Question

If the relative permeability and dielectric constant of a given medium are equal to \(\mu_{\mathrm{r}}\) and \(\mathrm{K}\) respectively, then the refractive index of the medium is equal to (A) \(\sqrt{\left(\mu_{\mathrm{T}} \mathrm{K}\right)}\) (B) \(\sqrt{\left(\mu_{1} E_{0}\right)}\)

Short answer

The correct expression for the refractive index based on the relative permeability \(\mu_r\) and dielectric constant \(K\) of the medium is \[n = \sqrt{(\mu_r \mu_0)(K \epsilon_0)}.\] None of the given options match this expression.

Step-by-step solution

Recall the relationships between permeability and permittivity

To find the refractive index, we need to work with the permeability and permittivity of the medium. Recall that: 1. Relative permeability (\(\mu_r\)) is the ratio of the material's permeability (\(\mu\)) to the permeability of free space (\(\mu_0\)): \[\mu = \mu_r \mu_0\] 2. Dielectric constant (\(K\)) is the ratio of the material's permittivity (\(\epsilon\)) to the permittivity of free space (\(\epsilon_0\)): \[\epsilon = K \epsilon_0\]

Use the formula for the speed of light in a medium

The speed of light in a medium (\(v\)) is given by the equation: \[v = \frac{1}{\sqrt{\mu \epsilon}}\]

Use the formula for the refractive index

The refractive index (\(n\)) of a medium is defined as the ratio of the speed of light in vacuum (\(c\)) to the speed of light in the medium (\(v\)): \[n = \frac{c}{v}\]

Substitute the values for relative permeability and dielectric constant into the formula for the speed of light

We have the relative permeability and the dielectric constant in terms of \(\mu_r\) and \(K\). We can now substitute these values into the speed of light equation: \[v = \frac{1}{\sqrt{(\mu_r \mu_0)(K \epsilon_0)}}\]

Substitute the expression for the speed of light in the refractive index formula

Now substitute the expression for \(v\) from Step 4 into the refractive index formula: \[n = \frac{c}{\frac{1}{\sqrt{(\mu_r \mu_0)(K \epsilon_0)}}}\]

Simplify the expression for the refractive index

After simplifying, the expression for the refractive index becomes: \[n = \sqrt{(\mu_r \mu_0)(K \epsilon_0)}\] Comparing this expression to the given options in the exercise, we see that it does not match any of the options directly. Are any of the given options equivalent to this expression?

Check the given options

Let's analyze the given options: (A) \(\sqrt{(\mu_{\mathrm{T}} \mathrm{K})}\): This option is missing the permittivity (\(\epsilon_0\)) and permeability (\(\mu_0\)) of free space. (B) \(\sqrt{(\mu_{1} E_{0})}\): This option involves an electric field (\(E_0\)) which is not relevant in this situation, and it is also missing the dielectric constant (\(K\)). Given the provided options, none of them correctly represent the refractive index based on the relative permeability and dielectric constant.