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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 is
A. \[3771 \times {10^{ - 7}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}\]
B. \[3771 \times {10^{ - 5}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}\]
C. \[3770 \times {10^{ - 6}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}\]
D. \[3771 \times {10^{ - 6}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}\]

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Last updated date: 21st Jun 2024
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Answer
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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.

Complete step by step answer:
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.

Additional information:
The permeability of a material is the resistance offered by the material to the external magnetic field.The magnetic susceptibility of a material is the ability of the material to get magnetized when it is exposed to an external magnetic field.The ratio of the magnetic moment per unit volume to the magnetizing field intensity is known as magnetic susceptibility of the material.

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.