Question

) An inductor is formed hy winding V tuns of a thin conducting wire into a circular f radius a. The inductor toop is in the plane with its center at the origin and connected to a resistor R. In the presence of a loop as shown below around a non-magnetic material r- y magnetio field B ธิ 2094 + Z)sin(at) wb/m.. Teslas. where ω-angular frequency Derive and calculate the following parameters (a) Derive the magnetic flux linking a single turn of the inductor (b) Calculate the transformer emf (ver), for N-20, a # 5cm, and f s| MHz. (c) What is the self-inductance L of this 20-turn inductor. (d) Draw an equivalent circuit showing the coil as a sinusoidal source, with two resistors. The resistance of the connected resistor is R = 10 Ω and the internal resistance of the coil is 5 0. Label the correct polarity of the Vthar when t T/4 where T 1/f is the period of the sinusoidal signal. (e) Calculate the induced current / and voltage drop Ve across the resistor R at time t T/4 Indicate on your schematic the direction of this current at this instant of time

Answer 1

Given magnetic field B = 2 (y^- 4 + z^-) sin wt wb/m^2 (a) magnetic flux through a single turn, phi = B vector . A vector as A vector = pi a^2 z^- (direction of a area element takes towards the direction of normal phi = 2(y^- phi + z6-) sin wt . pi a^2 z^- phi = 2 pi a^2 sin wt wb (b) for N = 20 turns total magnetic flux phi = N 2 pi a^2 sin wt wb phi = 40 pi a^2 sin wt transformer em^2 v_emf^tr = -d phi/dT = -d/dt (40 pi a^2 sin wt) = -40 pi a^2 w cos wt volt as w = 2 pi f and f = 1 MHZ = 10^6 HZ a = 5 cm = 5 times 10^-2 m \r\n v_emf^tr = -40 pi 25 times 10^-4 times 2 pi times 10^6 cos wt volt v_emf^tr = -(2 pi^2 times 10^5) cos wr volt