Question

If refractive index and dielectric constant of the material of a glass slab is \(\frac{3}{2}\) and 3 respectively then magnetic permeability of the glass is (In \(\mathrm{Sl}\) system) (1) \(5.32 \times 10^{-7}\) (2) \(3.94 \times 10^{-7}\) (3) \(10^{-7}\) (4) \(9.42 \times 10^{-7}\)

Short answer

Option (4) \(9.42 \times 10^{-7}\)

Step-by-step solution

- Understand the given data

Identify the given values: The refractive index () is \(\frac{3}{2}\) and the dielectric constant (\(\epsilon_r\)) is 3.

- Relate refractive index to dielectric constant and magnetic permeability

Use the formula that relates the refractive index (), dielectric constant (\(\epsilon_r\)), and magnetic permeability (\(\mu_r\)): \[ n = \sqrt{\epsilon_r \mu_r} \]

- Solve for the magnetic permeability

Rearrange the equation to solve for the relative magnetic permeability (\(\mu_r\)): \[ \mu_r = \frac{n^2}{\epsilon_r} \]Substitute the given values: \[ \mu_r = \frac{(\frac{3}{2})^2}{3} = \frac{\frac{9}{4}}{3} = \frac{9}{12} = \frac{3}{4} = 0.75 \]

- Find absolute magnetic permeability

The absolute magnetic permeability (\(\mu\)) in the SI system is given by \[ \mu = \mu_r \cdot \mu_0 \]where \(\mu_0\) is the permeability of free space \[ \mu_0 = 4\pi \times 10^{-7} \]So \[ \mu = 0.75 \times 4\pi \times 10^{-7} \]

- Calculate the final value

Substitute \(\pi = 3.14\): \[ \mu = 0.75 \times 4 \times 3.14 \times 10^{-7} = 9.42 \times 10^{-7} \]Thus, the correct answer is (4) \(9.42 \times 10^{-7}\).

Key concepts

Refractive Index

The refractive index of a material is a measure of how much it can bend light. This is important in optics and is denoted by the symbol \(n\). The refractive index relates to the speed of light in a vacuum compared to the speed of light in the material.

Mathematically, it's given as: \[ n = \frac{c}{v} \] where \(c\) is the speed of light in a vacuum and \(v\) is the speed of light in the material. Refractive index affects how light rays change direction when entering different materials, producing effects like bending or slowing down of light in media such as glass or water.

Dielectric Constant

The dielectric constant, also known as relative permittivity, measures a material's ability to store electrical energy in an electric field. It is denoted as \( \epsilon_r \).

This property is essential in understanding how materials interact with electric fields. A higher dielectric constant means the material can store more electrical energy.

Dielectric constant can be calculated using the equation: \[ \epsilon_r = \frac{C_{material}}{C_{vacuum}} \] where \(C_{material}\) is the capacitance with the dielectric material and \(C_{vacuum}\) is the capacitance with a vacuum. This concept helps in designing capacitors and other electronic components.

Relative Magnetic Permeability

Relative magnetic permeability, indicated by \(\mu_r\), measures how much a material can support the formation of a magnetic field within itself. This relates to how materials respond to magnetic fields compared to a vacuum.

The formula is given by: \[ \mu_r = \frac{B}{H} \] where \(B\) is the magnetic flux density and \(H\) is the magnetic field strength.

In our exercise, we use the given dielectric constant and refractive index to find \(\mu_r\) through the relation: \[ n = \sqrt{\epsilon_r \mu_r} \] Rearranging to solve for \(\mu_r\): \[ \mu_r = \frac{n^2}{\epsilon_r} \] This approach helps calculate the magnetic properties of the material using its known optical and electrical properties.

Absolute Magnetic Permeability

Absolute magnetic permeability, denoted by \(\mu\), quantifies a material's ability to conduct a magnetic field. It's a product of the relative permeability \( \mu_r \) and the permeability of free space \( \mu_0 \).

Mathematically, this is written as: \[ \mu = \mu_r \cdot \mu_0 \] where \( \mu_0 \) is a physical constant approximately equal to \( 4\pi \times 10^{-7} \) V\·s/A\·m.

Using our previous result for \( \mu_r \), we calculate \( \mu \) as follows: \[ \mu = 0.75 \times 4\pi \times 10^{-7} = 9.42 \times 10^{-7} \] Thus, the absolute permeability guides us in understanding how the material influences magnetic fields in real-world applications.