Faraday's law

Problem:

A small circular search coil is used to search for stray magnetic fields around a transformer.  The coil diameter is 1.5 cm and its output is connected to an ideal voltmeter.  For a stray field of 1 mT at 60 Hz, how many turns would be required for the search coil to produce a reading of 1 mV?

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Solution:

Problem:

(a)  A coil has an inductance of 2 mH, and a current through it changes from 0.2 A to 1.5 A in a time of 0.2 s.  Find the magnitude of the average induced emf in the coil during this time.
(b)  A 25 turn circular coil of wire has a diameter of 1 m.  It is placed with its axis along the direction of the Earth's magnetic field (magnitude 50 μT), and then, in 0.2 s, it is flipped 180o.  What is the average emf generated?

Solution:

Problem:

Clever farmers whose lands are crossed by large power lines have been known to steal power by stringing wire near the power line and making use of the induced current.
Suppose that the farmer places a rectangular loop with two sides of length a parallel to the power line, the closest one at distance 5 m from it.  The loop and the power line lie in the same plane.  Let the length of the sides perpendicular to the power line be b = 0.5 m.  The power line carries a 60 Hz alternating current with a peak current of 10 kA.
(a)  If the farmer wants a peak voltage of 170 V (which is the peak of standard 120 V AC power) what should be the length a?
(b)  If the equipment the farmer connects to the loop has a resistance R = 5 Ω, what is the farmer's average power consumption?
(c)  If the power company charges 10 cents per kWh, what is the monetary value of the energy stolen each day?
(Give a numerical answer!)

Solution:

Problem:

A non-conducting ring of mass M, radius R, and total charge Q uniformly distributed around the ring hangs horizontally, suspended by non-conducting thin (“massless”) strings as shown.

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A uniformly increasing magnetic field B = B0t pointing down is turned on at time t = 0.
(a)   Find the torque on the ring in terms of B0, M, R, and Q.
(b)   If the strings supporting the ring provide a torque –αφ, where φ is the angle through which the ring turns, write the equation of motion in terms of B0, M, R, Q and α.

Solution:

Problem:

An electron accelerator employs a time varying magnetic flux through a plane circular loop of radius R = 0.85 m, and the electrons always move in this circular path with this radius.  The magnetic induction in the loop plane

B(r) = B0 - Kr2,  r < R;    B = 0, r > R,

is everywhere normal to the loop plane with r the distance from the loop center.
(a)  Show that, at any instant, the average magnetic induction in the loop Bav, must be related to BR by Bav = 2BR.  Evaluate K.
(b)  B0 increases linearly from 0 to 1.2 Tesla in 5.3 sec.  Deduce the energy gain per turn for the electrons and the maximum electron energy achieved.

Solution:

Problem:

A spatially uniform current density j = j0 cosωt flows through the hole of a torus along the axis of the torus as shown.  The inner radius of the torus is r, and the cross section is square with sides a (a << r).  The torus is made of an insulating material with m = μ0.  A wire of resistance R wraps around the torus with a total of N turns.  Determine the current flowing in the wire.

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Solution:

Problem:

A loop of wire of resistance R and a coil of self-inductance L encloses an area A.  A spatially uniform magnetic field is applied perpendicular to the plane of the loop with the following time dependence:
For t < 0 the field is zero, for 0 < t < t0 B(t) = kt, for t > t0  the field remains constant at B0 = kt0
(a)  Calculate the current I in the loop for all times t > 0, given that I = 0 for t = 0. 
(b)  Make simple sketches of the current vs.  time for t0 < L/R and t0 > L/R..

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Solution:

Problem:

A square loop with side b is made of a wire of mass m and negligible electric resistance. The loop has a gap which can be closed with a switch.  Initially the switch is open.  The loop is pivoted along its top horizontal side and placed in the weak vertical uniform magnetic field B as shown.  The loop is then pulled to a horizontal position, the switch is closed and the loop is released. Eventually, the loop comes to rest due to air resistance. Find the angle θ that the plane of the loop makes with the vertical at the final position. The self inductance of the loop is L.

 

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Solution: