Answer any ten of the following questions (2 marks each). All questions solved below.
What is the unit of electric flux? When do we consider electric flux to be positive and when do we consider it to be negative?
Unit: The SI unit of electric flux is Volt-meter (V-m) or Newton-meter² per Coulomb (N-m²/C).
Sign Convention:
State uniqueness theorem.
The Uniqueness Theorem states that if a solution to Laplace's equation satisfies a given set of boundary conditions, then that solution is the unique (only) possible solution.
Write down the conditions of stable and unstable equilibrium for a dipole placed in a uniform electric field.
Define capacitance. On what factors does capacitance depend?
Definition: Capacitance is the ability of a system to store electric charge per unit potential difference. C = Q/V.
Factors:
Define dielectric constant of a material. Give two examples of any dielectric material.
Definition: The dielectric constant (K) is the ratio of the permittivity of the material to the permittivity of free space. K = epsilon / epsilon_0.
Examples: Glass, Mica, or Pure Water.
Define electric displacement vector D. What types of charges are associated with D?
Definition: The electric displacement vector D accounts for the effects of free and bound charges in a medium, defined as D = (epsilon_0 * E) + P.
Associated Charges: D is exclusively associated with free charges.
What is a magnetic dipole? Define dipole moment.
Magnetic Dipole: A pair of magnetic poles of equal strength and opposite sign separated by a small distance (e.g., a bar magnet or current loop).
Dipole Moment: The product of pole strength and the distance between poles (m = q_m * l), or for a current loop, m = I * A.
What is a toroid? What are toroids used for?
Toroid: A solenoid bent into the shape of a hollow doughnut or ring.
Uses: To create a uniform magnetic field in a confined space, used in particle accelerators and transformers.
Write the expression of torque acting on a current loop placed in a uniform magnetic field. Name all the parameters involved.
Torque (tau) = m x B = NIAB sin(theta)
Parameters: N = number of turns, I = current, A = area of loop, B = magnetic field, theta = angle between area vector and magnetic field.
What is thermoelectricity? Mention applications of Seebeck effect.
Thermoelectricity: The direct conversion of temperature differences to electric voltage and vice versa.
Applications: Thermocouples for temperature measurement, thermoelectric generators (TEGs).
State the law of intermediate temperature.
The thermo-emf produced in a thermocouple with junctions at temperatures T1 and T3 is the algebraic sum of the thermo-emfs produced when the junctions are at T1 and T2, and T2 and T3 respectively. (E13 = E12 + E23).
Write two differences between Peltier effect and Thomson effect.
| Peltier Effect | Thomson Effect |
|---|---|
| Occurs at the junctions of two dissimilar metals. | Occurs along the length of a single conductor with a temperature gradient. |
| Surface phenomenon. | Bulk/Volume phenomenon. |
Answer any five questions (10 marks each). All questions solved below.
(a) For a uniformly charged sphere of charge density rho, find the expression of electric field intensity at a point outside and inside the sphere. Draw the graph.
Solution: By Gauss Law, flux = Q_encl / epsilon_0.
(b) E = 6xy i + (3x² - 3y²) j. Find divergence and curl. Is it a possible electrostatic field?
(a) Show that electric field is the negative gradient of potential. Meaning of -ve sign?
Proof: Work done dW = -F.dl = -qE.dl. Also dW = q.dV. Thus q.dV = -qE.dl, which leads to E = -dV/dl. In vector form, E = -grad V.
Negative Sign: It indicates that the electric field points in the direction of decreasing electric potential.
(b) If V = 2x + 3y - z, find electric field strength.
E = -(i * dV/dx + j * dV/dy + k * dV/dz)
dV/dx = 2; dV/dy = 3; dV/dz = -1.
E = -2i - 3j + k
(a) Explain the principle of working of a capacitor.
A capacitor works on the principle that the capacitance of an insulated conductor is increased significantly when an earthed conductor is placed near it, as it reduces the potential of the first conductor for the same amount of charge.
(b) What is electrical potential energy? Derive expression for uniformly charged non-conducting sphere.
Definition: The work done in assembling a system of charges from infinity to their current positions.
Derivation: U = Integral of (1/2 * epsilon_0 * E² dV). For a sphere of radius R and charge Q:
U = (3/5) * (Q² / 4 * pi * epsilon_0 * R)
(a) Deduce D = epsilon_0*E + P. Draw field lines for D, E, and P.
Applying Gauss law to a dielectric medium: div(E) = (rho_free + rho_bound) / epsilon_0. Since rho_bound = -div(P), we get div(epsilon_0*E + P) = rho_free. Letting D = epsilon_0*E + P, we get div(D) = rho_free.
(b) Deduce Gauss' law in dielectrics.
The surface integral of the displacement vector D over a closed surface is equal to the net free charge enclosed by the surface. Integral(D.dS) = Q_free.
(a) State Biot-Savart law. Write in vector form.
dB = (mu_0 / 4*pi) * (I * dl x r) / r³
(b) Expression for magnetic field of a long straight wire. Does it encircle the conductor?
Using Ampere's Law: Integral(B.dl) = mu_0 * I. For a circular path B(2*pi*r) = mu_0 * I.
B = (mu_0 * I) / (2 * pi * r)
Yes, the magnetic field lines form concentric circles encircling the conductor.
(a) State and prove Ampere's circuital law.
The line integral of the magnetic field B around any closed loop is equal to mu_0 times the net current passing through the loop. Proof uses a circular path around a straight wire.
(b) What is Helmholtz coil? Show field rate of change is constant midway.
A pair of identical coaxial coils separated by a distance equal to their radius, used to produce a very uniform magnetic field.
At the midpoint, the second derivative of B with respect to x is zero, meaning dB/dx is constant.(a) Variation of thermo-emf with temperature. Define inversion and neutral temperature.
Thermo-emf (e) varies parabolically: e = aT + bT².
(b) Thermoelectric power and diagram.
Thermoelectric power (S) is the rate of change of thermo-emf with temperature, S = de/dT.
The diagram is a plot of S vs T, typically a straight line for most thermocouples.(a) Experimental determination of Peltier coefficient.
Measured using a calorimeter by passing a known current through a junction and measuring the heat evolved or absorbed.
(b) Apply thermodynamics to thermocouple to show pi = T * (de/dT).
By Second Law of Thermodynamics, dS = 0 for a reversible cycle. This leads to the relation involving Peltier coefficient (pi) and thermo-emf (e).
(a) Average power in series LCR circuit.
P_avg = V_rms * I_rms * cos(phi)
where cos(phi) is the power factor.
(b) Resonant frequency. Effect on current if R is replaced by 2R?
f_r = 1 / (2 * pi * sqrt(L * C))
At resonance, current I = V/R. If R becomes 2R, the resonant current will be halved.
(a) State and prove Thevenin's theorem.
Any linear bilateral network with voltage sources and resistances can be replaced by a single voltage source (Vth) in series with a single resistance (Rth).
(b) Calculate current in 6-ohm resistor using Thevenin's.