Question

The relative permeability of glass is $\dfrac{3}{8}$ and the dielectric constant of glass is 8. The refractive index of glass is?
A. $1.5$
B. $1.1414$
C. $1.732$
D. $1.6$

Answer 1

Here we are using the fact that the speed of light is dependent upon the relative electric permittivity. Also we take into consideration that the permittivity for each material will be different. Then we will use the definition of refractive index to find out the refractive index of glass.

Complete step by step answer:
The speed of light in glass is given by, $r = \,\dfrac{1}{{\sqrt {\varepsilon \,\mu } }}$
Here,${\varepsilon _g}$ is the permittivity of glass (The dielectric constant)
${\mu _g}$ is the relative magnetic permeability of glass
We put these values in the above equation of speed of light in glass.
$r = \,\dfrac{1}{{\sqrt {{\varepsilon _g}{\varepsilon _ \circ }\,{\mu _g}{\mu _ \circ }} }}$
If we separate out the factor of $\dfrac{1}{{\sqrt {{\varepsilon _ \circ }\,{\mu _ \circ }} }}$ from the above equation, we get
$r = \,\dfrac{1}{{\sqrt {{\varepsilon _ \circ }\,{\mu _ \circ }} }} \times \dfrac{1}{{\sqrt {{\varepsilon _g}\,{\mu _g}} }}$as we have the speed of light as $c = \,\dfrac{1}{{\sqrt {{\varepsilon _ \circ }{\mu _ \circ }} }}$. Here, ${\mu _o}$ is the magnetic permeability of free space and ${\varepsilon _o}$ is the permittivity of free space.

Now, we will put this in the equation and we get,
$r = \,c \times \dfrac{1}{{\sqrt {{\varepsilon _g}\,{\mu _g}} }}$
After rearranging the equation, the equation becomes,
$\dfrac{c}{r} = \,\sqrt {{\varepsilon _g}\,{\mu _g}}$
Now, putting values given in the Question, we get

$$\dfrac{c}{r} = \sqrt {\dfrac{3}{8} \times 8} \\\ \Rightarrow\dfrac{c}{r} = \sqrt 3 \\\ \Rightarrow\dfrac{c}{r} = 1.732$$

The refractive index of glass:
$\dfrac{{{\rm{speed}}\,{\rm{of}}\,{\rm{light in vacuum}}}}{{{\rm{speed of light in glass}}}} = \dfrac{c}{r}\\\ \therefore\dfrac{{{\rm{speed}}\,{\rm{of}}\,{\rm{light in vacuum}}}}{{{\rm{speed of light in glass}}}} = 1.732$
Therefore, the refractive index of glass is 1.732, and the correct option is C.

Additional information:
The Electromagnetic waves have wavelength and frequency whose product gives the speed of the light in free space. The refractive index for air is 1 for all wavelengths. When an EM wave travels from one medium to another medium with a different refractive index, then the speed of the wave changes as well as wavelength. But the frequency does not change.

Note: The calculation of speed of light is done with consideration of the electric permittivity of free space and magnetic permeability of free space. In this formula of refractive index of glass, the speed of a wave in a vacuum should be in the numerator at all times.