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Solved Problems-5

Problem-5
(a) A magnetic field is given in cylindrical coordinates by
Description: 18160.png
 
where, B0 is a constant
 
A rectangular plane is located in the yz-plane and parallel to the z-axis as shown in the figure. The loop is moving in the plane with a velocity Description: 16689.png where v0 is a constant. Find the circulation of induced electric field around the loop.

(b) Also, find the circulation of induced electric field around the loop, if the loop is moving in the plane with a velocity Description: 19601.png and if the magnetic field is given by
Description: 17222.png
 
where, Description: 16654.pngDescription: 20445.png and Description: 18787.png are constants.

Rectangular loop moving in a magnetic field
 
Solution
(a) Since the loop is moving and the magnetic field is constant, the circulation of induced electric field around the loop is,
 
Description: 19339.png
 
where c is an arbitrary constant.
 
Description: 26489.png
 
When r → y, it results,
 
Description: 16874.png
 
(b) Here, since the magnetic field is also time varying, in addition to the component of (a), another component will be as given.
 
From (a) one component of the circulation of induced electric field around the loop is,
Description: 26520.png
 
The other component is,
 
Description: 26529.png
 
When r → y, it results,
 
Description: 26557.png
 
Hence, the total circulation of induced electric field around the loop is,
 
Description: 26564.png
 





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