A one-turn rectangular loop oriented parallel to the x - y plane and having dime

Question

A one-turn rectangular loop oriented parallel to thex-y plane and having dimensions a = 3cm, b = 11 cm moves with a constant velocity v =11 m/s in the +x direction. An external agent (force)ensures that this velocity remains constant throughout the motion.The wire of which the loop is made has negligible resistance. Aresistor R = 29 W (not shown)is connected in the upper side of the loop.

At some point, the loop enters a region of constant, spatiallyuniform magnetic field B = 1.5 T pointing in the-z direction (into the page). The magnetic field covers aregion of "thickness" d = 28 cm in thex-direction.

The figure above shows three configurations: (1) loop enteringand half way into the field region; (2) loop entirely within thefield region; and (3) loop emerging from the field region andthree-fourths of the way out.

(a) Calculate the induced current in the loop for eachconfiguration. Give the magnitude and the direction, with apositive answer denoting a counterclockwise current as seen in theview above and a negative answer denoting a clockwisecurrent.

Configuration (1):   I = A    

HELP:  The special case of loopsmoving through time independent magnetic fields can be solved bythe method of "motion EMFs" (see textbook). However, the magneticflux method (Faraday's law) covers all cases and is no moredifficult than motional EMFs even in special cases where the latteris adequate. The following remarks will refer exclusively to aFaraday's law approach.

Write an algebraic expression for the flux through the loop whena long side of the loop is a distance x into the magneticfield with x < b. It is most natural tocalculate the flux in the direction of the B-field (-zdirection), so do so.

Configuration (2):   I = A     *
0   OK

Configuration (3):   I = A    

(b) You are a designer and wish to increase the current thatresults from the above device. Your summer engineering intern fromUIUC makes several suggestions. For each of these suggestionslisted below, indicate whether that suggestion, taken alone, (1) =increases, (0) = leaves unchanged, or (-1) = decreases themaximum value of the current that is achieved. Insert thecorresponding number (1 or 0 or -1) in each of the following answerboxes.

Increase the loop dimension a:      

Increase the loop dimension b:       *
0   OK

Increase the speed v:      

Increase the field strength B:      

Increase the field "thickness" d:       *
0   OK

Increase the number of turns of the loop:      

Orient the field in the y-direction:  

Explanation / Answer

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