Chapter 20, Problem 52P

(a)

To determine

The magnetic field inside the toroid.

Answer to Problem 52P

The magnetic field inside the toroid $\text{B =}\frac{{\mu }_0\text{NI}}{2\pi \text{R}}$ .

Explanation of Solution

Given:

The given toroid is shown below.

  

For path 1, each loop consists of the same magnetic field along the path and it is parallel to the path as well.

Formula used:

The magnetic field is calculated as

  ${\displaystyle \sum {\text{B}}_{\parallel }\cdot \text{dl}}={\mu }_0\text{I}$

Calculation:

Since each turn is enclosed by path 1. Thus, the enclosed current in the loop is $\text{NI}$ .

Thus, magnetic field in the loop is

  $\begin{array}{l}\text{B}(2\pi \text{R})={\mu }_0\text{NI}\\ \text{B =}\frac{{\mu }_0\text{NI}}{2\pi \text{R}}\end{array}$

Conclusion:

Thus, the magnetic field inside the toroid is $\text{B =}\frac{{\mu }_0\text{NI}}{2\pi \text{R}}$ .

(b)

To determine

The magnetic field outside the toroid.

Answer to Problem 52P

The magnetic field outside the toroid is zero.

Explanation of Solution

Given:

Given:

The given toroid is shown below.

  

For path 1, each loop consists of the same magnetic field along the path and it is parallel to the path as well.

For path 2, each loop pierces the enclosed area twice, once upwards and secondly downwards.

Formula used:

The magnetic field is calculated as

  ${\displaystyle \sum {\text{B}}_{\parallel }\cdot \text{dl}}={\mu }_0\text{I}$

Calculation:

Ampere’s law considers the magnetic field as the total sum of the enclosed current. So, the net current in the loop is zero.

Thus, magnetic field in the loop is

  $\begin{array}{l}\text{B}(2\pi \text{R})=0\\ \text{B =0}\end{array}$

Conclusion:

Thus, the magnetic field outside the toroid is zero.

(c)

To determine

To explain: Uniformity of the magnetic field inside the toroid.

Answer to Problem 52P

Magnetic field is not uniform. It varies inversely inside and zero outside.

Explanation of Solution

Introduction:

A field is constant when the factor on which it depends does not vary with time and space.

As seen in the above 2 sub-parts, the field inside is not uniform, and the outside does not exist. Over the loop 1, magnetic field is inversely proportional to the radius of the toroid and so,

  $\text{B}\propto \frac{1}{\text{R}}$

Conclusion:

Thus, magnetic field will be strongest at the inside wall of the toroid. However, it is weakest outside the toroid.