WAEC SSCE - Physics - 1989 (Essay)

(a) Meaning of a linear expansivity of \( 1.0 \times 10^{-5}\,\text{K}^{-1} \)

It means that each unit length of the solid increases by \( 1.0 \times 10^{-5} \) of its original length for every 1 K (1 °C) rise in temperature. For example, a 1 m length would lengthen by \( 1.0 \times 10^{-5}\,\text{m} \) for each degree rise.

(b)(i) Experiment to determine the linear expansivity of a steel rod

Measure the original length \( l_0 \) of the steel rod and its initial temperature \( \theta_1 \). Place the rod in a jacket through which steam is passed, with one end fixed and the other end pressing against a micrometer/spherometer (or an optical lever). Record the increase in length \( \Delta l \) as the rod is heated to the steam temperature \( \theta_2 \). The linear expansivity is

\[ \alpha = \frac{\Delta l}{l_0(\theta_2 - \theta_1)} \]

(b)(ii) Safety gap between rails

Expansion of one bar: \( \Delta l = l_0 \, \alpha \, \Delta\theta \)

\[ \Delta l = 3 \times (1.0 \times 10^{-5}) \times (41 - 29) \]

\[ \Delta l = 3 \times 1.0 \times 10^{-5} \times 12 = 3.6 \times 10^{-4}\,\text{m} = 0.36\,\text{mm} \]

A gap of about 0.36 mm must be left between successive bars.

(c) Advantages and disadvantages of thermal expansion of solids

Three advantages:

• Fitting a metal tyre or ring onto a wheel (heat to expand, then cools and grips tightly).
• Hot riveting of metal plates (rivet contracts on cooling and clamps the plates firmly).
• Use in bimetallic strips for thermostats, thermometers and fire alarms.

Two disadvantages:

• Railway lines and bridges may buckle or bend if no expansion gaps are provided.
• Overhead telephone/electric cables sag in hot weather and may snap when they contract in cold weather; tight-fitting parts may jam or crack.

(a)(i) Photoelectric emission and threshold frequency

• Photoelectric emission is the ejection of electrons from the surface of a metal when electromagnetic radiation (light) of sufficiently high frequency falls on it.
• Threshold frequency is the minimum frequency of the incident radiation below which no electrons are emitted, no matter how intense the light.

(a)(ii) Symbols in \( E = hf - hf_o \)

• \( E \) is the maximum kinetic energy of the emitted (photo)electron.
• \( hf \) is the energy of the incident photon (h = Planck's constant, f = its frequency).
• \( hf_o \) is the work function of the metal (\( f_o \) is the threshold frequency).

(b) Frequency of the incident radiation

By Einstein's equation, the photon energy equals the work function plus the kinetic energy given to the electron:

\[ hf = W + KE = (1.33 \times 10^{-16}) + (1.97 \times 10^{-16}) = 3.30 \times 10^{-16}\,\text{J} \]

\[ f = \frac{hf}{h} = \frac{3.30 \times 10^{-16}}{6.60 \times 10^{-34}} = 5.0 \times 10^{17}\,\text{Hz} \]

The frequency of the incident radiation is \( 5.0 \times 10^{17}\,\text{Hz} \) (treating the given energies in joules).

(c) Below the threshold frequency, each photon does not carry enough energy to free an electron; the light energy absorbed by the surface electrons is simply re-radiated or converted into heat, so no electron gains sufficient energy to overcome the work function and escape. The number of electrons emitted (once above threshold) depends on the intensity (brightness) of the incident light.

(d) Duality of matter

The duality of matter means that matter (such as electrons) can behave both as particles and as waves. It behaves as a particle in phenomena such as the photoelectric effect and collisions, and as a wave in phenomena such as electron diffraction and interference, where a beam of electrons produces a diffraction pattern like that of light. The de Broglie relation \( \lambda = \dfrac{h}{mv} \) links the two aspects.

(a) Laws of electromagnetic induction

• Faraday's law: whenever there is a change in the magnetic flux linking a circuit, an e.m.f. is induced, and the magnitude of the induced e.m.f. is directly proportional to the rate of change of the magnetic flux linkage.
• Lenz's law: the direction of the induced current is always such that it opposes the change producing it.

(b)(i) Simple experiment

Connect a coil of wire to a sensitive galvanometer. When a bar magnet is pushed into (or pulled out of) the coil, the galvanometer deflects, showing that an e.m.f. (and current) is induced. When the magnet is held still there is no deflection.

(b)(ii) Two factors on which the induced e.m.f. depends

• The rate at which the magnetic flux changes (the speed of the magnet).
• The number of turns on the coil (and the strength of the magnet).

(c) r.m.s. value of an alternating current

The root-mean-square value of an alternating current is the value of the steady direct current that would dissipate heat in a given resistor at the same rate as the alternating current does.

(d)(i) Meaning of the symbols in \( I = I_o \sin \omega t \)

• \( I \) is the instantaneous value of the current at time t.
• \( I_o \) is the peak (maximum) value of the current.
• \( \omega \) is the angular frequency, \( \omega = 2\pi f \).
• \( \omega t \) is the phase angle of the current at time t.

(d)(ii) Instantaneous value

Peak value: \( I_o = I_{rms}\sqrt{2} = 15.0 \times 1.414 = 21.2\,\text{A} \).

At phase angle \( \omega t = 30^\circ \):

\[ I = I_o \sin 30^\circ = 21.2 \times 0.5 = 10.6\,\text{A} \]

The instantaneous current is about 10.6 A.