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Question: The magnetic susceptibility of a material of rod is 299. Permeability \[{\mu _0}\] of vacuum is \[4\...

The magnetic susceptibility of a material of rod is 299. Permeability μ0{\mu _0} of vacuum is 4π×10−7 H⋅m−14\pi \times {10^{ - 7}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}. Absolute permeability of the material of the rod is
A. 3771×10−7 H⋅m−13771 \times {10^{ - 7}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}
B. 3771×10−5 H⋅m−13771 \times {10^{ - 5}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}
C. 3770×10−6 H⋅m−13770 \times {10^{ - 6}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}
D. 3771×10−6 H⋅m−13771 \times {10^{ - 6}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}

Explanation

Solution

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 μ\mu of the material is given by
μ=μ0(1+χm)\mu = {\mu _0}\left( {1 + {\chi _m}} \right) …… (1)
Here, μ0{\mu _0} is the permeability of the free space and χm{\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.
χm=299{\chi _m} = 299
The permeability of the free space is 4π×10−7 H⋅m−14\pi \times {10^{ - 7}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}.
μ0=4π×10−7 H⋅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 4π×10−7 H⋅m−14\pi \times {10^{ - 7}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}} for μ0{\mu _0} and 299299 for χm{\chi _m} in equation (1).
μ=(4π×10−7 H⋅m−1)(1+299)\mu = \left( {4\pi \times {{10}^{ - 7}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}} \right)\left( {1 + 299} \right)
⇒μ=(4π×10−7 H⋅m−1)(300)\Rightarrow \mu = \left( {4\pi \times {{10}^{ - 7}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}} \right)\left( {300} \right)

Substitute 227\dfrac{{22}}{7} for π\pi in the above equation.
⇒μ=(4×227×10−7 H⋅m−1)(300)\Rightarrow \mu = \left( {4 \times \dfrac{{22}}{7} \times {{10}^{ - 7}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}} \right)\left( {300} \right)
⇒μ=264007×10−7 H⋅m−1\Rightarrow \mu = \dfrac{{26400}}{7} \times {10^{ - 7}}\,{\text{H}} \cdot {{\text{m}}^{ - 1}}
∴μ=3771×10−7 H⋅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 3771×10−7 H⋅m−13771 \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 π\pi is substituted as 227\dfrac{{22}}{7} in the formula for absolute permeability of the material of the rod. There are two values of π\pi which are 3.14 as well as 227\dfrac{{22}}{7}. Here, the value 227\dfrac{{22}}{7} of π\pi is substituted to make the calculations simple.