Motional emf

Problem:

In the figure below, assume that R = 6 Ω, d = 1.2 m, and a uniform 2.5 T magnetic field is directed into the page.  At what speed should the bar be moved to produce 0.5 A in the resistor?

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Solution:

Problem:

A pair of parallel conducting rails a distance d apart is placed in a uniform magnetic field B which is perpendicular to the rails.  A resistance R is connected across the rails and a conducting bar of mass m and negligible resistance is placed at rest on the rails and perpendicular to them.  A constant force F is applied to the bar pulling it along the rails.
(a)  What is the value of v when the bar's acceleration becomes zero?
(b)  Derive an expression for the speed v(t) of the bar as a function of time.
(c)  If F is suddenly reduced to zero at time t' = ln(2) mR/(Bd)2, find the rate of decrease of the kinetic energy of the bar for t > t'.
(d)  Show that the rate of decrease of the kinetic energy of the bar is equal to the ohmic heating rate.

Solution:

Problem:

Two identical conducting bars rest on two horizontal parallel conducting rails.  The bars are perpendicular to the rails and parallel to each other as shown.  The distance between the bars is L.  At a certain moment, a uniform vertical upward magnetic field is turned on.  The field quickly reaches its maximum magnitude and then remains constant.
Neglecting friction, find the new distance between the bars.  Assume that the resistance of each bar is much greater than the resistance of the rails.

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Solution:

Problem:

Under the influence of the gravity near the surface of the earth a square wire of length l, mass m and resistance R slides without friction down very long parallel conducting rails of negligible resistance.  The rails are connected to each other at the bottom by a rail of negligible resistance, parallel to the wire, so that the wire and the rails form a closed rectangular conducting loop.  The plane of the rails makes an angle θ with the horizontal, and a uniform vertical magnetic field B exists throughout the region.
(a)  Show that the wire acquires a steady state speed for any fixed θ.
(b)  If the angle θ is lowered from θ1 = π/3 to θ2 = π/6, find the percent change of the steady-state speed of the wire.
(c)  Find the percent change of the corresponding power converted into joule energy.

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Solution:

Problem:

A rod of length L lying in the xy-plane pivots with constant angular velocity ω counterclockwise about the origin.  A constant magnetic field of magnitude B0 is oriented in the z-direction.  Find the motional emf in the rod.

 

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Problem:

(a)  A long solenoid of length L is wound with N turns of wire that carry a current I.  A copper disk of radius a rotates with angular velocity ω about a shaft at the center of the solenoid as shown in the figure below.  What is the potential difference between slip rings on the shaft and on the edge of the disk?
(b)  Assume L = 1 m, a = 10 cm, N = 1000, I = 2 A and the disk rotates 1200 revolutions per minute.  Calculate the potential difference in Volts.

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Solution:

Problem:

(a)  A thin copper plate of mass m has a shape of a square with a side b and thickness d.  The plate is suspended on a vertical spring with a force constant k in a uniform horizontal magnetic field B parallel to the plane of the plate.  Find the period of the small-amplitude vertical oscillations of the plate. 
Neglect the resistivity of the plate.
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(b)  A bar of mass m is suspended horizontally on two vertical springs with the force constants k and 3k.  The bar bounces up and down while remaining horizontal.  Find the period of oscillations of the bar.  Neglect the mass of the string that connects the springs and the friction between the string and the pulley.
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Solution: