Question

A current in the windings on a toroid is $2A$. There are $400$ turns and the mean circumferential length is $40cm$. If the magnetic field inside is $1T$, the relative permeability is:
A $200$
B $2500$
C $400$
D $150$

Answer 1

Toroid is a circular ring with a hole in the middle that has numerous turns of enameled wire. As toroid has no endpoint the current is enclosed by a circular loop. By Ampere’s law, the magnetic field inside the magnet can be found.
The formula used:
By Ampere’s law the magnetic field at the centreline of the toroid known as
$B = \dfrac{{\mu NI}}{{2\pi r}}$

Complete answer:
Let the current $I = 2.0A$, turn $N = 400$, length of wire $l = 40cm$. The magnetic field inside is $B = 1T$.
In this case, $2\pi r = l = 40 \times {10^{ - 2}}m$.
As we know, $B = \dfrac{{\mu NI}}{{2\pi r}}$
\Rightarrow $$$$\mu = \dfrac{{2\pi rB}}{{NI}}
$\Rightarrow \mu = \dfrac{{40 \times {{10}^{ - 2}} \times 1}}{{4\pi \times {{10}^{ - 7}} \times 400 \times 2}}$
\Rightarrow \mu = \dfrac{{{{10}^4}}}{{8\pi }}$$$$ = \dfrac{{10000}}{{8 \times 3.14}}$$$$ = \dfrac{{10000}}{{25}} = 400

Hence, the correct answer is option C

Note:
Toroid can be considered as circular solenoid using an electronic circuit. There are various uses of toroid. It’s a low-frequency inductor that needs large inductances. The change in the magnetic field resulted in the voltage which is known as Faraday’s law of induction.
Both Toroid and Solenoid are kind of similar, thus they are both designed on the principle of magnetism. But there are some differences between Solenoid and Toroid. Toroids have a circular shape where the solenoid is cylindrical. The formation of the magnetic field is within the toroid but in the solenoid, it is outside. Toroid has a nonuniform magnetic field but solenoid has a uniform magnetic field.
Toroid can be used variously. It’s used in telecommunication. Modern medical equipment uses toroid and torsion can be used in musical instruments also.