Problem 2: It is required to produce inside a toroid a field of 2 x 10^-3 Weber/m². The toroid has a radius of 15cm and 300turns. Find the current required for this purpose. If toroid is wound on an iron core of permeability 300times the permeability of free space what increase in B will occur for the same current?

Solution:

Step: 1 Overview

Toroid and toroidal field:
It is a coil of insulated copper wire wound on a circular core with closed turns. It is used to create a uniform magnetic field of induction.

Equation of magnetic field of induction produced by a toroid is given below:

B=(µ_oIN)/(2.Pi.r)

Step: 2 Calculation

Given:

B = 2 x 10^-3 W/m²
r=15cm=0.15m
N=300turns
µ_o= 4Pi x 10^-7

Required:

i) I=?
According to above equation:
B=(µ_oIN)/(2.Pi.r)
µ_oIN=B.2.Pi.r
I=(B.2.Pi.r)/(µ_oN)
I=((2 x 10^-3).2.(3.14).(0.15))/((4 x 3.14 x 10^-7)(300))
I=(0.001884)/(0.0003768)
I=4.99A (Ans)

ii) Magnetic field (B') at which µ=300µ_o =?
B'=(µIN)/(2.Pi.r)
when µ=300µ_o
B'=(300µ_oIN)/(2.Pi.r)
B'=300{(µ_oIN)/(2.Pi.r)}
since (µ_oIN)/(2.Pi.r)=B
B'=300B (Ans)
hence when permeability will become 300 times, magnetic field will also become 300 times.