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# The magnetic susceptibility of a material of rod is 299. Permeability ${\mu _0}$ of vacuum is $4\pi \times {10^{ - 7}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}$. Absolute permeability of the material of the rod isA. $3771 \times {10^{ - 7}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}$B. $3771 \times {10^{ - 5}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}$C. $3770 \times {10^{ - 6}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}$D. $3771 \times {10^{ - 6}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}$

Last updated date: 21st Jun 2024
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Hint: Use the expression for the absolute permeability of a material. This formula gives the relation between permeability of the free space and magnetic susceptibility of the material. Substitute all the given values in this equation and determine the absolute permeability of the material of the rod.

Formula used:
The absolute permeability $\mu$ of the material is given by
$\mu = {\mu _0}\left( {1 + {\chi _m}} \right)$ …… (1)
Here, ${\mu _0}$ is the permeability of the free space and ${\chi _m}$ is the magnetic susceptibility of the material.

We have given that the magnetic susceptibility of a material of rod is 299.
${\chi _m} = 299$
The permeability of the free space is $4\pi \times {10^{ - 7}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}$.
${\mu _0} = 4\pi \times {10^{ - 7}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}$
We can determine the absolute permeability of the material of the rod using equation (1).

Substitute $4\pi \times {10^{ - 7}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}$ for ${\mu _0}$ and $299$ for ${\chi _m}$ in equation (1).
$\mu = \left( {4\pi \times {{10}^{ - 7}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}} \right)\left( {1 + 299} \right)$
$\Rightarrow \mu = \left( {4\pi \times {{10}^{ - 7}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}} \right)\left( {300} \right)$

Substitute $\dfrac{{22}}{7}$ for $\pi$ in the above equation.
$\Rightarrow \mu = \left( {4 \times \dfrac{{22}}{7} \times {{10}^{ - 7}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}} \right)\left( {300} \right)$
$\Rightarrow \mu = \dfrac{{26400}}{7} \times {10^{ - 7}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}$
$\therefore \mu = 3771 \times {10^{ - 7}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}$
Therefore, the absolute permeability of the material of the rod is $3771 \times {10^{ - 7}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}$.

Hence, the correct option is A.

Note:The students may get confused that why the value of $\pi$ is substituted as $\dfrac{{22}}{7}$ in the formula for absolute permeability of the material of the rod. There are two values of $\pi$ which are 3.14 as well as $\dfrac{{22}}{7}$. Here, the value $\dfrac{{22}}{7}$ of $\pi$ is substituted to make the calculations simple.