Answer
Verified
432.6k+ views
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.
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.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Which are the Top 10 Largest Countries of the World?
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Difference Between Plant Cell and Animal Cell
Give 10 examples for herbs , shrubs , climbers , creepers
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you graph the function fx 4x class 9 maths CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE