Question

The space inside a toroid is filled with tungsten whose susceptibility is $6.8 \times {10^{ - 5}}$. The percentage increase in magnetic field will be:
A. $0.68\%$
B. $0.068\%$
C. $0.0068\%$
D. $6.8\%$

Answer 1

Use the formula for magnetic field inside the toroid when there is no material filled in the toroid and when there is a material filled in the toroid. Also, use the formula for permeability of the material in terms of susceptibility. Substitute all these values in the formula for percentage increase in the magnetic field.

Formula used:
The magnetic field $B$ inside the toroid is given by
$B = {\mu _0}nI$ …… (1)
Here, ${\mu _0}$ is the permeability of the free space, $n$ is the number of turns of the coil per unit length of toroid and $I$ is the current through the toroid.
The magnetic field $B'$ inside the toroid when a substance is placed inside it is
$B' = {\mu _r}B$ …… (2)
Here, ${\mu _r}$ is the relative permeability of the material inside the toroid and $B$ is the magnetic field inside the toroid when there is no material in the toroid.
The permeability ${\mu _r}$ of a material is given by
${\mu _r} = 1 + \chi$ …… (3)
Here, $\chi$ is susceptibility of the material.

Complete step by step answer:
The toroid is filled with tungsten whose susceptibility is $6.8 \times {10^{ - 5}}$.
$\chi = 6.8 \times {10^{ - 5}}$

The magnetic field inside $B$ the toroid when tungsten is not filled in it is
$B = {\mu _0}nI$
Let us now determine the magnetic field $B'$ in the toroid when tungsten is filled in it.
Substitute ${\mu _0}nI$ for $B$ in equation (2).
$B' = {\mu _r}{\mu _0}nI$

As the susceptibility of tungsten is positive, tungsten is paramagnetic in nature. When a paramagnetic material is filled in the toroid, the magnetic field in the toroid increases. The increase in magnetic field of the toroid will be $B' - B$ and the percentage increase in the magnetic field is $\dfrac{{B' - B}}{B} \times 100$.

Let us now determine the percentage increase in the magnetic field.
${\text{Percentage increase in B}} = \dfrac{{B' - B}}{B} \times 100$

Substitute ${\mu _r}{\mu _0}nI$ for $B'$ and ${\mu _0}nI$ for $B$ in the above equation.
${\text{Percentage increase in B}} = \dfrac{{{\mu _r}{\mu _0}nI - {\mu _0}nI}}{{{\mu _0}nI}} \times 100$
$\Rightarrow {\text{Percentage increase in B}} = \left( {{\mu _r} - 1} \right) \times 100$

Substitute $1 + \chi$ for ${\mu _r}$ in the above equation.
$\Rightarrow {\text{Percentage increase in B}} = \left( {1 + \chi - 1} \right) \times 100$
$\Rightarrow {\text{Percentage increase in B}} = \chi \times 100$

Substitute $6.8 \times {10^{ - 5}}$ for $\chi$ in the above equation.
$\Rightarrow {\text{Percentage increase in B}} = \left( {6.8 \times {{10}^{ - 5}}} \right) \times 100$
$\therefore {\text{Percentage increase in B}} = 0.0068\%$

Therefore, the percentage increase in the magnetic field is $0.0068\%$. Hence, the correct option is C.

Note: One can also solve the same question by another method. One can also use the formula for magnetic field inside the toroid in terms of magnetic field intensity inside the toroid when there is no material filled and when a material is filled inside the toroid. Also use the value of magnetic permeability in terms of susceptibility and solve.