Chapter 20, Problem 49P

To determine

Part (a) To Determine:

Expressions for Magnetic fields inside a toroid

To determine

To find magnetic field inside the toroid:

Consider a closed loop inside the toroid having radius r,

Let I be the current through the coil, then total current enclosed by the loop is given by,

$I_e\text{ = }NI$ ……….. (eq.2)

Where,

N → Total number of turns of wire in toroid

I → Current through Coil

Then Ampere’s law for loop 1 can be written as,

${\displaystyle \oint B·dl}={\mu }_{0}NI$

Here B and elemental length dl are in same direction and magnetic field is uniform along toroid. Therefore we can write,

$B{\displaystyle \oint dl}={\mu }_{0}NI$……….. (eq.3)

Then,

${\displaystyle \oint dl}=2\pi r_{}$

Appling on eq.3,

$B=\frac{{\mu }_{0}NI}{2\pi r_{}}$ ……….. (eq.4)

To determine

Part (b) To Determine:

Expressions for Magnetic fields outside a toroid

To determine

To find magnetic field outside the toroid:

Now consider a closed path outside the toroid as shown in figure.

It can be shown that total current enclosed by the loop, $I_e\text{ = }0$

Since the current coming outwards from the plane of paper is cancelled by current going into it.

Therefore, Eqn.1 can be written as

${\displaystyle \oint B·dl}=0$

I.e. Magnetic field outside the toroid, B=0

To determine

Part (c) To Determine:

How magnetic fields inside a toroid if varies