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497. Problem 36.33 (RHK) A rod of length L is caused to move at constant
speed v along horizontal conducting rails, as shown in the figure. In this case
the magnetic field in which the rod moves is not uniform but is provided
by a current
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in a
long parallel wire. We may assume that
,
,
, and
. We
have to calculate (a) the emf induced in the rod; and (b) find the
current in the conducting loop. We may assume that the resistance of the rod is
and
that the resistance of the rails is negligible. (c) We have to calculate
the rate at which internal energy is being generated in the rod. (d) We
have to calculate the force that must be applied to the rod by an external agent
to maintain its motion. (e) We have to find the rate at which the
external agent does work on the rod and compare our answer with that found in
(c).
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Solution: Click For PDF Version From Ampere’s law and considering that the magnetic field due to a long current carrying wire will be circular, we note that the magnitude of the field at a distance x from the wire will be
The flux enclosed in the loop formed by the rod and the conducting rails when the rod is at distance y from the left-hand end will be given by the integral
Flux enclosed is changing with time as the conducting rod is being made to
move with constant speed
Data for the problem are;
and
(a) The induced emf developed in the loop will be
(b) As the resistance in the circuit formed by the conducting rod is
(c) The rate at which internal energy is being generated in the rod will be given by
(d) The force that must be applied by an external agent to maintain the motion of
the conducting rod will be given by integrating the Lorentz force on element of
length
(e) The rate at which the external agent does work on the rod will be given by
It is equal to the rate at which internal energy is being generated in the rod.
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. By the
Faraday’s law the induced emf in the circuit will be given by
at a
distance x from the wire carrying current
over the length of the conductor,
