A rectangular conducting loop of width and negligible resistance is attached to an…
PHYSICS · Multi-part · A Level · Physics · Fields and their consequences · Fields and their consequences
Problem
A rectangular conducting loop of width w and negligible resistance is attached to an external resistor of resistance R. The loop is pulled at a constant speed v into and completely through a region that contains a uniform magnetic field B directed into the page. The field region has a length L in the direction of motion and extends infinitely in the other directions. Assume the front edge of the loop first enters the field region at time t=0 and exits the far side at time t=L/v. Ignore any mechanical friction.
Throughout the problem, take the positive direction for any induced current to be counter-clockwise when viewed from the observer’s side of the page.
(a) Derive an expression for the magnitude of the induced emf E(t) in the loop while any portion of it is inside the field region.
(b) Obtain an expression for the induced current I(t) as a function of time for 0≤t≤L/v. Express your answer in terms of the given quantities and fundamental constants.
(c) Determine the instantaneous electrical power P(t) dissipated in the resistor during the interval 0≤t≤L/v.
(d) Calculate the total electrical energy Ediss that is dissipated in the resistor from the moment the loop first enters until it has completely left the field region.
(e) The loop is kept moving at the constant speed v by an external force. Compute the total mechanical work Wext done by that external force over the same interval and compare it with Ediss. Explain any equality or inequality you obtain.