Problem 1:

Consider the LCR circuit shown. The circuit is driven by a sinusoidal voltage V(t) = V₀exp(iωt), where ω = ω₀ = 1/√(LC) is the resonant frequency of the circuit.
(a)  Solve for the current I(t) in the circuit.
(b)  Find the voltage V_AB(t) where V_AB = V_A - V_B.
 

Problem 2:

(a)  Derive the expression and calculate the capacitance per unit length for a piece of coaxial cable in SI units.  The coaxial cable has a diameter of an inner wire of 1 mm and an inner diameter (ID) of an outer shield of 5 mm. The dielectric constant of the insulator between the two conductors is κ_e = 1.5.  The dielectric permeability of the free space is  ε₀ = 8.8*10^-12 F/m.
(b)  Derive the expression and calculate the inductance for this cable in SI units, assuming that the relative magnetic permeability is unity κ_m  = 1.  The magnetic permeability of the free space is
μ₀ = 4π*10^-7 H/m.  For problems (a) and (b) assume that the skin depth of the wire and shield are zero (a high-frequency limit).
(c)  Derive the expression and calculate the wave impedance for this cable.

Problem 3:

In the circuit shown, the switch has been in position a for a very long time.  At t = 0, it abruptly switches to position b.  Find the expression for the capacitor voltage V_C as a function of time.

Let V = 50 V, R₁ = 500 Ω, R₂ = 125 Ω,  R₃ = 25 Ω, C = 2*10^-6 F, I₁ = 0.1 A, and I₂ = 0.2 A.

Problem 4:

In the circuit shown below, two conducting spheres of radius R each are located far away from each other and are connected by thin wires through a solenoid with inductance L.  Initially, one of the spheres is charged and the other is neutral.  How long after the switch is closed do the charges on the spheres become equal?