Question

If refractive index and dielectric constant of the material of a glass slab is \[\dfrac{3}{2}\] and \[3\] respectively then magnetic permeability of the glass is (in S.I. system)
A. \[5.37 \times {10^{ - 7}}\]
B. \[3.94 \times {10^{ - 7}}\]
C. \[{10^{ - 7}}\]
D. \[9.12 \times {10^{ - 7}}\]

Answer 1

Hint: The expression of relative permeability of glass in terms of its refractive index and dielectric constant will be used to calculate its value. We will also use the expression for magnetic permeability in terms of its relative permeability and permeability of free space.

Complete step by step answer:
Given:
The refractive index of the material of the glass slab is \[n = \dfrac{3}{2}\].
The dielectric constant of the material of the glass slab is \[\varepsilon = 3\].
We have to find the magnetic permeability of the material of glass in the S.I. system.
The glass's refractive index is equal to the square root of the product of relative permeability and dielectric constant of glass.
\[n = \sqrt {{\mu _r}\varepsilon } \]
Here \[{\mu _r}\] is the relative permeability of glass.

Substitute \[\dfrac{3}{2}\] for n and \[3\] for \[\varepsilon \] in the above expression.
\[\dfrac{3}{2} = \sqrt {{\mu _r}\left( 2 \right)} \]……(1)
On squaring both sides of equation (1), we get:
\[\begin{array}{c}
{\left( {\dfrac{3}{2}} \right)^2} = {\left( {\sqrt {{\mu _r}\left( 3 \right)} } \right)^2}\\
\Rightarrow \dfrac{9}{4} = 3{\mu _r}\\
\Rightarrow {\mu _r} = \dfrac{3}{4}
\end{array}\]

We know that relative permeability is the ratio of magnetic permeability of glass to free space's magnetic permeability.
\[\begin{array}{l}
{\mu _r} = \dfrac{\mu }{{{\mu _0}}}\\
\Rightarrow \mu = {\mu _r}{\mu _0}
\end{array}\]
Here \[{\mu _0}\] is the magnetic permeability in free space, and its value is given below.
\[{\mu _0} = 4\pi \times {10^{ - 7}}{{\rm{H}} {\left/
 {\vphantom {{\rm{H}} {\rm{m}}}} \right.
 } {\rm{m}}}\]

Substitute \[\dfrac{3}{4}\] for \[{\mu _r}\] and \[4\pi \times {10^{ - 7}}{{\rm{H}} {\left/
 {\vphantom {{\rm{H}} {\rm{m}}}} \right.
 } {\rm{m}}}\] for \[{\mu _0}\] in the above expression to get the value of magnetic permeability of glass.
\[\begin{array}{c}
\mu = \left( {\dfrac{3}{4}} \right)\left( {4\pi \times {{10}^{ - 7}}{{\rm{H}} {\left/
 {\vphantom {{\rm{H}} {\rm{m}}}} \right.
 } {\rm{m}}}} \right)\\
 = 9.42 \times {10^{ - 7}}{{\rm{H}} {\left/
 {\vphantom {{\rm{H}} {\rm{m}}}} \right.
 } {\rm{m}}}
\end{array}\]

We also know that the S.I. unit of magnetic permeability of a substance is \[{{\rm{H}} {\left/
 {\vphantom {{\rm{H}} {\rm{m}}}} \right.
 } {\rm{m}}}\].

The value of magnetic permeability calculated above is nearly equal to option (4). Therefore, the magnetic permeability of the given glass slab having a refractive index \[\dfrac{3}{2}\] and dielectric constant \[3\] is \[9.42 \times {10^{ - 7}}{{\rm{H}} {\left/
 {\vphantom {{\rm{H}} {\rm{m}}}} \right.
 } {\rm{m}}}\]

So, the correct answer is “Option D”.

Note:
We have to note that the value of magnetic permeability is asked, so do not just calculate the relative permeability. Also, substitute the value of magnetic permeability in free space in its S.I. because the final answer is asked in its S.I. system.