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Solved Problems-4

Problems-4
Derive an expression for the magnetising force at any point on the axis of a single-turn coil carrying a steady current ‘I’:
  1. When the coil is in the form of a circle of radius ‘r
  2. When the coil is in the form of a rectangle of sides a and b
  3. When it is square with sides of length ‘2a
Solution
  1. By the Biot–Savart law,
Description: 94990.png
∴ Description: 94977.png
∴ Description: 94969.png
 
By symmetry, the contribution along Description: 94962.png adds up to zero because the radial components produced by pairs of current elements 180° apart cancel.
 
∴ Hr = 0
 
Hence, the magnetic field is given as,
 
Description: 94955.png
 
Description: 150319.png
Circular current-carrying loop

 

Note: At the centre (h = 0), the field is, Description: 95523.png
  1. Let P be a point at a height h above the plane of the loop.
The magnetic field due to the side AB is given as,
Description: 151930.png
Description: 151936.png
Description: 151942.png
∴ Description: 151939.png
Description: 150511.png
Rectangular current-carrying loop
 
Similarly, the magnetic field due to the side CD is given as,
Description: 94908.png
 
Since the flow of current in the two elements is in opposite direction, their cosine components will cancel each other and thus, only the axial components will add together.
 
Hence, the resultant field due to sides AB and CD is given as,
Description: 94901.png
 
From the figure, it is seen that, Description: 94892.png
Description: 94883.png

Putting these values, we get,
Description: 94873.png
 
Similarly, the magnetic field due to the other two sides BC and DA is given as,
Description: 94862.png
 
Hence, total magnetic field due to the rectangular loop is given as,
Description: 94854.png
 
Description: 94846.png
  1. Along the axis of the coil there will be only a z-component of magnetic field by symmetry. In order to get the total field, it is only necessary to calculate the z-component of the field generated by one side of the coil and then multiply by four. Consider the right-hand side.
Let, Description: 94838.png
The position of the element of length Description: 94831.png is specified by the vector Description: 94823.png where, Description: 94815.png The position of the point of observation along the z-axis is specified by the vector Description: 94808.png.
Squarer current-carrying loop
 
Description: 94799.png

By the Biot–Savart law,
Description: 94791.png
 
Since all the x-components will add to zero, we have,
Description: 94781.png
 
By integration,
Description: 94770.png
 
As the coil has four sides, this must be multiplied by 4 to get the total magnetic field.
∴ Description: 94762.png
∴ Description: 94755.png

Alternative Method
 
From the result of Part (b), if a = b = 2a then the result becomes,
Description: 94748.png

 

Note: At the centre (z = 0), Description: 95509.png as obtained. This result can becompared with Description: 95502.png for a circular coil.

 





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