WAEC SSCE - Physics - 2001

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When an object appears a certain color, it is because it reflects that particular color of light and absorbs all other colors. Therefore, if a piece of cloth appears green in sunlight, it means that it reflects green light and absorbs all other colors. When the same piece of cloth is held in red light, it will appear black. This is because the red light being shone on the cloth contains only red light waves, and the cloth will absorb all the red light, as it does not reflect red light. Therefore, it appears black, as there is no reflected color for our eyes to see.

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Electrical resistance is a property of a material to oppose the flow of electrical current through it. When electrical current flows through a conductor, some of the electrical energy is converted into other forms of energy. One of the forms of energy that can be generated is heat energy, which is the result of collisions between the electrons and the atoms in the material. This is known as Joule heating, and the amount of heat generated is proportional to the resistance of the material and the square of the current flowing through it. Therefore, when electrical energy flows through a material with resistance, some of the energy is converted into heat energy due to the resistance of the material.

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The gravitational force between two particles is given by the formula F = Gm₁m₂/r², where G is the gravitational constant, m₁ and m₂ are the masses of the particles, and r is the distance between them. In this case, the distance between the particles is initially 0.10m and the force is 10N. If the distance is increased to 0.20m, we can use the formula to calculate the new force. Let's assume that the masses of the particles remain constant. The distance between them is doubled, so the new distance is 2r = 0.20m. Plugging this into the formula and solving for the new force, we get: F' = Gm₁m₂/(2r)² = Gm₁m₂/4r² Since the masses of the particles and the gravitational constant are constants, the only thing that has changed is the distance between the particles. Therefore, the new force is proportional to 1/r². Since r has doubled, the new force will be (1/2)² = 1/4 of the original force. So, the new force is: F' = (1/4)F = (1/4)10N = 2.5N Therefore, the magnitude of the new force is 2.5N. Option (C) is the correct answer.

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The amount of heat given out or absorbed when a substance changes its state at a constant temperature is known as latent heat. This means that even though the temperature remains constant during the change of state, heat is still being absorbed or released. For example, when ice is melted into water or water is evaporated into steam, there is no change in temperature, but heat is still absorbed by the substance during the change of state. The latent heat is specific to the substance and the change of state, and can be calculated by multiplying the mass of the substance by the specific latent heat of the substance for the particular change of state.

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The apparent weight of a body in a fluid is equal to the weight of the fluid displaced by the body. Using this principle, we can calculate the volume of the body as follows: Let V be the volume of the body Density of water = 1000kgm^-3 Weight of the body in air = 96N Weight of the body in water = 32N We know that: Weight of the body in air - Weight of the body in water = Upthrust = Weight of the water displaced Therefore, 96N - 32N = Weight of the water displaced 64N = Weight of the water displaced Weight of water displaced = Volume of body x Density of water x g 64N = V x 1000kgm^-3 x 10ms^-2 V = 64N / (1000kgm^-3 x 10ms^-2) V = 6.4 x 10^-3m³ Therefore, the volume of the body is 6.4 x 10^-3m³. Answer is correct.

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The power of an electrical appliance can be calculated by the formula P = V\(^2\)/R, where P is the power in watts, V is the voltage in volts, and R is the resistance in ohms. In this case, the bulb is rated 60W and 220V. So, we can rearrange the formula to find the resistance: R = V\(^2\)/P. Substituting the values gives us: R = 220\(^2\)/60. Evaluating this expression gives us: R = 806.7\(\Omega\). Therefore, the resistance of the filament when the bulb is operating normally is 806.7\(\Omega\). Note: It is important to remember that the actual resistance of the filament may vary due to factors such as temperature, age, and manufacturing tolerances.

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Electrons, which are usually considered as particles, can also exhibit wave-like properties, a concept known as wave-particle duality. When electrons are passed through crystals, their wave-like nature allows them to diffract, meaning they bend and scatter as they encounter the atoms in the crystal. This is similar to the way light diffracts when it passes through a narrow opening or encounters an obstacle. Therefore, the correct answer is that electrons passing through crystals are diffracted because they possess wave properties.

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In an elastic collision, both kinetic energy and momentum are conserved. This means that the total kinetic energy before the collision is equal to the total kinetic energy after the collision, and the total momentum before the collision is equal to the total momentum after the collision. There is no loss of kinetic energy or momentum in an elastic collision. Therefore, the statement "both kinetic energy and momentum are conserved" is correct. The other statements are incorrect as they imply a loss of either kinetic energy or momentum in an elastic collision, which is not the case.

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When surface waves travelling in deep water meet a shallow water boundary, they experience a change in speed and direction. This is known as refraction. The speed of the waves is dependent on the medium they are travelling through, so when they enter shallow water, their speed will change. We can use Snell's law, which states that the ratio of the sines of the angles of incidence and refraction is equal to the ratio of the velocities in the two media. In this case, we have: \begin{align*} \frac{\sin i}{\sin r} &= \frac{v_i}{v_r} \\ \frac{\sin 45}{\sin 30} &= \frac{15}{v_r} \\ \frac{\sqrt{2}}{0.5} &= \frac{15}{v_r} \\ v_r &= \frac{15}{\frac{\sqrt{2}}{0.5}} \\ v_r &= 10.6\text{ms}^{-1} \end{align*} Therefore, the speed of the waves in shallow water is approximately 10.6ms^-1. The correct option is (c) 10.6ms^-1.

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The correct option is "magnetic lines of force are closely packed together at neutral point". This statement is incorrect because neutral points are regions where the magnetic field is zero, and hence, there are no magnetic lines of force in these regions. Magnetic lines of force are an imaginary concept used to represent the direction and intensity of magnetic fields. They are assumed to be continuous and form closed loops, and are said to originate from the north pole and terminate at the south pole of a magnet. They also represent the path along which a free north pole would move in the presence of a magnetic field. Magnetic lines of force do not cross one another, and the presence of magnetic lines of force in a region indicates the presence of a magnetic field.

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The equation that represents the distance x, travelled by a body moving on a straight road with a constant speed is "x = Ut". In this equation, "U" represents the initial velocity of the body and "t" represents the time taken to cover the distance "x". Since the body is moving with a constant speed, its velocity remains the same throughout the journey and can be represented by "U". Therefore, the distance covered by the body can be calculated by multiplying the velocity "U" with the time taken "t", which is given by the equation "x = Ut". The other given equations represent the distance covered by a body in different scenarios like constant acceleration, initial velocity with constant acceleration, etc.

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The Earth's magnetic field has a north and south pole, and it is tilted with respect to the Earth's rotational axis. This means that at different points on the Earth's surface, the magnetic field has different characteristics. The magnetic equator is an imaginary line that circles the Earth, and it is the point where the magnetic field is horizontal. At Jos in Nigeria, the magnetic equator passes through, and this means that the magnetic field is horizontal at this point. Therefore, the angle of dip (the angle between the magnetic field lines and the Earth's surface) is zero.

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The pressure, volume, and temperature of an ideal gas are related by the ideal gas law, which is given by: P V = n R T where P is the pressure, V is the volume, n is the number of moles of gas, R is the gas constant, and T is the absolute temperature. If the volume of the gas remains constant, then the ideal gas law becomes: P = n R T / V Since the mass of the gas is constant, the number of moles n is also constant. Therefore, we can write: P/T = constant This means that the pressure of the gas is directly proportional to its temperature, when the volume is constant. To solve the problem, we can use the above formula to find the constant, and then use it to find the pressure at the new temperature: P1 / T1 = P2 / T2 where P1 is the initial pressure, T1 is the initial temperature, P2 is the final pressure, and T2 is the final temperature. Converting the temperatures to kelvin scale: T1 = 273 - 13 = 260 K T2 = 273 + 247 = 520 K Substituting the values: 500 / 260 = P2 / 520 Solving for P2: P2 = (500 / 260) x 520 = 1000 Nm^-2 Therefore, the pressure of the gas at the new temperature is 1000 Nm^-2. The correct option is: 1000Nm^-2.

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The physical processes of evaporation, thermal conduction, radiation of heat, and convectional current of fluids can all be explained by the molecular theory of matter. The theory explains that all matter is made up of tiny particles (atoms or molecules) which are constantly in motion. The motion of these particles can cause heat transfer through conduction, convection, and radiation. Evaporation is a process where some of the particles at the surface of a liquid have enough energy to escape and become a gas. Therefore, all the options listed can be explained by the molecular theory of matter.

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The time rate of change of momentum is called "force". Force is a vector quantity that causes an object to accelerate or change its velocity. When a force acts on an object for a certain amount of time, it produces a change in the object's momentum, which is called impulse. Therefore, force can also be defined as the rate of change of momentum, that is, the amount of impulse produced per unit of time.

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A screw jack is a simple machine used to lift heavy loads with little effort. The velocity ratio of a screw jack is defined as the ratio of the distance moved by the effort (Tommy bar) to the distance moved by the load (car). In this case, the car is raised through a vertical height of 25cm by turning the Tommy bar through 50 revolutions. The distance moved by the effort is the circumference of the circle traced out by the end of the Tommy bar, which is given by: circumference = 2πr = 2 x 3.142 x 0.12m = 0.754m (approx.) Since the Tommy bar is turned through 50 revolutions, the total distance moved by the effort is: distance moved by effort = 0.754m x 50 = 37.7m (approx.) Therefore, the velocity ratio is: velocity ratio = distance moved by effort / distance moved by load = 37.7m / 0.25m = 150.8 (approx.) So the approximate velocity ratio of the jack is 151, which is closest to.

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Humidity is used to describe the amount of water vapour present in the atmosphere. It is the measure of the water vapour content in the air, which can affect how comfortable we feel in a particular environment. Humidity can be measured in various ways, including absolute humidity, relative humidity, and specific humidity. Knowing the humidity level is important for a range of applications, from weather forecasting to indoor comfort control.

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The material used for constructing the core of an electromagnet is iron. An electromagnet is made by wrapping a coil of wire around a ferromagnetic core. When electric current flows through the wire, it creates a magnetic field around the wire. The ferromagnetic core helps to increase the strength of the magnetic field created by the current flowing through the wire. Iron is a good material for the core of an electromagnet because it is a ferromagnetic material that is easily magnetized and demagnetized. It also has a high magnetic permeability, which means it can conduct magnetic flux effectively.

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The maximum acceleration of a particle in simple harmonic motion is given by the product of the square of the angular frequency and the displacement of the particle from the equilibrium position. That is, a_max = -ω²A Where, A is the amplitude of the particle, and ω is the angular frequency. Substituting the given values, we get: a_max = - (10 rad/s)² x 0.05 m a_max = - 5 m/s² or 5 ms^-2 Since the maximum acceleration is a scalar quantity, we take its magnitude, which is simply 5 ms^-2. Therefore, the correct option is 5.00ms^-2.

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The force exerted on the block is 3N, and the block is at rest, meaning that the force of friction is equal in magnitude and opposite in direction to the applied force. Therefore, the friction force on the block is 3N. Answer: 3.0N

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When lamps are connected in parallel, each lamp has the same voltage across it. Therefore, the potential difference across each of the three lamps is 250V. The power of each lamp is given as 100W. We know that power (P) is equal to voltage (V) multiplied by current (I), i.e., P = VI. Therefore, the current through each lamp can be calculated as I = P/V = 100/250 = 0.4 A. Since the lamps are connected in parallel, the total current in the circuit is the sum of the currents through each lamp. Hence, the total current in the circuit is 3 times the current through one lamp, which is 3 x 0.4 A = 1.2 A. Therefore, the correct option is 1.2A.

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Convex spherical mirrors are preferred to plane mirrors as driving mirrors because they provide a wider field of view. Convex mirrors are curved outward and the reflected light rays diverge, resulting in a smaller image that appears farther away. This makes it easier for drivers to see objects behind them and in their blind spots. The wider field of view provided by convex mirrors is particularly useful for large vehicles such as buses and trucks, which have larger blind spots than smaller cars.

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The physical quality that affects the saturated vapour pressure of a liquid is temperature. Saturated vapour pressure is the pressure exerted by the vapour present in a system at equilibrium with its condensed phase, usually a liquid. As temperature increases, the average kinetic energy of the liquid molecules increases, causing more molecules to escape from the liquid phase and enter the gas phase. This leads to an increase in the number of gas molecules, which in turn results in an increase in pressure. As a result, the saturated vapour pressure of a liquid increases with increasing temperature, and decreases with decreasing temperature. Therefore, temperature is the correct answer. The other options - volume, mass, and density - do not directly affect the saturated vapour pressure of a liquid.

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(a)(i) An echo is a reflected sound wave heard distinctly after the original sound, produced when sound is reflected from a hard, distant surface back to the listener.

(a)(ii) Two useful applications:

• Depth sounding (echo sounding/SONAR) to measure the depth of the sea or locate shoals of fish and submarines.
• Detecting flaws in metals and, in medicine, ultrasonic scanning of the body.

(a)(iii) The walls, floors and ceilings of a recording studio are heavily padded with soft, porous material to absorb sound and prevent reflection (echoes and reverberation), so that only clear, direct sound is recorded.

(b)(i) Timbre (quality) is the property of a musical note that lets the ear distinguish two notes of the same pitch and loudness from different sources; it depends on the number and relative strength of the overtones present. Overtones are the higher frequencies (above the fundamental) that accompany a note.

(b)(ii) Resonance is the condition in which a body is set into vibration with large amplitude by a periodic force whose frequency equals the natural frequency of the body.

(c) Velocity of the ship

Distance of the cliff at the first sounding \(= \tfrac{1}{2}(340)(12) = 2040\,\text{m}\).

Distance at the second sounding \(= \tfrac{1}{2}(340)(8) = 1360\,\text{m}\).

Distance travelled by the ship between the two soundings:

Time between soundings \(= 2.1\,\text{min} = 126\,\text{s}\). Hence:

The ship approaches the cliff at about \(5.4\,\text{ms}^{-1}\).

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(a) Specific heat capacity is the quantity of heat required to raise the temperature of unit mass (1 kg) of a substance by one kelvin (1 K). Its unit is \(\text{J kg}^{-1}\,\text{K}^{-1}\).

(b)(i) Experiment to determine the specific heat capacity of copper (method of mixtures)

Procedure:

• Weigh the empty copper calorimeter together with its stirrer; record the mass \(m_c\).
• Half-fill the calorimeter with water, weigh again, and obtain the mass of water \(m_w\) by subtraction. Note the initial temperature of the water and calorimeter, \(\theta_1\).
• Weigh the copper ball; record its mass \(m_b\). Suspend it by a thread inside a beaker of water and heat until the water boils steadily. Record the steady temperature of the ball, \(\theta\) (about \(100^\circ\text{C}\)).
• Shake off the adhering water and quickly transfer the hot ball into the calorimeter. Stir gently and record the highest steady temperature of the mixture, \(\theta_2\).

Since heat losses are negligible, heat lost by the copper ball = heat gained by the water + heat gained by the calorimeter and stirrer:

Making \(c_b\), the specific heat capacity of copper, the subject:

(b)(ii) Two precautions

• Transfer the hot copper ball to the calorimeter quickly and shake off any adhering water first, to minimise heat loss to the surroundings.
• Lag (insulate) the calorimeter and stir the water gently before reading the highest steady temperature.

(c) Final steady temperature of the mixture

Data: copper block \(m_b=24\text{ g}=0.024\text{ kg}\) at \(230^\circ\text{C}\); calorimeter \(m_c=60\text{ g}=0.060\text{ kg}\); water \(m_w=54\text{ g}=0.054\text{ kg}\) at \(31^\circ\text{C}\); \(c_w=4200\ \text{J kg}^{-1}\text{K}^{-1}\); \(c_b=400\ \text{J kg}^{-1}\text{K}^{-1}\). Let the final steady temperature be \(\theta\).

Heat lost by copper block = heat gained by water + heat gained by copper calorimeter:

The final steady temperature of the mixture is \(\theta \approx 38.3^\circ\text{C}\).

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(a) A beam of polarised light

Light is a transverse wave. In an ordinary (unpolarised) beam the electric-field vibrations occur in all planes perpendicular to the direction in which the light travels. A beam of polarised light is a beam in which these vibrations have been restricted to one single plane perpendicular to the direction of travel. In other words, the vibration of the wave takes place in only one plane (the plane of polarisation), so the light is said to be plane-polarised.

(b) Action of a polaroid spectacle on a beam of sunlight

Sunlight reaching the eye is unpolarised: its vibrations lie in every plane at right angles to the beam. A polaroid lens contains long parallel molecular chains that create a single transmission (polarising) axis. When sunlight strikes the polaroid:

• only the vibration parallel to the transmission axis is allowed through, and
• the vibrations in every other plane are absorbed.

The light that emerges is therefore plane-polarised and much reduced in intensity. This is why polaroid spectacles cut down the dazzling glare of sunlight (especially reflected glare, which is already partly polarised), so the wearer sees more comfortably. The action is illustrated below.

In the diagram, the incoming sunlight vibrates in all directions (shown by the double-headed arrows pointing every way). After passing through the polaroid, whose transmission axis is vertical, only the vertical component of vibration survives, so the beam reaching the eye is plane-polarised in the vertical plane and dimmer than the original sunlight.

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(a) Transmutation equation

Nitrogen-14 bombarded by a neutron emits an alpha particle:

(Mass numbers: \(14+1 = 4+11\); atomic numbers: \(7+0 = 2+5\). The residual nucleus is boron-11.)

(b)(i) A radioactive atom has an unstable nucleus that spontaneously disintegrates, emitting radiation (alpha, beta or gamma), whereas a stable atom has a nucleus that does not decay.

(b)(ii) Half life is the time taken for half the atoms (or nuclei) in a given sample of a radioactive material to decay.

(b)(iii) \(105\,\text{days} = 3\times 35\,\text{days} = 3\) half lives. Fraction remaining:

(c) Photoelectric calculations (\(\lambda = 5.00\times10^{-7}\,\text{m}\), \(W_0 = 1.90\,\text{eV}\)):

(i) Photon energy:

(ii) Maximum kinetic energy: work function \(W_0 = 1.90\times1.6\times10^{-19} = 3.04\times10^{-19}\,\text{J}\).

(iii) Stopping potential:

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Given: maximum height \(h = 50\ \text{m}\), \(g = 10\ \text{m s}^{-2}\), and velocity at the top \(= 0\).

(ii) Speed of throw (find first): using \(v^{2} = u^{2} - 2gh\) with \(v = 0\):

(i) Time to reach maximum height: using \(v = u - gt\) with \(v = 0\):

The ball takes about 3.16 s to reach the top, having been thrown at about 31.6 m s\(^{-1}\).

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(a) To say that the capacitance of a capacitor is \(5\,\mu\text{F}\) means that the capacitor stores a charge of \(5\,\mu\text{C}\) \((5\times10^{-6}\,\text{C})\) on each plate for every \(1\,\text{V}\) of potential difference applied across its plates. In other words, the ratio of the charge stored to the potential difference across the plates is \(5\,\mu\text{C V}^{-1}\), since \(C=\dfrac{Q}{V}\).

(b)(i) & (ii) The capacitance of a parallel-plate capacitor is given by \(C=\dfrac{\varepsilon A}{d}\). The factors on which it depends and the form of dependence are:

Factor	How the capacitance depends on it
Common (overlapping) area of the plates, \(A\)	Directly proportional: \(C\propto A\). Increasing the area increases the capacitance.
Distance between the plates, \(d\)	Inversely proportional: \(C\propto \dfrac{1}{d}\). Increasing the separation decreases the capacitance.
Permittivity (nature) of the dielectric, \(\varepsilon\)	Directly proportional: \(C\propto \varepsilon\). A dielectric of higher permittivity increases the capacitance.

(c)(i) Open-circuit diagram (switch open) showing the three capacitors in series with the \(90\,\text{V}\) battery:

(c)(ii) Effective capacitance. For capacitors in series the reciprocals add:

\[\frac{1}{C}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}=\frac{1}{8}+\frac{1}{12}+\frac{1}{24}=\frac{3+2+1}{24}=\frac{6}{24}=\frac{1}{4}\]\[C=4\,\mu\text{F}\]

(c)(iii) Charge on each capacitor. In a series arrangement the same charge flows onto every capacitor, and it equals the charge on the equivalent capacitor:

\[Q=CV=4\times10^{-6}\times90=3.6\times10^{-4}\,\text{C}=360\,\mu\text{C}\]

Therefore the charge on each of the \(8\,\mu\text{F}\), \(12\,\mu\text{F}\) and \(24\,\mu\text{F}\) capacitors is the same: \(Q=360\,\mu\text{C}\).

(c)(iv) P.d. across the \(8\,\mu\text{F}\) capacitor.

\[V_{8}=\frac{Q}{C_1}=\frac{360\,\mu\text{C}}{8\,\mu\text{F}}=45\,\text{V}\]

The potential difference across the \(8\,\mu\text{F}\) capacitor is \(45\,\text{V}\). (As a check: \(V_{12}=360/12=30\,\text{V}\) and \(V_{24}=360/24=15\,\text{V}\); \(45+30+15=90\,\text{V}\), which equals the battery voltage.)

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(a) Gain in the long jump

The horizontal range of a projectile is \(R = \dfrac{v^2\sin 2\theta}{g}\). With \(v = 8\,\text{ms}^{-1}\) and \(g = 10\,\text{ms}^{-2}\), \(\dfrac{v^2}{g} = \dfrac{64}{10} = 6.4\,\text{m}\).

First attempt (\(\theta = 30^\circ\)): \(R_1 = 6.4\sin 60^\circ = 6.4(0.8660) = 5.5424\,\text{m}\).

Second attempt (\(\theta = 45^\circ\)): \(R_2 = 6.4\sin 90^\circ = 6.4(1) = 6.4000\,\text{m}\).

(b)(i) Hooke's law: Provided the elastic limit is not exceeded, the extension of an elastic material is directly proportional to the applied force (load).

(b)(ii) Experiment to verify Hooke's law: A spiral spring is clamped vertically with a pointer against a metre rule. The unstretched pointer reading is noted. Known masses are added one at a time; for each load the new pointer reading is taken and the extension found by subtraction. A graph of load against extension is plotted; a straight line through the origin verifies that extension is proportional to load, confirming Hooke's law.

(b)(iii) Two precautions:

• Do not exceed the elastic limit of the spring (avoid overloading so it returns to its original length).
• Take readings with the eye level to the pointer to avoid parallax error; allow the spring to settle before reading.

(c) Mass of the scale pan

Extension is proportional to the total load (pan mass \(m\) plus object). Natural length \(= 20.00\,\text{cm}\).

With 40 g: length \(21.80\,\text{cm}\), extension \(e_1 = 1.80\,\text{cm}\), load \(=(m+40)\).

With 60 g: length \(22.05\,\text{cm}\), extension \(e_2 = 2.05\,\text{cm}\), load \(=(m+60)\).

The mass of the scale pan is \(104\,\text{g}\ (0.104\,\text{kg})\).

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The fall in e.m.f. with use is caused by two defects of the simple voltaic cell:

• Polarisation. As the cell works, hydrogen gas is liberated and clings to the copper (positive) plate, forming an insulating layer. This layer increases the internal resistance and sets up a back e.m.f., so the useful e.m.f. falls.
• Local action. Impurities (such as carbon and iron) in the commercial zinc plate form tiny local cells with the zinc, so the zinc is eaten away even when no current is being drawn, weakening the cell.

Corrections:

• Polarisation is removed by adding a depolariser (an oxidising agent such as manganese(IV) oxide or potassium dichromate) which oxidises the hydrogen to water.
• Local action is prevented by amalgamating the zinc plate, i.e. coating it with mercury so that the surface presents a clean layer of pure zinc only.

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(a) Stress and strain

Stress is the force acting per unit cross-sectional area of the material:

Strain is the extension produced per unit original length of the material:

Thus stress measures the intensity of the applied load while strain measures the relative deformation it produces.

(b) Work done on the cord

For an elastic cord obeying Hooke's law, the work done equals the elastic potential energy stored, given by the area under the force-extension graph:

With \(F = 40\,\text{N}\) and \(e = 5\,\text{cm} = 0.05\,\text{m}\):

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(a) Two properties of cathode rays

• They travel in straight lines from the cathode and cast sharp shadows of objects placed in their path.
• They are streams of negatively charged particles (electrons), so they are deflected by both electric and magnetic fields (towards the positive plate).

Other acceptable properties: they carry energy and momentum, they produce heat and fluorescence on striking a screen, and they can produce X-rays on hitting a metal target.

(b) Increasing the intensity and energy of cathode rays

Intensity (number of electrons per second) is increased by raising the filament (heater) current. A hotter cathode emits more electrons by thermionic emission, so the beam current and hence the brightness increase.

Energy (speed of each electron) is increased by raising the accelerating potential difference between the cathode and the anode. Each electron gains kinetic energy \(eV\) as it is accelerated, so a larger anode voltage gives faster, more energetic electrons.

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For the vertical motion of the projectile, the initial vertical velocity is \(u\sin\theta\) and the acceleration due to gravity is \(-g\).

Using

At maximum height,

Therefore,

Hence, the greatest height reached is

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The graph is a straight line of the maximum kinetic energy \(E\) of the emitted photoelectrons (vertical axis, in joules) against the frequency \(f\) of the incident radiation (horizontal axis, in hertz). The line crosses the frequency axis at the point \(K\) and, when produced backwards, cuts the energy axis below the origin at the point \(C\). The graph is a plot of Einstein's photoelectric equation,

\[ E = hf - hf_0 \]

where \(h\) is Planck's constant and \(f_0\) is the threshold frequency of the metal. Comparing this with the equation of a straight line \(y = mx + c\): \(m = h\) is the gradient and \(c = -hf_0\) is the intercept on the energy axis.

(i) The slope of the graph

Comparing \(E = hf - hf_0\) with \(y = mx + c\), the gradient is

\[ \text{slope} = \frac{\Delta E}{\Delta f} = h \]

So the slope represents Planck's constant, \(h\) \(\left(\approx 6.6\times10^{-34}\,\text{J s}\right)\). It has the same value for every metal.

(ii) The intercept OC

OC is the intercept on the energy axis, obtained when \(f = 0\). From the equation, \(E = -hf_0\), so this intercept is negative and equal in magnitude to the work function \(\varphi\) of the metal:

\[ OC = -hf_0 = -\varphi \]

Thus \(|OC|\) is the minimum energy needed to release an electron from the surface of the metal, i.e. the work function.

(iii) The intercept OK

OK is the intercept on the frequency axis, obtained when the photoelectron energy \(E = 0\). Setting \(0 = hf - hf_0\) gives \(f = f_0\). Therefore OK represents the threshold (cut-off) frequency \(f_0\), the minimum frequency of the incident radiation below which no electrons are emitted from the metal.

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None

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Kinetic energy of the electron from its de Broglie wavelength

The de Broglie relation links wavelength to momentum:

Substituting \(h = 6.6\times10^{-34}\,\text{Js}\), \(m = 9.0\times10^{-31}\,\text{kg}\), \(\lambda = 1.0\times10^{-10}\,\text{m}\):

Then the kinetic energy is:

Equivalently, using \(E_k = \dfrac{h^2}{2m\lambda^2}\) gives the same result, \(E_k \approx 2.4\times10^{-17}\,\text{J}\).

Détails de la réponse

(a) How a gas can be made to conduct electricity

At ordinary pressure a gas is an insulator. It can be made to conduct by ionising it, that is by producing charged particles within it. This is done by applying a very high potential difference across the gas while it is at low pressure (as in a discharge tube), or by heating it strongly, or by exposing it to X-rays, ultraviolet or radioactive radiation. The high field/energy strips electrons from some of the gas atoms, producing free electrons and positive ions; under the applied field these charges drift to the electrodes and so a current flows.

(b) Electric charge carriers in gases

• Positive ions.
• Electrons (together with negative ions).