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Solved Problems-5

Problems-5
Find the field inside a solenoid of length L having N turns uniformly wound round a cylinder of radius a and carrying a current I.
or,
 
Derive a general expression for the field Description: Description: 93807.png at any point along the axis of a solenoid (uniform cylindrical coil wound on a nonmagnetic frame). Sketch the variation of Description: Description: 93797.png from point to point along the axis.
or,
 
A solenoid of length L consists of N turns of wire carrying a current I. Show that the field at any point along the axis,
Description: Description: 93787.png
 
where θ1 and θ2 are the angles subtended at the point by the end turns. Also, show that if L is very large compared to the radius of the solenoid then at the centre of the solenoid, Description: Description: 93778.png.
 
Solution
We consider the cross section of the solenoid (Figure).
 
Description: Description: 158217.png
Solenoid of finite length
 
Since the solenoid consists of circular loops, the contribution to the magnetic field by an element of the solenoid of length dz is,
 
Description: Description: 93695.png
From the figure, Description: Description: 159395.png
 
∴ Description: Description: 93669.png
∴ Description: Description: 93662.png
 
So, the field is given as,
Description: Description: 93654.png
 
∴ Description: Description: 93647.png
 
We consider three cases:
  1. When P is at One End of the Solenoid:
Here, Description: Description: 93639.png 
The magnetic field is given as, Description: Description: 93631.png 
Description: Description: 93623.png

 

Note: If L >> a then Description: Description: 95459.png
  1. When P is at the Centre of the Solenoid:
Here, Description: Description: 93589.png
The magnetic field is given as, Description: Description: 93579.png
Description: Description: 93570.png

 

Note: If L >> a then Description: Description: 95448.png
  1. When P is at the Midway between One End and Centre of the Solenoid:
Here,
Description: Description: 93542.png 
 
The magnetic field is given as Description: Description: 93526.png

 

Note: If L >> a then Description: Description: 159404.png and the magnetic field is given as, Description: Description: 95427.png

 

The variation of the field from point to point is shown in Figure.
 
Description: Description: 93493.png
Variation of magnetic field from point to point along the axis of a finite solenoid




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