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Solved Problems-4

An infinite straight wire carrying a current I is placed to the left of a rectangular loop of wire with width w and length l, as shown in the figure.
(a) Determine the magnetic flux through the rectangular loop due to the current I.
(b) Suppose that the current is a function of time with Description: 17025.png where a and b are positive constants. What is the induced emf in the loop and the direction of the induced current?
Rectangular loop near a wire
(a) Using Ampere’s law,
Description: 17983.png
the magnetic field due to a current-carrying wire at a distance r away is,
Description: 19228.png 
The total magnetic flux ϕ through the loop can be obtained by summing over contributions from all differential area elements Description: 17259.png
Description: 16696.png
NB: Here, the area vector has been chosen to point into the page so that φ > 0

(b) According to Faraday’s law, the induced emf is,
Description: 24582.png
where Description: 19213.png

The straight wire carrying a current I produces a magnetic flux into the page through the rectangular loop. By Lenz’s law, the induced current in the loop must be flowing counterclockwise in order to produce a magnetic field out of the page to counteract the increase in inward flux.

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