Electromagnetic Induction

Consider a circuit consisting of a capacitor of capacitance C and a coil with N turns per unit length, cross sectional area S and length d, where d2 S. There is another coil of length d/2, cross sectional area S/2 and 2N turns per unit length completely inside the larger coil, as shown in the figure. The ends of this smaller coil are connected with each other by an insulated conducting wire. The self-inductance of the larger coil is L. Neglecting edge effects and all the Ohmic resistances, the resonant frequency of the circuit is: [figure]

1. 4 15 LC
2. 6 5 LC
3. 2 3 LC
4. 23 LC

A conducting square loop initially lies in the XZ plane with its lower edge hinged along the X-axis. Only in the region y 0, there is a time dependent magnetic field pointing along the Z-direction, B(t)=B0( t) k, where B0 is a constant. The magnetic field is zero everywhere else. At time t=0, the loop starts rotating with constant angular speed about the X axis in the clockwise direction as viewed from the +X axis (as shown in the figure). Ignoring self-inductance of the loop and gravity, which of the following plots correctly represents the induced e.m.f. (V) in the loop as a function of time: [figure]

1. [figure]
2. [figure]
3. [figure]
4. [figure]

A region in the form of an equilateral triangle (in x-y plane) of height L has a uniform magnetic field B pointing in the +z-direction. A conducting loop PQR, in the form of an equilateral triangle of the same height L, is placed in the x-y plane with its vertex P at x = 0 in the orientation shown in the figure. At t = 0, the loop starts entering the region of the magnetic field with a uniform velocity v along the +x-direction. The plane of the loop and its orientation remain unchanged throughout its motion. [figure] Which of the following graph best depicts the variation of the induced emf (E) in the loop as a function of the distance (x) starting from x = 0?

1. [figure]
2. [figure]
3. [figure]
4. [figure]