Question

A toroid has a core (non-ferromagnetic) of inner radius $25cm$ and outer radius $26cm$, around which $3500$ turns of a wire are wound. If the current in the wire is $11A$ ,

(1) what is the magnetic field outside the toroid?
(2) what is the magnetic field inside the core of the toroid?
(3) what is the magnetic field in the empty space surrounded by the toroid?

Answer 1

(1) Magnetic Field outside the Toroid
Toroid: Toroid is basically a solenoid which is bent in a circular shape and the ends are joined.

Magnetic field outside the toroid is always zero, because the current flows at outer circumference and inner circumference of toroid are in opposite direction to each other. Therefore, current enclosed by Amperian loop is zero. Hence the magnetic field outside the toroid is zero.

Final answer: Zero

(2) Magnetic field inside the core of toroid
Formula used: $B={\mu }_{0}nI$

Step 1: Find number of turns per unit length.
Given, Inner radius of the toroid, $r_1=25cm=0.25m$ Outer radius of the toroid, $r_2=26cm=0.26m$
Number of turns on the coil,$N=3500$
Length of the toroid, $l=2\pi \left(\frac{r_1+r_2}{2}\right)$
$l=\pi (0.25+0.26)$
$l=0.51\pi$
So, number of turns per unit length $n=\frac{N}{l}$ $n=\frac{3500}{0.51\pi }$

Step 2: Find magnetic field inside the core of toroid. Given, Current in the coil, $I=11A$
Magnetic field inside the core of a toroid $B={\mu }_{0}nI$
Here, ${\mu }_0$=permeability of the free space$=4\pi \times {10}^{-7}T\frac{m}{A}$
$B=4\pi \times {10}^{-7}\times \frac{3500}{0.51\pi }\times 11$
$B=3.019\times {10}^{-2}T\approx 3.0\times {10}^{-2}T$
Final answer: $3.0\times {10}^{-2}T$

(3) Magnetic field in the empty space surrounded by the toroid
Magnetic field is non-zero only inside the core surrounded by the windings of the toroid. So, the field in the empty space surrounded by the toroid will be zero, as there is no current enclosed by an Amperian loop considered in the empty space.
Final answer: Zero