Mutual, Self Inductance, Inductors in Circuits PDF

Preview

Summary

Practice problems on Mutual, Self Inductance, Inductors in Circuits...

Description

PHYS272 Problem Session: Mutual and Self Inductance The following is from the textbook and describes a general strategy for solving for the self-inductance of a particular inductor:

While these instructions are for calculating self inductance, they are rather similar for finding mutual inductance. In the case of mutual inductance, rather than , we use , so we have a choice of which flux to calculate. We generally choose the easier of the two. In terms of the self inductance, L, or the mutual inductance, M, we can describe the induced emf due to a changing current using

for self

inductance, or for mutual inductance, where the latter formula can also have the indices 1 and 2 swapped, depending on the emf of interest.

PHYS272 Problem Session

Page 1

P1. A toroidal coil has a mean radius of 14 cm and a cross-sectional area of 0.50 cm2; it is wound uniformly with 1,600 turns. A second toroidal coil of 850 turns is wound uniformly over the first coil. Ignoring the variation of the magnetic field within a toroid, determine the mutual inductance of the two coils (in H). Recommended procedure: a) Recall what geometry a toroid describes, sketch as needed. b) Use Ampere’s Law to find the magnetic field due to a toroid. c) Make the recommended approximation where you ignore the variation in magnetic field within the toroid (i.e., just use the mean radius of the inner toroid to find the mean magnetic field in the inner toroid. d) Find the flux in the outer toroid due to the inner toroid. e) Combine with the definition of mutual inductance to obtain the final answer.

PHYS272 Problem Session

Page 2

P2. A coil of 35 turns is wrapped around a long solenoid of cross-sectional area 8.0 ✕ 10−3 m2. The solenoid is 0.40 m long and has 600 turns. a) What is the mutual inductance of this system (in H)? b) The outer coil is replaced by a coil of 35 turns whose radius is three times that of the solenoid. What is the mutual inductance of this configuration (in H)?

P3. a) Calculate the self-inductance of a 45.8 cm long, 10.0 cm diameter solenoid having 1000 loops. b) How fast can it be turned off if the induced emf cannot exceed 3.00 V and a current of 18.2 Aflows through this inductor?

PHYS272 Problem Session

Page 3

Stored Energy and Inductors in Circuits (RL, LC)

Like capacitors, inductors serve to store energy in circuits. Whereas capacitors store energy in the electric field, inductors store energy in the magnetic field. The total energy stored is

, or in terms of energy density, energy per unit volume,

. When analyzing inductors in circuits, we can use the recommended conventions in class for applying loop rules (KVL):

PHYS272 Problem Session

Page 4

P1. Write out loop rules (KVL) for the following two circuits. Show that they result in the same form of differential equation, only one is in terms of the charge Q, and the other in terms of the current I.

P2. Write out the solution for a charging capacitor, and use the similarity in the above equations to write out the equivalent solution for a “charging” (i.e., current ramping up) inductor. [Note you can perform the same exercise for a discharging capacitor and an inductor ramping down… try it!]

PHYS272 Problem Session

Page 5

Inductors and capacitors both store energy, one in charge, the other in current. If these two elements are connected in a circuit the resulting behavior is a periodic transfer of energy from one to the other, and back again.

P3. Write out the loop rule (KVL) for the above circuit. Rewrite the resulting equation entirely in terms of the charge, using the definition of current. Assume the solution has the following form: Q(t) = A sin(ωt + φ) Plug this into your loop rule equation and solve for the angular frequency of the oscillation. Since the energy oscillates from being entirely stored in the electric field of the capacitor, then entirely stored in the magnetic field of the inductor, you can use conservation of energy to find a relationship between the maximum current and maximum stored charge. What is this relation?

PHYS272 Problem Session

Page 6

P4. The 4.84 A current through a 1.50 H inductor is dissipated by a 1.86 Ω resistor in a circuit like that in the figure below, with the switch in position 2.

a) What is the initial energy in the inductor? b) How long will it take the current to decline to 5.00% of its initial value? c) Calculate the average power dissipated, and compare it with the initial power dissipated by the resistor.

P5. In an oscillating LC circuit, the maximum charge on the capacitor is 4.0 ✕ 10−6 C and the maximum current through the inductor is 5.0 mA. a) b)

What is the period of the oscillations (in s)? How much time (in s) elapses between an instant when the capacitor is uncharged and the next instant when it is fully charged?

PHYS272 Problem Session

Page 7...