current by \varphi; that is, the current lags.

Phasor diagram (2 marks)

• The current I is the reference phasor (horizontal).
• V_R=IR is drawn along I (OA).
• V_L=IX_L is drawn at 90^\circ ahead of I (AB).
• The resultant V (OB) makes the angle \varphi with I: \tan\varphi=\dfrac{V_L}{V_R}=\dfrac{X_L}{R}.

Impedance of the RL series circuit (2 marks)

Photoelectric effect (1 mark)

The photoelectric effect is the emission of electrons from a metal surface when electromagnetic radiation of sufficiently high frequency falls on it. The emitted electrons are called photoelectrons.

No emission below threshold frequency (2 marks)

According to Einstein, light consists of photons of energy hf, and one photon is absorbed by one electron. The electron needs at least the work function \Phi to escape:

If f<f_0, the energy of each photon is less than the work function (hf<\Phi), so no single photon can free an electron. An electron cannot store the energy of one photon and add the energy of another, so increasing the intensity (sending more photons) still gives no emission. Hence no electrons are emitted below the threshold frequency, however bright the light.

Effect of intensity on photoelectric current (3 marks)

Keeping the frequency (above f_0) and the potential difference fixed, the photoelectric current increases linearly with the intensity. Intensity is the number of photons arriving per second; each photon releases one electron, so the number of photoelectrons per second, and hence the current, is proportional to intensity.

When the current is plotted against the collector potential for different intensities, brighter light gives a larger saturation current, but the current falls to zero at the same stopping potential V_0 for all intensities. So intensity increases the number of electrons but not their maximum kinetic energy, K.E_{\max}=eV_0.

From F_c = 2mv\omega\sin\theta: v = \dfrac{F_c}{2m\omega\sin\theta} = \dfrac{274}{2(2\times10^{5})(7.3\times10^{-5})\sin25^\circ} = \dfrac{274}{12.3} = 22.2\,\mathrm{m\,s^{-1}}.

This equals 80\,\mathrm{km\,h^{-1}} (the book's Example 24.2), so the km/h options A and B are also wrong.

Gamma rays have the shortest wavelength (about 10^{-12}\,\mathrm{m}) in the electromagnetic spectrum, so from f=c/\lambda they have the highest frequency (and photon energy). The order of increasing frequency is microwaves, infrared, ultraviolet, gamma rays.

F=G\dfrac{m_1m_2}{r^{2}} is directly proportional to each mass. Doubling one mass (same distance) doubles the force: F'=G\dfrac{(2m_1)m_2}{r^{2}}=2F. B (4F) would need both masses doubled.

g=\dfrac{F}{m}=\dfrac{GM}{r^{2}}: the test mass m cancels, so g depends only on the mass M causing the field and the distance r from it. The shape, velocity and mass of the test body do not matter.

Luminosity is power (energy emitted per second), so its unit is \mathrm{J\,s^{-1}}, the watt. \mathrm{W\,m^{-2}} (A) is the unit of radiant flux intensity, the kelvin (B) is the unit of temperature, and the joule (D) is energy, not power.

In SHM a=-\omega^{2}x. With \omega=10\,\mathrm{rad\,s^{-1}}, \omega^{2}=100, so a=-100x. Options B and D are not proportional to x, and C has the wrong sign and does not square \omega.

f=\dfrac{1}{T}=\dfrac{1}{2\,\mathrm{s}}=0.5\,\mathrm{Hz}.

A diffraction grating gives very sharp, widely separated maxima whose angles can be measured accurately. With the grating element d known, \lambda=d\sin\theta/m (book section 18.2.1). A capacitor stores charge, and Melde's apparatus studies standing waves on a string.

V=\dfrac{1}{4\pi\varepsilon_0}\dfrac{Q}{r}, so V\propto1/r. The tempting option A, 1/r^{2}, is how the electric field (and the force) varies, not the potential.

C=\dfrac{\varepsilon_0\varepsilon_rA}{d}, so C is directly proportional to the plate area. Increasing d (C) or decreasing A (D) reduces C, and the battery voltage (B) changes the charge stored, not the capacitance.

D_1 and D_2 conduct in the positive half-cycle and D_3 and D_4 in the negative half-cycle, and both pairs send the current through the load in the same direction. Options C and D describe a half-wave rectifier.

For a pure inductor, I=I_0\sin\omega t gives V=V_0\sin(\omega t+\pi/2): the voltage leads the current by \pi/2 (90^\circ). Remember ELI: in L the emf comes before I.

This is the book's own example in section 21.6. Watch the unit: the answer is in metres, not millimetres.

Each photon releases one electron, so the number of electrons emitted per second is proportional to the number of photons per second, that is, to the intensity. Frequency, wavelength and photon energy decide the kinetic energy of the electrons (and whether emission happens at all), not their number.

Each fission of uranium-235 releases 2 or 3 neutrons. These neutrons cause further fissions, which release more neutrons, so the reaction keeps itself going. Photons, protons and electrons do not cause further fissions.

F=L/4\pi d^{2}: the radiant flux intensity is inversely proportional to the square of the distance. It is directly (not inversely) proportional to the luminosity, and 'distance' alone misses the square.

The ozone layer lies in the stratosphere (about 16 km to 50 km). By absorbing the Sun's ultraviolet radiation it makes the temperature rise with height in this layer.

Effect (1 mark). The orbital velocity increases: a satellite in a lower orbit must move faster.

Mathematical explanation (2 marks). For a circular orbit of radius r, gravity provides the centripetal force:

In a lower orbit r is smaller and the Earth's pull is stronger, so a larger speed is needed to keep the satellite on its circle. If r is reduced to one quarter, v_o doubles. The fastest circular orbit is the one just above the surface: v_o=\sqrt{gR_E}=7.9\,\mathrm{km\,s^{-1}}.

Marks: spring constant (1), angular frequency (1), acceleration (1). These are the data of Example 17.2 of the book.

Relation (2 marks)

At constant temperature, the pressure of a gas is directly proportional to its density, P\propto\rho.

Reason: density \rho=\dfrac{Nm}{V} tells how many molecules (how much mass) there are in each unit volume. In a denser gas more molecules strike each unit area of the wall every second. At constant temperature each molecule delivers the same momentum per collision, so more collisions per second give a larger force per unit area, that is, a larger pressure. Doubling the density doubles the pressure.

Mathematical expression (1 mark)

\rho: density of the gas; \langle v^{2}\rangle: mean square speed of the molecules, which is constant at constant temperature.

Any three of the following (1 mark each):

1. Coherent sources: the two sources must have the same frequency and a constant phase difference. (In Young's experiment both slits are lit by the same wavefront from one single slit.)
2. Monochromatic light: light of a single wavelength; otherwise fringes of different colours overlap and the pattern blurs.
3. Narrow, closely spaced slits and a distant screen: each slit must be narrow enough to diffract the light so that the two beams overlap, and d must be small and L large so that the fringe spacing \Delta y=\lambda L/d is big enough to see.
4. Nearly equal amplitudes: the two waves should be of equal intensity so that the dark fringes are completely dark.
5. Stable arrangement: no vibrations or air currents that would shift the fringes.

Many radio stations broadcast at the same time, each at its own frequency. All their waves reach the aerial and try to drive small electrical oscillations in the receiver.

The tuning circuit of the receiver (a coil and a variable capacitor) is an electrical oscillator with its own natural frequency. When we turn the tuning knob, we change the capacitance and so change this natural frequency.

When the natural frequency of the circuit becomes equal to the transmission frequency of a particular station, resonance occurs. The circuit then oscillates with maximum amplitude at that frequency and absorbs maximum energy from that station only. Signals of other stations are far from resonance and produce very small responses, so we hear only the tuned station. For good selection the tuning circuit has little damping, which makes the resonance sharp.

One mark for each appliance with the role of its capacitor:

1. Ceiling fan: a capacitor connected with the fan's auxiliary (starting) winding shifts the phase of its current. This produces a rotating magnetic field, which starts the motor and keeps it turning; a capacitor of suitable value increases the speed of the fan.
2. Refrigerator: a start (or run) capacitor with the compressor motor gives the extra starting torque the compressor needs and keeps it running smoothly.
3. Camera flash (flash gun): the small battery charges a capacitor slowly; the capacitor then releases its stored energy in about a millisecond through the flash tube, producing a bright flash.
4. Also acceptable: in TV and radio sets capacitors filter unwanted frequencies and tune to a station; in a mobile charger or any d.c. power supply a capacitor smooths the rectified voltage by reducing the ripple.

Mutual inductance (1 mark)

When the current in one coil changes, an emf \varepsilon_2=-M\,\Delta I_1/\Delta t is induced in a neighbouring coil (mutual induction). The constant M is the mutual inductance: the ratio of the emf induced in one coil to the time rate of change of current in the other coil.

SI unit and factors (2 marks)

SI unit: the henry (H), 1\,\mathrm{H}=1\,\mathrm{V\,s\,A^{-1}}. The mutual inductance of two coils is 1 H if a current changing at 1\,\mathrm{A\,s^{-1}} in one coil induces an emf of 1 V in the other.

For two coaxial coils M=\dfrac{\mu_r\mu_0N_1N_2A}{l}, so M depends on:

• the number of turns of both coils (N_1N_2);
• the area of cross-section of the coils;
• the distance between the coils and their relative orientation (largest when close and coaxial);
• the core material, its relative permeability \mu_r (a common iron core increases M).

35\,700\,\mathrm{km} is the well-known height of communication (geostationary) satellites above the Earth's surface (Table 15.1), so we take it as the height h above the surface. This is Assignment 15.2 of the book.

Braking radiation (2 marks)

Braking radiation (Bremsstrahlung) is the X-radiation emitted when a fast-moving electron is decelerated (braked) by the Coulomb attraction of a nucleus in the target of an X-ray tube. The attraction deflects and slows the electron, and the kinetic energy it loses is emitted as an X-ray photon, hf=\mathrm{K.E.}_{\text{before}}-\mathrm{K.E.}_{\text{after}}. Since an electron can lose any amount of energy, braking radiation forms the continuous X-ray spectrum.

Minimum wavelength (1 mark)

The shortest wavelength is produced when an electron, accelerated through p.d. V, gives all its kinetic energy eV to one photon:

Here h is Planck's constant, c the speed of light, e the charge of the electron and V the accelerating potential difference.

This is Example 20.2 of the book, which also finds the peak current I_0=V_0/X_C=16\,\mathrm{mA}.

The atmosphere keeps the Earth's climate balanced in these ways:

1. Keeps the Earth warm (greenhouse effect): greenhouse gases such as water vapour and \mathrm{CO_2} let sunlight in but absorb part of the outgoing infrared radiation and send some of it back to the surface. This keeps the average temperature near 15^\circ\mathrm{C} instead of -18^\circ\mathrm{C} and prevents extreme differences between day and night temperatures.
2. Shields life: the ozone layer in the stratosphere absorbs harmful ultraviolet radiation, and meteors burn up in the mesosphere.
3. Redistributes heat and moisture: solar heating causes convection; the resulting winds and circulation cells carry heat from the equator towards the poles and water vapour from the oceans over the land, producing weather and evening out temperatures across the globe.

Clouds and air also reflect part of the sunlight back to space, so the atmosphere helps to keep the energy the Earth absorbs equal to the energy it radiates (the energy budget).

Contrast is the range of brightness, from lightest to darkest, in an image. In an X-ray image it is the difference in brightness between neighbouring structures, which comes from the difference in how much X-radiation they absorb.

Bone absorbs X-rays much more than soft tissue, so bone on soft tissue gives high contrast and shows clearly. Neighbouring soft tissues absorb almost equally, so they appear in nearly the same shade of grey (low contrast) and are hard to see on a conventional X-ray.

Contrast is improved with a contrast medium of high atomic number: a barium meal (barium sulfate) for the stomach and intestines, or an iodine-based dye injected into blood vessels. The filled organ then appears white. Contrast in X-ray imaging allows the radiologist to evaluate structures that are not clearly evident on conventional X-ray exams.

Pair annihilation (1 mark)

Pair annihilation is the process in which an electron and a positron (its antiparticle) combine and disappear, producing two gamma-ray photons. Mass is converted into energy according to E=mc^{2}. It is the reverse of pair production.

Conservation of energy and momentum (2 marks)

Energy: the total energy of the pair (rest mass energy plus kinetic energy) appears as the energy of the photons:

For a pair at rest each photon has hf=m_0c^{2}=0.511\,\mathrm{MeV}, so the total energy before and after is 1.02\,\mathrm{MeV}.

Momentum: the electron and positron are nearly at rest, so their total momentum is about zero. A single photon always carries momentum hf/c, so one photon cannot be produced. Two photons of equal energy moving in opposite directions have equal and opposite momenta (+hf/c and -hf/c), giving zero total momentum. Hence momentum is conserved. Charge is also conserved: (+e)+(-e)=0.

Climate inertia is the slow response of the climate to changes in factors such as greenhouse gases. The oceans are its main cause because:

1. High heat capacity: water has a high specific heat capacity (about 4200\,\mathrm{J\,kg^{-1}\,K^{-1}}) and the oceans have an enormous mass. From Q = mc\Delta T, a huge amount of heat produces only a small change in temperature; the oceans can store about 1000 times more heat than the atmosphere.
2. Slow absorption and release: the oceans absorb heat slowly and release it slowly, and the deep ocean mixes with the surface only over centuries. So changes in ocean temperature, and in the climate patterns they control, happen gradually.
3. Stored heat keeps the warming going: even if greenhouse gas emissions were stabilised immediately, the heat already stored in the oceans would keep the Earth warming for decades. The oceans also absorb \mathrm{CO_2} and exchange it slowly with the air, which adds to the delay.

1. Diagnostic imaging (sonography). Short pulses of ultrasound from a piezoelectric probe enter the body and are partly reflected at boundaries between tissues. The time of each echo gives the depth of the boundary (d=vt/2) and its strength shows the type of tissue, so a computer builds a real-time image of organs such as the liver, kidneys, heart and the unborn baby. It uses no ionising radiation, so it is safe in pregnancy.

2. Measuring blood flow (Doppler ultrasound). Ultrasound reflected from moving blood cells returns with a changed frequency (Doppler shift). The shift gives the speed of the blood, so doctors can find blood clots and blocked arteries and check the working of the heart in adults and fetuses.

Other applications that may be given instead: focused intense ultrasound in neurosurgery to destroy a small region of tissue without cutting, and ultrasonic heating in physiotherapy for arthritis, rheumatism and sciatica.

Reason (1.5 marks)

The resolution of any microscope is limited by the wavelength of the radiation it uses: details smaller than about one wavelength cannot be seen. An electron microscope uses fast electrons whose de Broglie wavelength \lambda=h/\sqrt{2meV} is extremely short (about 0.01\,\mathrm{nm} at 10\,\mathrm{kV}), thousands of times shorter than visible light (400 to 700\,\mathrm{nm}). So it can resolve much finer detail.

Comparison (1.5 marks)

The binding energy per nucleon E_b/A is the average energy needed to remove one nucleon from a nucleus. Its variation with mass number (the binding energy curve) explains the stability of nuclei and the release of nuclear energy.

High binding energy per nucleon (1.5 marks)

Medium nuclei (A\approx50 to 100) have the highest values, with a peak of about 8.8\,\mathrm{MeV} for iron-56. Their nucleons are most tightly bound, so these nuclei are the most stable; they undergo neither fission nor fusion, and no energy can be obtained from them.

Low binding energy per nucleon (1.5 marks)

Very light nuclei (such as \ce{^{2}H}, 1.1\,\mathrm{MeV}) and very heavy nuclei (such as U-235, 7.6\,\mathrm{MeV}) have lower values and are less stable. They can gain stability by moving towards the peak, and energy is released when they do: light nuclei by fusion (hydrogen into helium in the Sun) and heavy nuclei by fission (U-235 into two medium nuclei, about 200\,\mathrm{MeV}). This is the basis of nuclear power and of stellar energy.

Energy imbalance: the equator receives direct sunlight and gains more energy than it loses, while the poles receive oblique sunlight and lose more energy than they gain.

1. Winds and circulation cells: warm air rises at the equator (low pressure) and cold air sinks at the poles (high pressure). The pressure differences drive the trade winds, westerlies and polar easterlies, which the Coriolis effect organises into the Hadley, Ferrel and polar cells.
2. Heat transport: these winds, together with ocean currents (gyres and the thermohaline circulation), carry heat from the tropics towards the poles and reduce the temperature difference.
3. Climate zones: the circulation creates wet equatorial regions (rising air), dry deserts near 30^\circ (sinking air) and stormy mid-latitudes near 60^\circ. Changes in the balance shift weather systems, ocean currents and climate zones.

Use (2 marks): Wien's displacement law states that the black body radiation curve peaks at a wavelength which is inversely proportional to the absolute temperature. A star radiates almost like a black body, so:

1. Record the star's spectrum and find the peak wavelength \lambda_{\max}, where the intensity is greatest.
2. Convert it to metres and divide Wien's constant by it: T=\dfrac{2.9\times10^{-3}\,\mathrm{m\,K}}{\lambda_{\max}}.
3. A shorter peak wavelength means a hotter star, so blue-white stars are hotter than red stars.

Formula (1 mark):

For example, Rigel peaks at 263\,\mathrm{nm}, so T=2.9\times10^{-3}/(263\times10^{-9})\approx1.1\times10^{4}\,\mathrm{K}.

At height h above the surface the field strength is

using the binomial approximation. So the fractional decrease in g is only 2h/R_E. The Earth's radius is very large (R_E=6.4\times10^{6}\,\mathrm{m}), so for ordinary heights this is tiny:

Even at the top of Mount Everest (h\approx8.8\,\mathrm{km}) g falls by less than 0.3\%, from 9.80 to about 9.77\,\mathrm{m\,s^{-2}}. Hence near the surface g is practically constant.

Function of control rods (1.5 marks)

Control rods, made of neutron-absorbing boron or cadmium, control the rate of the chain reaction by absorbing some of the neutrons. Inserting them further lowers the number of fissions and the power; withdrawing them raises it. In normal running they are adjusted so that exactly one neutron per fission causes a further fission, so the reaction neither dies out nor runs out of control. In an emergency they are dropped fully in to shut the reactor down.

Function of moderators (1.5 marks)

A moderator is a material that slows down fast neutrons. Fission neutrons are fast (about 2\,\mathrm{MeV}), but uranium-235 is split most easily by slow (thermal) neutrons of about 0.025\,\mathrm{eV}. In repeated collisions with the light nuclei of the moderator (water or graphite) the neutrons lose most of their energy and become thermal, which keeps the chain reaction going.

This is Assignment 16.1 of the book (320\,\mathrm{K}). The paper's formula sheet gives it as T=\frac{2}{3k}\left(\frac12mv^{2}\right); rearranging gives the same \frac32kT. Marks: formula 1, substitution 1, answer with unit 1.

Free oscillations (1 mark). A body is said to be executing free oscillations when it oscillates under the influence of its restoring force alone, without any external force acting on it. It then vibrates at its own natural frequency, and (ideally, with no damping) its amplitude and period stay constant.

Examples (2 marks):

1. A simple pendulum displaced slightly from its mean position and released oscillates freely at its natural frequency f_0=\dfrac{1}{2\pi}\sqrt{\dfrac{g}{l}}.
2. A mass attached to a spring, pulled aside and released, oscillates freely at f_0=\dfrac{1}{2\pi}\sqrt{\dfrac{k}{m}}.
3. (Also acceptable: a plucked guitar string, a struck tuning fork, a swing given a single push.)

Effect (2 marks)

Increasing the number of slits per unit length N decreases the grating element d=1/N. From the grating equation \sin\theta=m\lambda/d:

• each order appears at a larger angle, so the maxima are more widely spaced;
• fewer orders can be seen, because \sin heta=m\lambda/d cannot be more than 1;
• the maxima become sharper and brighter, because more slits share the same beam.

Sketch (1 mark)

Peak value (1.5 marks)

The peak value V_0 is the maximum value of the alternating voltage. It is reached twice in each cycle, as +V_0 and -V_0, and only for an instant. For the 220 V mains, V_0=311\,\mathrm{V}.

r.m.s. value (1.5 marks)

The r.m.s. value V_{\mathrm{rms}} is the steady (DC) voltage that produces the same heat in a resistor as the alternating voltage in the same time. It is the square root of the mean of V^2 over a cycle. AC meters read r.m.s. values, and ratings such as 220 V are r.m.s. values. For a sinusoidal voltage:

Ways to increase the fringe spacing (2 marks)

The fringe spacing is \Delta y=\lambda L/d, so it can be increased by:

• moving the screen farther from the slits (increasing L);
• using light of a longer wavelength, for example red instead of blue (increasing \lambda);
• bringing the two slits closer together (decreasing d).

Mathematical calculation (1 mark)

With \lambda=600\,\mathrm{nm}, L=1.0\,\mathrm{m} and d=0.50\,\mathrm{mm} the spacing is 1.2\,\mathrm{mm}. Doubling L to 2.0\,\mathrm{m}, or halving d to 0.25\,\mathrm{mm}, doubles it to 2.4\,\mathrm{mm}.

Gravitational potential (2 marks)

The gravitational potential \varphi (or V) at a point is the work done per unit mass in bringing a small test mass from infinity to that point. It equals the gravitational potential energy per unit mass, \varphi=U/m. It is a scalar with SI unit \mathrm{J\,kg^{-1}}. Its zero is at infinity, so it is negative at every finite distance from the Earth, and it increases (becomes less negative) with distance.

Derivation (4 marks)

Here M is the mass of the Earth (M=M_E). The force changes with distance, so the path is split into small steps and the work in each step is added.

Sketch (1 mark)

The labelled diagram of the path (above) is the sketch that goes with the derivation. The graph of \varphi against r shows the result:

Consider a mass m attached to a spring of spring constant k, oscillating on a smooth horizontal surface with amplitude x_0. Let its displacement be x at some instant.

Kinetic energy (2 marks)

Potential energy (2 marks)

Total energy remains constant (2 marks)

Sketch (1 mark)

K.E. and P.E. change continuously, one increasing as the other decreases, but their sum is the horizontal line \tfrac12 kx_0^{2}: the total energy of the spring system remains constant.

Working of Young's double-slit experiment (4 marks)

Young's experiment (1801) produces interference of light from two coherent sources and shows that light behaves as a wave.

1. Monochromatic light (for example from a sodium lamp or a laser) falls on a narrow single slit S_0. Light spreading out from S_0 by diffraction reaches two narrow, parallel slits S_1 and S_2, placed close together (separation d, about 0.5\,\mathrm{mm}) and at equal distances from S_0.
2. S_1 and S_2 receive light from the same wavefront, so they act as two coherent sources vibrating in phase.
3. Each slit diffracts the light, and the two sets of waves overlap and interfere in the region beyond the slits. A screen is placed at a distance L (about 1\,\mathrm{m}) from the slits.
4. At points where the path difference S_2P-S_1P is 0,\ \lambda,\ 2\lambda,\dots the waves arrive in phase and interfere constructively, giving bright fringes.
5. At points where the path difference is \tfrac12\lambda,\ \tfrac32\lambda,\dots the waves arrive in anti-phase and interfere destructively, giving dark fringes.
6. So equally spaced, alternate bright and dark fringes parallel to the slits appear on the screen. The central fringe at O is bright because the path difference there is zero. With white light the central fringe is white and the others are coloured.

Diagram (1 mark)

Derivation of fringe spacing (2 marks)

Circuit diagram (2 marks)

Four diodes D_1, D_2, D_3 and D_4 form a bridge. The AC input (secondary of a step-down transformer) is connected across corners A and B, and the load resistor R across the other two corners.

Working (4 marks)

Positive half-cycle: end A is positive and B negative. D_1 and D_2 are forward biased and conduct; D_3 and D_4 are reverse biased. Current flows from A through D_1, through R from top to bottom, through D_2 and back to B.

Negative half-cycle: end A is negative and B positive. D_3 and D_4 are forward biased and conduct; D_1 and D_2 are reverse biased. Current flows from B through D_3, through R again from top to bottom, through D_4 and back to A.

So the current through the load is in the same direction during both half-cycles, and both halves of the AC appear across R as pulsating DC (full-wave rectification). The output pulses at twice the supply frequency and can be smoothed with a capacitor across R.

Photoelectric effect (1 mark)

The photoelectric effect is the emission of electrons from a metal surface when electromagnetic radiation of sufficiently high frequency falls on it. The emitted electrons are called photoelectrons.

Work function and Einstein's equation (3 marks)

The work function \Phi of a metal is the minimum energy needed to remove an electron from its surface. It is different for different metals (for example about 2.3\,\mathrm{eV} for sodium and 4.3\,\mathrm{eV} for zinc). It is related to the threshold frequency f_0 by \Phi=hf_0.

Effect of frequency on K.E of photoelectrons (2 marks)

From K.E_{\max}=hf-\Phi: below the threshold frequency f_0 no electrons are emitted. Above f_0, the maximum kinetic energy increases linearly with the frequency; the graph is a straight line of slope h, cutting the frequency axis at f_0. Increasing the frequency (for example from red to violet to ultraviolet light) gives faster photoelectrons, whatever the intensity.

In a medium the Coulomb force is divided by the relative permittivity: F_{med}=\dfrac{1}{4\pi\varepsilon_0\varepsilon_r}\dfrac{q_1q_2}{r^{2}}=\dfrac{F_{air}}{\varepsilon_r}=\dfrac{10\,\mathrm{N}}{2}=5\,\mathrm{N}. The medium's polarisation weakens the field between the charges, the same effect that raises the capacitance of a capacitor by \varepsilon_r.

RC is the time constant, so its unit is the second: \Omega\times\mathrm{F}=\dfrac{\mathrm{V}}{\mathrm{A}}\times\dfrac{\mathrm{C}}{\mathrm{V}}=\dfrac{\mathrm{C}}{\mathrm{A}}=\mathrm{s}. Weber, henry and tesla are units of magnetic flux, inductance and magnetic field.

Comparing with V=V_0\sin(2\pi ft) gives \omega=2\pi f=314\,\mathrm{rad\,s^{-1}}, so f=\dfrac{314}{2\times3.14}=50\,\mathrm{Hz}. (Book exercise MCQ 7 is the same type with \omega=157, giving 25 Hz.)

The nucleus has Z=56 protons and mass number A=138, so N=A-Z=138-56=82. (192 is A+Z, 138 is the number of nucleons and 56 the number of protons.)

The decay constant is the fraction of nuclei that decay per unit time, \lambda=\dfrac{1}{N}\dfrac{\Delta N}{\Delta t}, so its unit is \mathrm{s^{-1}} (per second). This also follows from \lambda=0.693/T_{1/2}.

The mutual inductance is the flux linkage of the secondary per unit current in the primary: M=\dfrac{N\Delta\Phi}{\Delta I}=\dfrac{2\times10^{-2}}{0.01}=2\,\mathrm{H} (taking the given flux change as the change in flux linkage, N=1). This matches the paper's formula sheet, M=N\varphi/I.

The induced current opposes the change that produces it, so work must be done against it, and this work appears as electrical energy. If it helped the change, energy would be created from nothing. So Lenz's law follows from the law of conservation of energy.

In a pure inductor the current lags the voltage by 90^\circ. Energy stored in the magnetic field during one quarter cycle is returned to the source in the next, so the average power is V_{\mathrm{rms}}I_{\mathrm{rms}}\cos90^\circ=0. Option C, V_{\mathrm{rms}}I_{\mathrm{rms}}, is the average power in a pure resistor.

Bosons are particles that do not obey the Pauli exclusion principle (they have whole-number spin), so any number of them can share one quantum state. The photon (spin 1) is the standard example, and the book names photons as bosons. The proton and the neutrino have spin \frac12, so they are fermions.

Cobalt-60 emits penetrating gamma rays (and has a convenient half-life of 5.27 years), so it is the standard source for radiotherapy of many cancers. Carbon-14 is used for dating, nickel-63 in small batteries and detectors, and strontium-90 only for a few surface treatments.

Definition (1 mark)

One electronvolt is the energy gained (or lost) by an electron when it moves through a potential difference of one volt.

Relation with the joule (2 marks)

Also 1\,\mathrm{keV}=1.6\times10^{-16}\,\mathrm{J} and 1\,\mathrm{MeV}=1.6\times10^{-13}\,\mathrm{J}. The electronvolt is a convenient small unit for atomic and nuclear energies.

The maximum kinetic energy of photoelectrons is found by measuring the stopping potential V_0. In the photoelectric set-up the collector is made negative with respect to the emitter (battery reversed), and the retarding potential is increased until the photoelectric current just becomes zero. At this potential even the fastest electrons are stopped, so their kinetic energy has all been used in doing work eV_0 against the field:

For example, if V_0=2\,\mathrm{V} then K.E_{\max}=2\,\mathrm{eV}=3.2\times10^{-19}\,\mathrm{J}. If the frequency and the work function are known instead, the same energy follows from Einstein's equation K.E_{\max}=hf-\Phi.

Wien's displacement law: the black body radiation curve for different temperatures peaks at a wavelength which is inversely proportional to the absolute temperature of the body.

Explanation: a hot body emits all wavelengths, but not equally; the intensity is greatest at the peak wavelength \lambda_{\max}. As the temperature rises the peak is displaced to shorter wavelengths, and the intensity rises at every wavelength. That is why a heated iron rod glows red, then yellow, then white, and why blue stars are hotter than red stars. Measuring \lambda_{\max} gives a star's surface temperature, T=b/\lambda_{\max}.

Consider two points A and B on a field line, a small distance \Delta r apart, so the field E between them is uniform. A positive test charge q_0 is moved slowly from A to B, along the field.

\Delta V/\Delta r is the potential gradient. The negative sign shows that the field points in the direction of decreasing potential.

A reactor starts and keeps a controlled chain reaction, in which on average exactly one neutron from each fission causes another fission:

1. The core contains fuel rods of uranium enriched to about 3% U-235, surrounded by a moderator (water or graphite). A few free neutrons are always present, from spontaneous fission or a small neutron source.
2. The moderator slows these neutrons to thermal energies (about 0.025\,\mathrm{eV}), at which they readily cause fission of U-235. Each fission releases 2 or 3 more fast neutrons, which are slowed in turn.
3. Control rods of cadmium or boron absorb neutrons. They are slowly withdrawn until the reactor becomes critical (one neutron per fission causes a further fission), and are then adjusted to keep the rate of fission, and so the power, constant.
4. The coolant removes the heat, and safety rods drop in to stop the reaction at once in an emergency.

The self-inductance of a long coil (solenoid) is

So it is affected by three factors (one mark each):

1. Number of turns N: L\propto N^2, so doubling the turns makes L four times larger.
2. Size of the coil: L is proportional to the area of cross-section A and, for the same number of turns, inversely proportional to the length l (a closely wound coil has more inductance).
3. Nature of the core: a core of high relative permeability \mu_r, such as soft iron, increases L many times compared with air.

The inductance does not depend on the current flowing in the coil.

A GM tube is a metal cylinder (cathode) with a thin central wire (anode) at about 400\,\mathrm{V} positive, filled with argon at low pressure and a little quenching gas, with a thin mica window at one end.

1. Radiation entering the window ionises a few gas atoms.
2. The electrons are accelerated towards the wire by the strong field and ionise more atoms: an avalanche of electrons reaches the anode.
3. This gives a pulse of current, which makes a voltage pulse across the resistor R.
4. Each pulse is amplified and counted by a scaler or ratemeter: one pulse for each particle or photon, so the count rate measures the radiation.
5. The quenching gas stops the discharge so the tube is ready for the next particle (after a dead time of about 10^{-4}\,\mathrm{s}).

Impedance Z is the total opposition of an AC circuit, Z=V/I, measured in ohms. In an RLC series circuit the same current I flows through R, L and C, but V_R=IR is in phase with I, V_L=IX_L leads it by 90^\circ and V_C=IX_C lags it by 90^\circ. V_L and V_C are opposite, so the net reactive voltage is V_L-V_C, and the supply voltage is the phasor sum of V_R and V_L-V_C.

with \tan\theta=(X_L-X_C)/R. If X_L>X_C the current lags; if X_C>X_L it leads. At resonance (X_L=X_C) the impedance is minimum, Z=R, and the current is maximum.

The capacitance of a parallel plate capacitor is

So the capacitance can be increased in three ways:

1. **Increase the area A of the plates** (C\propto A): larger plates hold more charge at the same voltage. Long foils are rolled up to get a large area in a small space.
2. **Decrease the separation d between the plates** (C\propto1/d): halving the gap doubles the capacitance.
3. **Insert a dielectric of high relative permittivity \varepsilon_r** between the plates (C\propto\varepsilon_r): for example, mica (\varepsilon_r\approx6) makes the capacitance about six times its value in air. The polarised dielectric weakens the field between the plates, so more charge is stored per volt.

The effective capacitance of a circuit can also be increased by connecting more capacitors in parallel, since C_e=C_1+C_2+\cdots.

Inductor (1.5 marks)

Doubling the frequency doubles the inductive reactance, because the current changes twice as fast and the back emf is twice as large.

Capacitor (1.5 marks)

Doubling the frequency halves the capacitive reactance, because the capacitor charges and discharges twice as often and twice as much charge flows per second.

Photoelectric effect (2 marks)

The photoelectric effect is the emission of electrons from a metal surface when electromagnetic radiation of sufficiently high frequency falls on it. The emitted electrons are called photoelectrons and the current they form is the photoelectric current.

Failure of classical physics (2 marks)

Classical physics treats light as a continuous wave whose energy depends on intensity. It fails to explain:

• Threshold frequency: no emission below f_0 however intense the light (wave theory predicts emission at any frequency if the light is bright enough).
• Kinetic energy independent of intensity: K.E_{\max} does not increase with intensity.
• Kinetic energy depends on frequency: K.E_{\max} increases linearly with f.
• No time lag: emission is instantaneous even in very dim light.

Einstein's photoelectric equation (3 marks)

Half-life (3 marks)

The half-life T_{1/2} of a radioactive element is the time taken for half of the radioactive nuclei present in a sample to decay.

Starting with N_0 undecayed nuclei, N_0/2 remain after one half-life, N_0/4 after two, N_0/8 after three, and so on: N=N_0(1/2)^{n} after n half-lives. The half-life is a fixed property of each isotope. It does not depend on the amount of the sample or on temperature, pressure or chemical state. Examples: radon-222, 3.82 days; cobalt-60, 5.27 years; carbon-14, 5730 years; uranium-238, 4.5 billion years.

Derivation of T_{1/2}=0.693/\lambda (4 marks)

Phenomenon of self-induction

A coil carrying a current produces a magnetic field whose flux passes through the coil itself. If the current is changed, for example by moving the rheostat quickly or by switching on or off, the flux through the coil changes, and by Faraday's law an emf is induced in the same coil. This is self-induction: the phenomenon in which a changing current induces an emf in the coil itself.

The constant L is the self-inductance of the coil: the ratio of the induced emf to the rate of change of current. The minus sign is Lenz's law: the induced emf opposes the change in current, so it is called the back emf. It slows the growth of current when the circuit is switched on and tries to maintain the current when it is switched off.

Unit

The unit is the henry. The self-inductance of a coil is one henry if an emf of one volt is induced in it when the current changes at the rate of one ampere per second.

Factors

• number of turns N (L\propto N^2);
• area of cross-section A of the coil;
• length l of the coil for the same number of turns (L\propto1/l);
• nature of the core, its relative permeability \mu_r (soft iron increases L greatly).