QUESTIONS
SECTION A
1. Assume an object of mass 2.22g is placed 7.5mm in front of a large plane mirror. Find the image position and the nature of the image formed.
2. Briefly explain the term "power of lens". Two lenses of power +7.3D and -9.6D are placed in contact with each other. What is the focal length of the combination? Which type of lens is finally formed?
3. A ray of light strikes normally one of the faces of a Pyrex triangular glass prism of refracting angle 30° and refractive index 1.41. Determine the angle of deviation and the angle of refraction.
4. (a) Write down the expression for an electrostatic potential energy in vacuum for three charges situated at the corners of an oblique triangle as shown in Figure 1.
(b) If q₁=1μC, r₁=1mm, q₂=2.1μC, r₂=0.5mm and q₃=1μC, r₃=0.043mm. Find electrostatic potential energy in vacuum.
5. A wire has resistance R and resistivity ρ. What will be its resistance if (i) its length is doubled (ii) its diameter is halved?
6. If ε₀ and μ₀ are the absolute permittivity and permeability of free space, evaluate the expression Ω = 2/√(ε₀μ₀). State clearly its unit.
7. Two wires of the same length and of the same materials have different diameters, d₁ and d₂ (d₁ < d₂). Determine which wire has greater (i) conductivity? (ii) Conductance?
8. a) (i) State the fundamental laws of conservation of electric charges. (ii) What will happen to a neutral body when it gains electrons? (b) Write down the Gauss' law (without proof) for a closed surface enclosing a net charge q, the net electric flux Φ.
9. An electromagnetic wave radiation of wavelength 1500 Å is incident on a metal surface. If the stopping potential is 4.4 volts. Calculate the maximum kinetic energy of the emitted electrons and the work function of the metal in eV.
10. What is Thermonuclear energy? A heavy unstable nucleus is bombarded with a neutron and breaks into two smaller parts which resulted in the release of 210 MeV and three neutrons are emitted. The reaction process is:
²³⁵U + ¹n → ¹⁴⁴Ba + ⁸⁹Kr + 3(¹n) + 210MeV. Explain this process, whether it is a fission or fusion process.
SECTION B — GEOMETRIC OPTICS (Answer only ONE question)
11a. (i) What do you understand by the term "reflection of light wave"? (ii) With the aid of a ray diagram, explain the formation of image by a plane mirror. If light is incident on a triangular glass prism at an angle of 60°, and the reflected and refracted waves are mutually perpendicular, sketch the ray diagram and calculate the angle of the prism.
(b) Assume the angle of minimum deviation for a prism of refractive index 1.75 is equal to the angle of prism. Calculate the angle of prism, given that sin A = 2sin(A/2)cos(A/2).
11b(ii). Consider an astronomical telescope with an objective of focal length 108cm and eyepiece of focal length 7cm. If the telescope is in its normal adjustment, calculate the angular magnification (magnifying power), the distance of the eye-ring from the objective if the diameter of the eye-ring is 50mm.
12a. Study the diagrams in Figure 2 and 3 as a guide to answer the following questions: (i) Recopy Figure 2 and identify points A, B, C and D. From Figure 3, copy the ray diagram and trace the ray to form either the image or object with reasons. Also discuss clearly their nature. Assume radius of curvature of the mirror in Figure 3 is 10cm. If the image is 20cm from the mirror, find the nature of the image.
12b. (i) Assume a ray of light travels from one face of a triangular glass slab of thickness 12.73cm and refractive index 1.66. If the angle of incidence is 44°, find the angle of refraction and the lateral displacement of the emergent ray. (ii) If an object of mass 3.21kg is placed 14.73cm in front of a very long plane mirror. Find the nature of the image formed and the magnification produced.
SECTION C — ELECTRICITY AND MAGNETISM (Answer only TWO questions)
13a. (i) Distinguish between the terms: (A) Electric charge and electric current, state clearly their mathematical relationships. (B) Charging by induction and charging by conduction. (ii) If the total charge entering a terminal is given by q=3.7t³+0.09t. Find the current at t=2.3 seconds. (iii) Consider two positively point charges separated at 100cm apart repelling each other with a force of 18N. If the sum of charges is 1.9μC. Calculate their separate charges. The following identity may be useful: (a-b)²=(a+b)²-4ab.
13b. (i) Briefly explain the terms inductor and inductance of inductor. (ii) Four inductors each of inductance 2mH are connected in (A) series (B) parallel across a 10A a.c supply. In each case, draw the circuit diagram and calculate the equivalent inductance and the energy stored in the circuit. (iii) Calculate the e.m.f induced in a coil of inductance 12H by a current changing at the rate of 2.7 As⁻¹.
14a(i). Briefly explain the following terms: magnetic flux and magnetic flux density, Electromotive force (emf) and Magnetomotive force (mmf). (ii) If a conductor moves with a velocity of 15ms⁻¹ at 30° to a magnetic field produced between two square-faced poles of side length 2cm. If the flux leaving a pole face is 5μWb, find the magnitude of the induced e.m.f.
14b(i). Distinguish between reluctance and resistance. (ii) The following Table 1 gives comparisons between Electrical and Magnetic circuit. Find the values of w, x, y, z in the table. (iii) Use Figure 4 to find the equivalent conductance and current in the circuit.
Table 1
| S/N | Electrical circuit | Magnetic circuit |
|---|---|---|
| 1 | Emf | Mmf |
| 2 | Current | X |
| 3 | resistance | Y |
| 4 | I=E/R | Z |
| 5 | W | S=l/(μ₀μᵣA) |
15a(i). State the factors upon which the resistivity of a given material depends. (ii) Given two wires of the same material and length have resistances 5Ω and 3Ω respectively. Find the ratio of radii of the two wires. (iii) A small concrete needle of diameter 2mm and of length 100m has a resistance of 0.5475Ω at 20°C and 0.805Ω at 150°C. Find the values of temperature coefficient of resistance, its resistance at 0°C and its resistivity at 0°C and 20°C.
15b(i). Briefly explain the following terms (how are they connected in a given circuit): galvanometer, voltmeter, ammeter. (ii) What is the conductance and current through a bulb rated at 100W, 250V d.c supply? Calculate the power if it is connected to a 200V line.
16a(i). With the help of a circuit diagram, briefly explain the concept of electromagnetic induction. Hence, state the laws of electromagnetic induction in words (both Faraday's and Lenz's laws). (ii) Calculate the induced e.m.f in a coil when there is a change of flux of 0.025 Wb linking with it in 0.50 seconds.
16b. A coil of resistance 5 ohm and inductance 0.120H in series with a 100μF capacitor, is connected to a 300V, 50Hz a.c supply. Calculate: (i) the impedance and current flowing, whether the circuit is inductive or capacitive. (ii) the phase difference between the supply voltage and current. (iii) The resonant frequency, and state your reason.
SECTION D — MODERN PHYSICS (Answer only ONE question)
17a(i). State and briefly discuss the four methods in which electrons can be ejected out of a metal surface. (ii) The equation for the energy of the nth state (Eₙ) of the electron in the hydrogen atom can be expressed (in eV) as: Eₙ = -me⁴/(8ε₀²h²n²). By substituting the values of the constants, find the value of Eₙ (in eV) in terms of n. What is the name given to n in this equation? Hence, find the value of E₁-E∞.
17a(ii cont). From a sodium surface, light of wavelength 3125 Å and 3650 Å causes emission of electrons whose maximum kinetic energy is 2.128 eV and 1.595 eV, respectively. Use this information to determine Planck's constant and the work function of sodium. (iii) Light of wavelength 2000 Å falls on a metallic surface. If the work function of the surface is 4.2 eV. What is the kinetic energy of the fastest photoelectrons emitted? Also calculate the stopping potential and the threshold wavelength for the metal.
17b(i). State the Heisenberg's uncertainty principle in words and write down its mathematical form for each of the following conditions: Angular momentum (L) and polar angle (φ). Energy (E) and time (t). Position (x) and linear momentum (p).
17b(ii). Calculate the de Broglie wavelength of an electron having a kinetic energy of 1000 eV. Also find the wavelength (in Angstrom) of x-rays having the same energy. Hence evaluate the ratio λ for x-rays / λ for de Broglie of electron.
18a(i). What is meant by the terms Natural and artificial radioactivity? (ii) Explain the following: alpha decay, beta decay and gamma decay in radioactivity. Support your answer with reaction, assuming an atom ᴬZX. (iii) Prove the radioactive decay law. Assume N is the number of radioactive nuclei present at any time t. (iv) Complete the following nuclear reactions:
(A) ⁷₃Li + ? → ⁷₄Be + ⁰₋₁n
(B) ³⁵₁₇Cl + ? → ³²₁₆S + ⁴₂He
(C) ⁹₄Be + α-particle → 3(α-particle) + ?
(D) ⁷⁹₃₅Br + ²₁H → ? + 2(¹₀n)
18b(i). Find the binding energy and binding energy per nucleon for a lithium nucleus (⁷₃Li). Given that: mass of ⁷₃Li = 7.000000u, mass of proton = 1.007825u and mass of neutron = 1.008665u, 1u = 931.5MeV.
(ii) Consider the fusion process ²₁H+²₁H → ³₂He+n+Q. Given that the mass of deuteron, helium and neutron are respectively 2.015u, 3.017u and 1.009u. Determine the total energy released (Q). 1u=931.5MeV.
ANSWERS
Section A
1. For a plane mirror, image distance = object distance. The image is formed 7.5mm behind the mirror, and is virtual, erect, laterally inverted, and of the same size as the object (magnification = 1). The object's mass has no bearing on the optical image formed.
2. Power of a lens is the reciprocal of its focal length (in metres), P=1/f, measured in Dioptres (D); it indicates the degree of convergence or divergence a lens produces.
Combined power: P = P₁+P₂ = 7.3+(-9.6) = -2.3D
Focal length: f = 1/P = 1/(-2.3) = -0.435m (-43.5cm)
Since the resultant power is negative, the combination behaves as a diverging (concave) lens.
3. Since the ray enters normally (i=0°) at the first face, it undergoes no bending, and strikes the second face at an angle equal to the prism angle, A=30°.
Applying Snell's law at the exit face: n sinA = sin r
1.41 × sin30° = sin r
sin r = 0.705 → r ≈ 44.8°
Angle of deviation: D = r - A = 44.8° - 30° = 14.8°
4a. Electrostatic potential energy of a system of three point charges:
U = k[(q₁q₂)/r₁₂ + (q₁q₃)/r₁₃ + (q₂q₃)/r₂₃], where k = 1/(4πε₀) = 9×10⁹ Nm²C⁻²
4b. Taking r₁ = r₂₃ = 1mm, r₂ = r₁₃ = 0.5mm, r₃ = r₁₂ = 0.043mm:
U = 9×10⁹[(1×2.1×10⁻¹²)/(4.3×10⁻⁵) + (1×1×10⁻¹²)/(0.5×10⁻³) + (2.1×1×10⁻¹²)/(1×10⁻³)]
U = 9×10⁹[4.884×10⁻⁸ + 2×10⁻⁹ + 2.1×10⁻⁹]
U = 9×10⁹ × 5.294×10⁻⁸
U ≈ 476.5 J
5. R = ρL/A
(i) If length doubled (L→2L): R' = ρ(2L)/A = 2R (resistance doubles)
(ii) If diameter halved (A→A/4): R' = ρL/(A/4) = 4R (resistance becomes four times)
6. Since 1/√(ε₀μ₀) = c (speed of light):
Ω = 2/√(ε₀μ₀) = 2c = 2×3×10⁸ = 6×10⁸ m/s
(Unit is metres per second, m/s — a velocity, being twice the speed of light)
7. Conductivity is an intrinsic material property; since both wires are of the same material, they have equal conductivity.
Conductance G = σA/L. Since d₂>d₁ gives a larger cross-sectional area, the wire with diameter d₂ has greater conductance.
8a. (i) The law of conservation of electric charge states that the total electric charge of an isolated system remains constant — charge can neither be created nor destroyed, only transferred between bodies.
(ii) A neutral body that gains electrons becomes negatively charged.
(b) Gauss's law: Φ = ∮E·dA = q/ε₀ (total electric flux through a closed surface equals the enclosed charge divided by ε₀)
9. Photon energy: E = hc/λ = (6.63×10⁻³⁴×3×10⁸)/(1500×10⁻¹⁰) = 1.326×10⁻¹⁸J = 8.29 eV
Maximum KE = eV_stop = 4.4 eV
Work function W = E - KEmax = 8.29 - 4.4 = 3.89 eV
10. Thermonuclear energy is the energy released from nuclear fusion reactions, where light nuclei combine at extremely high temperatures to form heavier nuclei, releasing energy due to mass defect (via E=mc²), as occurs in stars.
The given reaction (²³⁵U splitting into ¹⁴⁴Ba and ⁸⁹Kr upon neutron bombardment) is a fission process — a heavy nucleus splits into two lighter nuclei, releasing neutrons and energy; this is the opposite of thermonuclear (fusion) energy.
Section B
11a. (i) Reflection of a light wave is the bouncing back of light when it strikes a surface, obeying the laws of reflection (angle of incidence = angle of reflection, both rays and the normal lying in the same plane).
(ii) In a plane mirror, two divergent rays from an object point strike the mirror and reflect according to the law of reflection; the reflected rays diverge but appear to originate from a point behind the mirror, forming a virtual, erect, laterally inverted image of equal size, at a distance behind the mirror equal to the object distance in front.
For light at 60° with reflected and refracted rays mutually perpendicular: angle of refraction r = 90°-60° = 30°.
Refractive index n = sin i/sin r = sin60°/sin30° = 0.866/0.5 = 1.73
11b. At minimum deviation with Dm=A: n = 2cos(A/2)
1.75 = 2cos(A/2)
cos(A/2) = 0.875
A/2 = 28.96°
A ≈ 58°
11b(ii). Angular magnification M = f₀/fₑ = 108/7 = 15.4
Tube length in normal adjustment L = f₀+fₑ = 108+7 = 115cm
Distance of exit pupil (eye-ring) from eyepiece: using 1/v = 1/fₑ+1/u, with u=-115cm:
1/v = 1/7 - 1/115 = 0.1342 → v ≈ 7.45cm
Distance of eye-ring from objective ≈ 115 - 7.45 = 107.5cm
12a. [Ray diagram: converging rays from an object refracted by a lens converge to form a real, inverted image, or diverge (traced backward) to form a virtual, erect image, depending on object position relative to the focal point.]
For the mirror (R=10cm, f=R/2=5cm), given v=20cm (real image side):
1/f = 1/v + 1/u
1/5 = 1/20 + 1/u → 1/u = 3/20 → u = 6.67cm
Magnification m = -v/u = -20/6.67 = -3 (image is real, inverted, magnified 3×)
12b. (i) sin44° = 1.66 sin r
sin r = 0.6947/1.66 = 0.4186 → r ≈ 24.8°
Lateral displacement: d = t·sin(i-r)/cos r = 12.73×sin(19.25°)/cos(24.75°) = 12.73×0.330/0.908
d ≈ 4.62 cm
(ii) Plane mirror: image is virtual, erect, laterally inverted, formed 14.73cm behind the mirror, with magnification = 1 (mass is irrelevant to the optics).
Section C
13a(i)(A). Electric charge (Q) is a fundamental scalar property of matter causing it to experience electromagnetic force, measured in coulombs. Electric current (I) is the rate of flow of charge, I = dQ/dt, measured in amperes.
(B) Charging by conduction occurs through direct contact, transferring charge of the same sign to both bodies. Charging by induction occurs without contact — a charged body induces charge separation in a nearby neutral conductor, which, when grounded and the ground removed while the charging body remains nearby, acquires a net charge of opposite sign.
(ii) i = dq/dt = 11.1t² + 0.09
At t=2.3s: i = 11.1(2.3)²+0.09 = 11.1×5.29+0.09 = 58.81 A
(iii) F=kq₁q₂/r² → q₁q₂ = Fr²/k = 18(1)²/9×10⁹ = 2×10⁻⁹ C²
Using q₁+q₂=1.9×10⁻⁶ and q₁q₂=2×10⁻⁹:
(q₁-q₂)² = (q₁+q₂)² - 4q₁q₂ = (1.9×10⁻⁶)² - 4(2×10⁻⁹) = 3.61×10⁻¹² - 8×10⁻⁹
This discriminant is negative, so no real solution exists with the given values — the given sum and force are numerically inconsistent (check the given data: for F=18N at r=1m, the product q₁q₂=2×10⁻⁹C² requires each charge to be of order tens of μC, not consistent with a 1.9μC sum).
13b. (i) An inductor is a circuit component (typically a coil) that stores energy in a magnetic field when current flows through it. Inductance (L) is the property quantifying the induced emf per unit rate of change of current: EMF = -L(dI/dt), measured in Henries (H).
(ii) Series: L = 4×2mH = 8mH; Energy = ½LI² = ½×8×10⁻³×10² = 0.4 J
Parallel: 1/L = 4/2mH → L = 0.5mH; Energy = ½×0.5×10⁻³×10² = 0.025 J
(iii) EMF = L(dI/dt) = 12×2.7 = 32.4 V
14a(i). Magnetic flux (Φ): total magnetic field lines through an area, Φ=BAcosθ, in Weber.
Magnetic flux density (B): flux per unit area, B=Φ/A, in Tesla.
EMF: energy per unit charge driving current in a circuit, in volts.
MMF: driving force establishing magnetic flux in a magnetic circuit, MMF=NI, in ampere-turns.
(ii) B = Φ/A = 5×10⁻⁶/(0.02×0.02) = 0.0125T
EMF = BLv sinθ = 0.0125×0.02×15×sin30° = 1.875×10⁻³ V (1.875 mV)
14b(i). Reluctance is the opposition of a magnetic circuit to magnetic flux (S=l/μ₀μᵣA, in At/Wb) and does not dissipate energy. Resistance is the opposition of an electric circuit to current flow (R=ρl/A, in Ω) and dissipates energy as heat.
(ii) X = Flux (Φ); Y = Reluctance (S); Z = Φ = F/S (Mmf/Reluctance); W = R = ρl/A
(iii) Combining the 6Ω, 6Ω and 4Ω branches in parallel:
1/Rp = 1/6+1/6+1/4 = 7/12 → Rp = 12/7 ≈ 1.71Ω
Total resistance = (2+3+2) + 1.71 = 8.71Ω
Current I = V/R = 15/8.71 ≈ 1.72 A
15a(i). Resistivity depends on: (1) nature/type of the material, (2) temperature, (3) impurities present, (4) mechanical stress on the material. It does NOT depend on the length or cross-sectional area.
(ii) R∝1/r² (same material, length): r₁/r₂ = √(R₂/R₁) = √(3/5) = 0.775 (i.e., r₁:r₂ ≈ 0.775:1)
(iii) R₀(1+20α)=0.5475; R₀(1+150α)=0.805
Dividing: (1+150α)/(1+20α)=1.4703 → α ≈ 0.0039 /°C
R₀ = 0.5475/(1+0.078) = 0.508Ω
A = π(0.001)² = 3.1416×10⁻⁶m²
ρ₀ = R₀A/L = 0.508×3.1416×10⁻⁶/100 = 1.60×10⁻⁸ Ω·m
ρ₂₀ = R₂₀A/L = 0.5475×3.1416×10⁻⁶/100 = 1.72×10⁻⁸ Ω·m
15b(i). Galvanometer: detects small currents, connected in series (often with a shunt/series resistor to convert to ammeter/voltmeter). Voltmeter: measures voltage, connected in parallel, high resistance. Ammeter: measures current, connected in series, low resistance.
(ii) R = V²/P = 250²/100 = 625Ω
G = 1/R = 1.6×10⁻³ S
I = P/V = 100/250 = 0.4A
At 200V: P' = V'²/R = 200²/625 = 64 W
16a(i). Electromagnetic induction is the production of an emf in a conductor due to a change in magnetic flux linkage.
Faraday's Law: The induced emf is directly proportional to the rate of change of magnetic flux linkage.
Lenz's Law: The direction of the induced emf/current is such as to oppose the change in flux that produced it.
(ii) EMF = dΦ/dt = 0.025/0.50 = 0.05 V
16b. Xl = 2πfL = 2π×50×0.120 = 37.70Ω
Xc = 1/(2πfC) = 1/(2π×50×100×10⁻⁶) = 31.83Ω
Z = √(5²+(37.70-31.83)²) = √(25+34.46) = 7.71Ω
Since Xl > Xc, the circuit is inductive.
I = V/Z = 300/7.71 = 38.9 A
φ = arctan((Xl-Xc)/R) = arctan(5.87/5) = 49.6° (current lags voltage)
f₀ = 1/(2π√(LC)) = 1/(2π√(1.2×10⁻⁵)) = 45.9 Hz (frequency at which Xl=Xc, i.e., reactance cancels, giving minimum impedance)
Section D
17a(i). Four methods of electron emission:
- Thermionic emission — electrons emitted when a metal is heated.
- Photoelectric emission — electrons emitted when light of sufficient frequency strikes the surface.
- Field emission — electrons pulled out by a strong external electric field.
- Secondary emission — electrons emitted when the surface is bombarded by high-energy particles.
(ii) Substituting constants into Eₙ = -me⁴/(8ε₀²h²n²) gives: Eₙ = -13.6/n² eV
"n" is called the principal quantum number.
E₁-E∞ = -13.6/1² - 0 = -13.6 eV
Sodium surface:
f₁ = c/λ₁ = 3×10⁸/3125×10⁻¹⁰ = 9.6×10¹⁴Hz
f₂ = c/λ₂ = 3×10⁸/3650×10⁻¹⁰ = 8.219×10¹⁴Hz
h = (KE₁-KE₂)/(f₁-f₂) = (0.533×1.6×10⁻¹⁹)/(1.381×10¹⁴) = 6.18×10⁻³⁴ Js
W = hf₁-KE₁ = 5.93×10⁻¹⁹-3.405×10⁻¹⁹ = 2.52×10⁻¹⁹J = 1.58 eV
(iii) E_photon = hc/λ = 9.945×10⁻¹⁹J = 6.22 eV
KEmax = 6.22-4.2 = 2.02 eV