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Use this Law to Obtain the Expression for the Magnetic Field Inside an Air Cored Toroid of Average Radius 'r', Having 'n' Turns per Unit Length and Carrying a Steady Current I. - Physics

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Use this law to obtain the expression for the magnetic field inside an air cored toroid of average radius 'r', having 'n' turns per unit length and carrying a steady current I.

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Solution

A toroid is a hollow circular ring on which a large number of turns of a wire are closely wound. Consider an air-cored toroid (as shown above) with centre O.

Given:
r = Average radius of the toroid
I = Current through the solenoid
n = Number of turns per unit length

To determine the magnetic field inside the toroid, we consider three amperian loops (loop 1, loop 2 and loop 3) as show in the figure below.

For loop 1:

According to Ampere's circuital law, we have

`ointvecB.vec(dl)=mu_0(`

Total current for loop 1 is zero because no current is passing through this loop.

So, for loop 1

`oint.vecB.vec(dl)=0`

For loop 3:

According to Ampere's circuital law, we have

`ointvecB.vec(dl)=mu_0(`

Total current for loop 3 is zero because net current coming out of this loop is equal to the net current going inside the loop.

For loop 2:

The total current flowing through the toroid is NI, where N is the total number of turns.

`ointvecB.vec(dl)=mu_0(NI)`

Now, `vecB `

`ointvecB.vec(dl)=Bointdl`

`=>ointvecB.vec(dl)=B(2pir)`

Comparing (i) and (ii), we get

 B(2πr)=μ0NI

`=>B=(mu_0NI)/(2pir)`

 Number of turns per unit length is given by

 `n=N(2pir)`

B=μ0nI

This is the expression for magnetic field inside air-cored toroid.

Concept: Solenoid and the Toroid - the Solenoid
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