WAEC SSCE - Physics - 2016 (Essay)
Tambaya 1 Rahoto
State three different materials that can be used to demonstrate Brownian motion.
Materials that can demonstrate Brownian motion:
- Smoke particles suspended in air (in a smoke cell), viewed under a microscope.
- Pollen grains suspended in water.
- Fine dust or graphite particles suspended in water or air.
Note: a suspension of gamboge or sulphur particles in water is also acceptable. In each case the tiny visible particles are seen to move in continuous, random, zig-zag paths because they are bombarded unequally by the much smaller, fast-moving molecules of the surrounding fluid.
Bayanin Amsa
Materials that can demonstrate Brownian motion:
- Smoke particles suspended in air (in a smoke cell), viewed under a microscope.
- Pollen grains suspended in water.
- Fine dust or graphite particles suspended in water or air.
Note: a suspension of gamboge or sulphur particles in water is also acceptable. In each case the tiny visible particles are seen to move in continuous, random, zig-zag paths because they are bombarded unequally by the much smaller, fast-moving molecules of the surrounding fluid.
Tambaya 2 Rahoto
(a) Define critical angle.
(b) How are anti-nodes created in a stationary wave?
(c) The angle of minimum deviation of an equilateral triangular glass prism is 46.2°. Calculate the refractive index of the glass.
(d) An illuminated object is placed in front of a concave mirror and the position of a screen is adjusted in front of the mirror but no image is obtained on the screen. Give two possible reasons for this observation.
(e) An illuminated object is placed at a distance of 75 cm from a converging lens of focal length 30 cm.
(i) Determine the image distance.
(ii) If the lens is replaced by another converging lens, the object has to be moved 25 cm further away to have its sharp image on the screen. Determine the focal length of the second lens.
(a) Critical angle: the angle of incidence, in the denser medium, for which the angle of refraction in the less dense medium is \(90^{\circ}\). For angles greater than this, total internal reflection occurs.
(b) Formation of antinodes: antinodes are points of maximum displacement in a stationary wave. They are formed where the incident wave and the reflected wave meet in phase, so their displacements always add up (constructive superposition), giving maximum vibration.
(c) Refractive index of the prism. For a prism of refracting angle \(A=60^{\circ}\) (equilateral) at minimum deviation \(D=46.2^{\circ}\):
\[ n = \frac{\sin\!\left(\frac{A+D}{2}\right)}{\sin\!\left(\frac{A}{2}\right)} = \frac{\sin\!\left(\frac{60+46.2}{2}\right)}{\sin 30^{\circ}} = \frac{\sin 53.1^{\circ}}{0.5} = \frac{0.7997}{0.5} = 1.60 \](d) Concave mirror gives no image on the screen - two possible reasons:
- The object is placed at (or inside) the focal point, so the reflected rays are parallel or diverge and form a virtual image that cannot be caught on a screen.
- The screen is not placed at the correct image distance (it is nearer or farther than the position where the rays actually converge).
(e) Converging lens.
(i) \(u=75\,\text{cm}\), \(f=30\,\text{cm}\). Using \(\frac{1}{f}=\frac{1}{u}+\frac{1}{v}\):
(ii) The object is moved \(25\,\text{cm}\) farther, so \(u_2 = 75+25 = 100\,\text{cm}\), and the sharp image is formed on the same screen, i.e. \(v_2 = 50\,\text{cm}\).
\[ \frac{1}{f_2}=\frac{1}{u_2}+\frac{1}{v_2}=\frac{1}{100}+\frac{1}{50}=\frac{1+2}{100}=\frac{3}{100} \]\[ f_2 = \frac{100}{3} = 33.3\,\text{cm} \]Bayanin Amsa
(a) Critical angle: the angle of incidence, in the denser medium, for which the angle of refraction in the less dense medium is \(90^{\circ}\). For angles greater than this, total internal reflection occurs.
(b) Formation of antinodes: antinodes are points of maximum displacement in a stationary wave. They are formed where the incident wave and the reflected wave meet in phase, so their displacements always add up (constructive superposition), giving maximum vibration.
(c) Refractive index of the prism. For a prism of refracting angle \(A=60^{\circ}\) (equilateral) at minimum deviation \(D=46.2^{\circ}\):
\[ n = \frac{\sin\!\left(\frac{A+D}{2}\right)}{\sin\!\left(\frac{A}{2}\right)} = \frac{\sin\!\left(\frac{60+46.2}{2}\right)}{\sin 30^{\circ}} = \frac{\sin 53.1^{\circ}}{0.5} = \frac{0.7997}{0.5} = 1.60 \](d) Concave mirror gives no image on the screen - two possible reasons:
- The object is placed at (or inside) the focal point, so the reflected rays are parallel or diverge and form a virtual image that cannot be caught on a screen.
- The screen is not placed at the correct image distance (it is nearer or farther than the position where the rays actually converge).
(e) Converging lens.
(i) \(u=75\,\text{cm}\), \(f=30\,\text{cm}\). Using \(\frac{1}{f}=\frac{1}{u}+\frac{1}{v}\):
(ii) The object is moved \(25\,\text{cm}\) farther, so \(u_2 = 75+25 = 100\,\text{cm}\), and the sharp image is formed on the same screen, i.e. \(v_2 = 50\,\text{cm}\).
\[ \frac{1}{f_2}=\frac{1}{u_2}+\frac{1}{v_2}=\frac{1}{100}+\frac{1}{50}=\frac{1+2}{100}=\frac{3}{100} \]\[ f_2 = \frac{100}{3} = 33.3\,\text{cm} \]Tambaya 3 Rahoto
(a) Explain the term net force.
(b) Define the principle of conservation of linear momentum and state one example of it.
(c) A ball of mass 200 g released from a height of 2.0 m hits a horizontal floor and rebounds to a height of 1.8 in. Calculate the impulse received by the floor. (g = 10 ms\(^{-2}\)).
(d) A body of mass 20 g performs a simple harmonic motion at a frequency of 5 Hz. At a distance of 10 cm from the mean position, its velocity is 200 cms\(^{-1}\). Calculate its:
(i) maximum displacement from the mean position;
(ii) maximum velocity;
(iii) maximum potential energy. (g = 10 ms\(^{-2}\) \(\pi\) = 3.14)
(a) Net force is the single resultant force obtained when all the forces acting on a body are combined (added as vectors). It is the force that determines the body's acceleration, \(F_{net}=ma\).
(b) Principle of conservation of linear momentum: in a closed system on which no external force acts, the total linear momentum before an interaction (collision or explosion) equals the total linear momentum after it. Example: the recoil of a gun when a bullet is fired - the forward momentum of the bullet equals the backward momentum of the gun.
(c) Impulse received by the floor.
Mass \(m = 200\,\text{g}=0.2\,\text{kg}\).
Speed on hitting the floor: \(v_1=\sqrt{2gh_1}=\sqrt{2\times10\times2.0}=\sqrt{40}=6.32\,\text{ms}^{-1}\) (downward).
Speed on rebound: \(v_2=\sqrt{2gh_2}=\sqrt{2\times10\times1.8}=\sqrt{36}=6.0\,\text{ms}^{-1}\) (upward).
Taking upward as positive, impulse on the ball
By Newton's third law the impulse received by the floor is equal and opposite: 2.46 Ns directed downward.
(d) Simple harmonic motion.
\(m=20\,\text{g}=0.02\,\text{kg}\), \(f=5\,\text{Hz}\Rightarrow \omega=2\pi f=2\times3.14\times5=31.4\,\text{rads}^{-1}\).
At \(x=10\,\text{cm}=0.1\,\text{m}\), \(v=200\,\text{cms}^{-1}=2.0\,\text{ms}^{-1}\).
(i) Maximum displacement (amplitude) A. Using \(v=\omega\sqrt{A^2-x^2}\):
\[ v^2=\omega^2(A^2-x^2)\Rightarrow A^2 = x^2 + \frac{v^2}{\omega^2}=0.01+\frac{4}{(31.4)^2}=0.01+0.00406=0.01406 \]\[ A = 0.119\,\text{m}\;(\approx 11.9\,\text{cm}) \](ii) Maximum velocity.
\[ v_{max}=\omega A = 31.4\times0.119 = 3.73\,\text{ms}^{-1} \](iii) Maximum potential energy (equals the total energy of the oscillation):
\[ P.E._{max}=\tfrac{1}{2}m\omega^2A^2=\tfrac{1}{2}(0.02)(31.4)^2(0.01406)=0.139\,\text{J} \]So \(P.E._{max}\approx 0.14\,\text{J}\).
Bayanin Amsa
(a) Net force is the single resultant force obtained when all the forces acting on a body are combined (added as vectors). It is the force that determines the body's acceleration, \(F_{net}=ma\).
(b) Principle of conservation of linear momentum: in a closed system on which no external force acts, the total linear momentum before an interaction (collision or explosion) equals the total linear momentum after it. Example: the recoil of a gun when a bullet is fired - the forward momentum of the bullet equals the backward momentum of the gun.
(c) Impulse received by the floor.
Mass \(m = 200\,\text{g}=0.2\,\text{kg}\).
Speed on hitting the floor: \(v_1=\sqrt{2gh_1}=\sqrt{2\times10\times2.0}=\sqrt{40}=6.32\,\text{ms}^{-1}\) (downward).
Speed on rebound: \(v_2=\sqrt{2gh_2}=\sqrt{2\times10\times1.8}=\sqrt{36}=6.0\,\text{ms}^{-1}\) (upward).
Taking upward as positive, impulse on the ball
By Newton's third law the impulse received by the floor is equal and opposite: 2.46 Ns directed downward.
(d) Simple harmonic motion.
\(m=20\,\text{g}=0.02\,\text{kg}\), \(f=5\,\text{Hz}\Rightarrow \omega=2\pi f=2\times3.14\times5=31.4\,\text{rads}^{-1}\).
At \(x=10\,\text{cm}=0.1\,\text{m}\), \(v=200\,\text{cms}^{-1}=2.0\,\text{ms}^{-1}\).
(i) Maximum displacement (amplitude) A. Using \(v=\omega\sqrt{A^2-x^2}\):
\[ v^2=\omega^2(A^2-x^2)\Rightarrow A^2 = x^2 + \frac{v^2}{\omega^2}=0.01+\frac{4}{(31.4)^2}=0.01+0.00406=0.01406 \]\[ A = 0.119\,\text{m}\;(\approx 11.9\,\text{cm}) \](ii) Maximum velocity.
\[ v_{max}=\omega A = 31.4\times0.119 = 3.73\,\text{ms}^{-1} \](iii) Maximum potential energy (equals the total energy of the oscillation):
\[ P.E._{max}=\tfrac{1}{2}m\omega^2A^2=\tfrac{1}{2}(0.02)(31.4)^2(0.01406)=0.139\,\text{J} \]So \(P.E._{max}\approx 0.14\,\text{J}\).
Tambaya 4 Rahoto
(a) Explain briefly dielectric strength.
(b) An electromagnetic wave has its wavelength shorter than those of radiowave and microwave but longer than that of visible light.
(i) Identify the wave.
(ii) Name one suitable detector for the wave.
(iii) Name one source of the wave.
(c) An oil drop carrying a charge of 1.0 x 10\(^{-19 }\)C is found to remain at rest in a uniform electric field of intensity 1200 NC\(^{-1}\). Calculate the weight of the oil drop.
(d) An RLC series circuit consists of a 100\(\Omega\) resistor, 0.05 H inductor and a 25 \(\mu\) capacitor. A 220 V, 50 Hz mains voltage is applied across the circuit. Calculate the:
(i) impedance;
(ii) current. (\(\pi\) = 3.14)
(a) Dielectric strength: the maximum electric field (or maximum p.d. per unit thickness) that an insulator (dielectric) can withstand without breaking down and allowing charge to conduct through it.
(b) The wave is infrared radiation.
- (i) Infrared (heat) radiation.
- (ii) Detector: a thermopile / bolometer / blackened thermometer (a special infrared-sensitive photodiode is also acceptable).
- (iii) Source: any hot body, e.g. the Sun, an electric heating element or a warm object.
(c) Weight of the oil drop. Since the drop is at rest, the electric force balances the weight:
\[ W = qE = (1.0\times10^{-19})(1200) = 1.2\times10^{-16}\,\text{N} \](d) RLC series circuit. \(R=100\,\Omega\), \(L=0.05\,\text{H}\), \(C=25\,\mu\text{F}=25\times10^{-6}\,\text{F}\), \(f=50\,\text{Hz}\).
\[ X_L = 2\pi f L = 2(3.14)(50)(0.05)=15.7\,\Omega \]\[ X_C = \frac{1}{2\pi f C}=\frac{1}{2(3.14)(50)(25\times10^{-6})}=\frac{1}{7.85\times10^{-3}}=127.4\,\Omega \](i) Impedance:
\[ Z=\sqrt{R^2+(X_C-X_L)^2}=\sqrt{100^2+(127.4-15.7)^2}=\sqrt{10000+12477}=\sqrt{22477}=150\,\Omega \](ii) Current:
\[ I=\frac{V}{Z}=\frac{220}{150}=1.47\,\text{A} \]Bayanin Amsa
(a) Dielectric strength: the maximum electric field (or maximum p.d. per unit thickness) that an insulator (dielectric) can withstand without breaking down and allowing charge to conduct through it.
(b) The wave is infrared radiation.
- (i) Infrared (heat) radiation.
- (ii) Detector: a thermopile / bolometer / blackened thermometer (a special infrared-sensitive photodiode is also acceptable).
- (iii) Source: any hot body, e.g. the Sun, an electric heating element or a warm object.
(c) Weight of the oil drop. Since the drop is at rest, the electric force balances the weight:
\[ W = qE = (1.0\times10^{-19})(1200) = 1.2\times10^{-16}\,\text{N} \](d) RLC series circuit. \(R=100\,\Omega\), \(L=0.05\,\text{H}\), \(C=25\,\mu\text{F}=25\times10^{-6}\,\text{F}\), \(f=50\,\text{Hz}\).
\[ X_L = 2\pi f L = 2(3.14)(50)(0.05)=15.7\,\Omega \]\[ X_C = \frac{1}{2\pi f C}=\frac{1}{2(3.14)(50)(25\times10^{-6})}=\frac{1}{7.85\times10^{-3}}=127.4\,\Omega \](i) Impedance:
\[ Z=\sqrt{R^2+(X_C-X_L)^2}=\sqrt{100^2+(127.4-15.7)^2}=\sqrt{10000+12477}=\sqrt{22477}=150\,\Omega \](ii) Current:
\[ I=\frac{V}{Z}=\frac{220}{150}=1.47\,\text{A} \]Tambaya 5 Rahoto
(a) Explain the terms:
(i) thermal equilibrium;
(ii) fundamental interval.
(b) List two uses of the hydraulic press.
(c) Name the material used to reset the steel index in the Six's maximum and minimum thermometer.
(d)(i) A nursing mother prepared her baby's milk mixture at 85°C, in a feeding bottle. In order to cool it to 40°C, she immersed the bottle in an aluminium bowl of heat capacity 90 JK\(^{-1}\) containing 500 g of water at 26°C. If the mass of the mixture is 300g, calculate the specific heat capacity of the mixture. [Neglect heat losses and heat capacity of the bottle; specific heat capacity of water = 4200 J kg\(^{-1}\) K\(^{-1}\)]
(ii) (\(\alpha\)) Name two ways through which the bottle losses heat.
(\(\beta\)) Name two industrial processes in which heat exchanger is used.
(a)(i) Thermal equilibrium: the state reached when two (or more) bodies in thermal contact are at the same temperature so that there is no net flow of heat between them.
(a)(ii) Fundamental interval: the temperature range between the lower fixed point (ice point) and the upper fixed point (steam point) of a thermometer scale.
(b) Two uses of the hydraulic press:
- Lifting heavy loads (e.g. hydraulic jack / car lift).
- Compressing or baling materials such as cotton, scrap metal and refuse (also forging and stamping metals).
(c) Material used to reset the steel index in the Six's maximum and minimum thermometer: a magnet (the steel indices are drawn back with a small magnet).
(d)(i) Specific heat capacity of the mixture.
Heat lost by the hot milk mixture = heat gained by the water + heat gained by the aluminium bowl.
Mixture: \(m=300\,\text{g}=0.3\,\text{kg}\), cools \(85^{\circ}\text{C}\to40^{\circ}\text{C}\), \(\Delta\theta=45\,\text{K}\).
Water: \(0.5\,\text{kg}\), warms \(26^{\circ}\text{C}\to40^{\circ}\text{C}\), \(\Delta\theta=14\,\text{K}\).
Bowl heat capacity \(=90\,\text{JK}^{-1}\), warms by \(14\,\text{K}\).
So \(c \approx 2.27\times10^{3}\,\text{Jkg}^{-1}\text{K}^{-1}\).
(d)(ii)(\(\alpha\)) Two ways the bottle loses heat: conduction and convection (radiation and evaporation are also acceptable).
(d)(ii)(\(\beta\)) Two industrial processes using a heat exchanger: electricity generation in power stations (steam condensers/boilers) and refrigeration / air-conditioning (also petroleum refining and food pasteurization).
Bayanin Amsa
(a)(i) Thermal equilibrium: the state reached when two (or more) bodies in thermal contact are at the same temperature so that there is no net flow of heat between them.
(a)(ii) Fundamental interval: the temperature range between the lower fixed point (ice point) and the upper fixed point (steam point) of a thermometer scale.
(b) Two uses of the hydraulic press:
- Lifting heavy loads (e.g. hydraulic jack / car lift).
- Compressing or baling materials such as cotton, scrap metal and refuse (also forging and stamping metals).
(c) Material used to reset the steel index in the Six's maximum and minimum thermometer: a magnet (the steel indices are drawn back with a small magnet).
(d)(i) Specific heat capacity of the mixture.
Heat lost by the hot milk mixture = heat gained by the water + heat gained by the aluminium bowl.
Mixture: \(m=300\,\text{g}=0.3\,\text{kg}\), cools \(85^{\circ}\text{C}\to40^{\circ}\text{C}\), \(\Delta\theta=45\,\text{K}\).
Water: \(0.5\,\text{kg}\), warms \(26^{\circ}\text{C}\to40^{\circ}\text{C}\), \(\Delta\theta=14\,\text{K}\).
Bowl heat capacity \(=90\,\text{JK}^{-1}\), warms by \(14\,\text{K}\).
So \(c \approx 2.27\times10^{3}\,\text{Jkg}^{-1}\text{K}^{-1}\).
(d)(ii)(\(\alpha\)) Two ways the bottle loses heat: conduction and convection (radiation and evaporation are also acceptable).
(d)(ii)(\(\beta\)) Two industrial processes using a heat exchanger: electricity generation in power stations (steam condensers/boilers) and refrigeration / air-conditioning (also petroleum refining and food pasteurization).
Tambaya 6 Rahoto
List three uses of rockets.
Uses of rockets:
- Launching artificial satellites and spacecraft into orbit (space exploration).
- As military weapons, i.e. carrying missiles/warheads.
- Scientific research, e.g. carrying instruments to study the upper atmosphere and outer space.
Other acceptable uses: signalling and distress flares, and fireworks/display.
Bayanin Amsa
Uses of rockets:
- Launching artificial satellites and spacecraft into orbit (space exploration).
- As military weapons, i.e. carrying missiles/warheads.
- Scientific research, e.g. carrying instruments to study the upper atmosphere and outer space.
Other acceptable uses: signalling and distress flares, and fireworks/display.
Tambaya 7 Rahoto
The diagram above illustrates a cathode ray tube. Identify the components X, Y, and Z.
Identifying the components of the cathode ray tube. Following the electron beam from left (the electron gun) to right (the screen end):
- X - the filament/cathode (heater of the electron gun). X is connected to the coil at the far left. When heated by the low-voltage supply it emits electrons by thermionic emission, providing the source of electrons for the beam.
- Y - the anode (accelerating and focusing anodes). Y points to the cylindrical electrodes held at a high positive potential. They accelerate the electrons and focus them into a fine beam directed along the axis of the tube.
- Z - the deflecting plates. Z points to the pair of parallel plates. A potential difference applied across them sets up an electric field that deflects the electron beam, steering the spot on the fluorescent screen.
In summary: X = cathode/filament (electron source), Y = accelerating anode, Z = deflecting plates.
Bayanin Amsa
Identifying the components of the cathode ray tube. Following the electron beam from left (the electron gun) to right (the screen end):
- X - the filament/cathode (heater of the electron gun). X is connected to the coil at the far left. When heated by the low-voltage supply it emits electrons by thermionic emission, providing the source of electrons for the beam.
- Y - the anode (accelerating and focusing anodes). Y points to the cylindrical electrodes held at a high positive potential. They accelerate the electrons and focus them into a fine beam directed along the axis of the tube.
- Z - the deflecting plates. Z points to the pair of parallel plates. A potential difference applied across them sets up an electric field that deflects the electron beam, steering the spot on the fluorescent screen.
In summary: X = cathode/filament (electron source), Y = accelerating anode, Z = deflecting plates.
Tambaya 8 Rahoto
A projectile is fired with a velocity of 20 ms\(^{-1}\) at an angle of 40° to the horizontal. Determine the components of the velocity of the projectile at its maximum height.
The initial velocity components are \(u_x = u\cos\theta\) (horizontal) and \(u_y = u\sin\theta\) (vertical), with \(u = 20\ \text{m s}^{-1}\), \(\theta = 40^\circ\).
At maximum height the vertical component of velocity is momentarily zero (the projectile stops rising). Only the horizontal component remains, and it is unchanged throughout the flight:
\[ v_x = u\cos\theta = 20 \times \cos 40^\circ = 20 \times 0.766 = 15.3\ \text{m s}^{-1}. \] \[ v_y = 0. \]At maximum height: horizontal component \(\approx 15.3\ \text{m s}^{-1}\), vertical component \(= 0\).
Bayanin Amsa
The initial velocity components are \(u_x = u\cos\theta\) (horizontal) and \(u_y = u\sin\theta\) (vertical), with \(u = 20\ \text{m s}^{-1}\), \(\theta = 40^\circ\).
At maximum height the vertical component of velocity is momentarily zero (the projectile stops rising). Only the horizontal component remains, and it is unchanged throughout the flight:
\[ v_x = u\cos\theta = 20 \times \cos 40^\circ = 20 \times 0.766 = 15.3\ \text{m s}^{-1}. \] \[ v_y = 0. \]At maximum height: horizontal component \(\approx 15.3\ \text{m s}^{-1}\), vertical component \(= 0\).
Tambaya 9 Rahoto
(a) Explain the following terms:
(i) mass defect;
(ii) binding energy of a nucleus.
(b)(i) Assuming the wave nature of an electron, what is the effect of decreasing the speed of a photoelectron on its; (\(\alpha\)) wavelength? (\(\beta\)) energy?
(ii) A particle of friasS 4.4 x 10\(^{-23}\) kg moves with a velocity of 10\(^5\)ms\(^{-1}\). Calculate its wavelength. (h = 6.6 x 10\(^{-34}\) Js)
The diagram above shows part of a radioactive decay series. Use it to answer the following questions.
(i) Name a pair of isotopes.
(ii) Name the isotopes with which the series starts.
(iii) Write down a nuclear equation for two ekgmples of each of: (\(\alpha\)) alpha decay; (\(\beta\)) beta decay.
(a)(i) Mass defect. The mass defect is the difference between the sum of the separate masses of all the protons and neutrons (nucleons) in a nucleus and the actual (smaller) measured mass of the nucleus:
\[ \Delta m = \big(Z m_p + (A-Z) m_n\big) - M_{\text{nucleus}} \](a)(ii) Binding energy of a nucleus. It is the energy equivalent of the mass defect (\(E = \Delta m\, c^2\)); that is, the energy released when the free nucleons come together to form the nucleus, or equivalently the energy that must be supplied to separate the nucleus completely into its individual nucleons.
(b)(i) Effect of decreasing the speed of a photoelectron (de Broglie relation \(\lambda = h/mv\), kinetic energy \(E = \tfrac{1}{2}mv^2\)):
- (\(\alpha\)) Wavelength: since \(\lambda = \dfrac{h}{mv}\), decreasing \(v\) increases the wavelength.
- (\(\beta\)) Energy: since \(E = \tfrac{1}{2}mv^2\), decreasing \(v\) decreases the kinetic energy.
(b)(ii) de Broglie wavelength. Data: \(m = 4.4 \times 10^{-23}\ \text{kg}\), \(v = 10^{5}\ \text{m s}^{-1}\), \(h = 6.6 \times 10^{-34}\ \text{J s}\).
\[ \lambda = \frac{h}{mv} = \frac{6.6 \times 10^{-34}}{(4.4 \times 10^{-23})(10^{5})} \]\[ \lambda = \frac{6.6 \times 10^{-34}}{4.4 \times 10^{-18}} = 1.5 \times 10^{-16}\ \text{m} \](c) The decay-series graph. The graph plots nucleon number \(A\) (vertical, 208 to 232) against proton number \(Z\) (horizontal, Pb 82, Bi 83, Po 84, At 85, Rn 86, Fr 87, Ra 88, Ac 89, Th 90). Each down-left diagonal step (\(Z\) falls by 2, \(A\) falls by 4) is an alpha decay; each horizontal step to the right (\(Z\) rises by 1, \(A\) unchanged) is a beta decay. This is the thorium (4n) series running from Th-232 down to Pb-208.
(c)(i) A pair of isotopes. Isotopes have the same proton number but different nucleon numbers. From the graph, examples are the two lead points \(^{208}_{82}\text{Pb}\) and \(^{212}_{82}\text{Pb}\) (both at Z = 82); equally acceptable: \(^{216}_{84}\text{Po}\) and \(^{212}_{84}\text{Po}\) (Z = 84), or \(^{228}_{88}\text{Ra}\) and \(^{224}_{88}\text{Ra}\) (Z = 88), or \(^{232}_{90}\text{Th}\) and \(^{228}_{90}\text{Th}\) (Z = 90).
(c)(ii) Isotope with which the series starts. The top-right point, at Z = 90 and A = 232, is thorium-232, \(^{232}_{90}\text{Th}\).
(c)(iii) Nuclear equations (two examples of each).
(\(\alpha\)) Alpha decay (emits \(^{4}_{2}\text{He}\); Z falls by 2, A by 4):
\[ ^{232}_{90}\text{Th} \rightarrow\ ^{228}_{88}\text{Ra} + ^{4}_{2}\text{He} \]\[ ^{224}_{88}\text{Ra} \rightarrow\ ^{220}_{86}\text{Rn} + ^{4}_{2}\text{He} \](\(\beta\)) Beta decay (emits \(^{0}_{-1}e\); Z rises by 1, A unchanged):
\[ ^{228}_{88}\text{Ra} \rightarrow\ ^{228}_{89}\text{Ac} + ^{0}_{-1}e \]\[ ^{212}_{82}\text{Pb} \rightarrow\ ^{212}_{83}\text{Bi} + ^{0}_{-1}e \]Bayanin Amsa
(a)(i) Mass defect. The mass defect is the difference between the sum of the separate masses of all the protons and neutrons (nucleons) in a nucleus and the actual (smaller) measured mass of the nucleus:
\[ \Delta m = \big(Z m_p + (A-Z) m_n\big) - M_{\text{nucleus}} \](a)(ii) Binding energy of a nucleus. It is the energy equivalent of the mass defect (\(E = \Delta m\, c^2\)); that is, the energy released when the free nucleons come together to form the nucleus, or equivalently the energy that must be supplied to separate the nucleus completely into its individual nucleons.
(b)(i) Effect of decreasing the speed of a photoelectron (de Broglie relation \(\lambda = h/mv\), kinetic energy \(E = \tfrac{1}{2}mv^2\)):
- (\(\alpha\)) Wavelength: since \(\lambda = \dfrac{h}{mv}\), decreasing \(v\) increases the wavelength.
- (\(\beta\)) Energy: since \(E = \tfrac{1}{2}mv^2\), decreasing \(v\) decreases the kinetic energy.
(b)(ii) de Broglie wavelength. Data: \(m = 4.4 \times 10^{-23}\ \text{kg}\), \(v = 10^{5}\ \text{m s}^{-1}\), \(h = 6.6 \times 10^{-34}\ \text{J s}\).
\[ \lambda = \frac{h}{mv} = \frac{6.6 \times 10^{-34}}{(4.4 \times 10^{-23})(10^{5})} \]\[ \lambda = \frac{6.6 \times 10^{-34}}{4.4 \times 10^{-18}} = 1.5 \times 10^{-16}\ \text{m} \](c) The decay-series graph. The graph plots nucleon number \(A\) (vertical, 208 to 232) against proton number \(Z\) (horizontal, Pb 82, Bi 83, Po 84, At 85, Rn 86, Fr 87, Ra 88, Ac 89, Th 90). Each down-left diagonal step (\(Z\) falls by 2, \(A\) falls by 4) is an alpha decay; each horizontal step to the right (\(Z\) rises by 1, \(A\) unchanged) is a beta decay. This is the thorium (4n) series running from Th-232 down to Pb-208.
(c)(i) A pair of isotopes. Isotopes have the same proton number but different nucleon numbers. From the graph, examples are the two lead points \(^{208}_{82}\text{Pb}\) and \(^{212}_{82}\text{Pb}\) (both at Z = 82); equally acceptable: \(^{216}_{84}\text{Po}\) and \(^{212}_{84}\text{Po}\) (Z = 84), or \(^{228}_{88}\text{Ra}\) and \(^{224}_{88}\text{Ra}\) (Z = 88), or \(^{232}_{90}\text{Th}\) and \(^{228}_{90}\text{Th}\) (Z = 90).
(c)(ii) Isotope with which the series starts. The top-right point, at Z = 90 and A = 232, is thorium-232, \(^{232}_{90}\text{Th}\).
(c)(iii) Nuclear equations (two examples of each).
(\(\alpha\)) Alpha decay (emits \(^{4}_{2}\text{He}\); Z falls by 2, A by 4):
\[ ^{232}_{90}\text{Th} \rightarrow\ ^{228}_{88}\text{Ra} + ^{4}_{2}\text{He} \]\[ ^{224}_{88}\text{Ra} \rightarrow\ ^{220}_{86}\text{Rn} + ^{4}_{2}\text{He} \](\(\beta\)) Beta decay (emits \(^{0}_{-1}e\); Z rises by 1, A unchanged):
\[ ^{228}_{88}\text{Ra} \rightarrow\ ^{228}_{89}\text{Ac} + ^{0}_{-1}e \]\[ ^{212}_{82}\text{Pb} \rightarrow\ ^{212}_{83}\text{Bi} + ^{0}_{-1}e \]Tambaya 10 Rahoto
State the dimension of; (a) impulse; (ii) acceleration; (iii) work
(a) Impulse. Impulse \(=\) force \(\times\) time \(=\) change in momentum. Its dimension is that of momentum:
\[ [\text{impulse}] = MLT^{-1}. \](ii) Acceleration. Acceleration \(=\) velocity/time \(= \dfrac{LT^{-1}}{T}\):
\[ [\text{acceleration}] = LT^{-2}. \](iii) Work. Work \(=\) force \(\times\) distance \(= (MLT^{-2}) \times L\):
\[ [\text{work}] = ML^{2}T^{-2}. \]Bayanin Amsa
(a) Impulse. Impulse \(=\) force \(\times\) time \(=\) change in momentum. Its dimension is that of momentum:
\[ [\text{impulse}] = MLT^{-1}. \](ii) Acceleration. Acceleration \(=\) velocity/time \(= \dfrac{LT^{-1}}{T}\):
\[ [\text{acceleration}] = LT^{-2}. \](iii) Work. Work \(=\) force \(\times\) distance \(= (MLT^{-2}) \times L\):
\[ [\text{work}] = ML^{2}T^{-2}. \]Tambaya 11 Rahoto
(a) What is doping?
(b) Explain how doping improves the conductivity of a semiconductor
(a) Doping is the deliberate addition of a small, controlled amount of a suitable impurity (a pentavalent or trivalent element) to a pure (intrinsic) semiconductor in order to increase the number of free charge carriers and hence its electrical conductivity.
(b) How doping improves conductivity:
A pure semiconductor such as silicon has very few free charge carriers, so it conducts poorly.
- Adding a pentavalent impurity (e.g. phosphorus, arsenic) donates extra free electrons to the crystal, producing an n-type semiconductor in which electrons are the majority carriers.
- Adding a trivalent impurity (e.g. boron, aluminium) creates vacancies called holes that act as positive carriers, producing a p-type semiconductor in which holes are the majority carriers.
In both cases the impurity greatly increases the concentration of mobile charge carriers, so more charge can flow for a given voltage and the conductivity rises sharply.
Bayanin Amsa
(a) Doping is the deliberate addition of a small, controlled amount of a suitable impurity (a pentavalent or trivalent element) to a pure (intrinsic) semiconductor in order to increase the number of free charge carriers and hence its electrical conductivity.
(b) How doping improves conductivity:
A pure semiconductor such as silicon has very few free charge carriers, so it conducts poorly.
- Adding a pentavalent impurity (e.g. phosphorus, arsenic) donates extra free electrons to the crystal, producing an n-type semiconductor in which electrons are the majority carriers.
- Adding a trivalent impurity (e.g. boron, aluminium) creates vacancies called holes that act as positive carriers, producing a p-type semiconductor in which holes are the majority carriers.
In both cases the impurity greatly increases the concentration of mobile charge carriers, so more charge can flow for a given voltage and the conductivity rises sharply.
Tambaya 12 Rahoto
An electron enters perpendicularly into a uniform magnetic field which has a flux density of 0.12 T This results in a magnetic force of 9.6 x 10\(^{-2}\) N on the electron. Calculate the speed of the electron as it enters the magnetic field. (e =1.6 x \(10^{19}\) C)
An electron moving perpendicular to a uniform magnetic field experiences a force
\[F=Bev\sin\theta,\qquad \theta=90^\circ,\ \sin 90^\circ=1.\]Making the speed the subject:
\[v=\frac{F}{Be}.\]Substituting \(F=9.6\times10^{-12}\text{ N}\), \(B=0.12\text{ T}\) and \(e=1.6\times10^{-19}\text{ C}\):
\[v=\frac{9.6\times10^{-12}}{0.12\times1.6\times10^{-19}}=\frac{9.6\times10^{-12}}{1.92\times10^{-20}}=5.0\times10^{8}\text{ m s}^{-1}.\]The speed of the electron as it enters the field is \(5.0\times10^{8}\text{ m s}^{-1}\).
Bayanin Amsa
An electron moving perpendicular to a uniform magnetic field experiences a force
\[F=Bev\sin\theta,\qquad \theta=90^\circ,\ \sin 90^\circ=1.\]Making the speed the subject:
\[v=\frac{F}{Be}.\]Substituting \(F=9.6\times10^{-12}\text{ N}\), \(B=0.12\text{ T}\) and \(e=1.6\times10^{-19}\text{ C}\):
\[v=\frac{9.6\times10^{-12}}{0.12\times1.6\times10^{-19}}=\frac{9.6\times10^{-12}}{1.92\times10^{-20}}=5.0\times10^{8}\text{ m s}^{-1}.\]The speed of the electron as it enters the field is \(5.0\times10^{8}\text{ m s}^{-1}\).
