Question

In terms of resistance $R$ and time $T$, the dimensions of ratio $\frac{\mu}{\epsilon}$ of the permeability ${\mu}$ and permittivity ${\epsilon}$ is :

• A. $[RT^{-2}]$
• B. $[R^2 T^{-1}]$
• C. $[R^2]$
• D. $[R^2 T^2]$

Answer 1

The Correct Option is C

To find the dimensions of the ratio $\frac{\mu}{\epsilon}$, we need to analyze the dimensions of both permeability $\mu$ and permittivity $\epsilon$, and then take their ratio.

1. **Permeability (\(\mu\))**:

• The permeability of a medium is defined from the relation \( B = \mu H \), where \( B \) is the magnetic flux density and \( H \) is the magnetic field strength.
• The dimensional formula for magnetic flux density \( B \) is \([M T^{-2} A^{-1}]\).
• The dimensional formula for magnetic field strength \( H \) is \([A L^{-1}]\).
• Thus, the dimensional formula for permeability \(\mu\) is:
  \[ \mu = \frac{B}{H} = \frac{[M T^{-2} A^{-1}]}{[A L^{-1}]} = [M L T^{-2} A^{-2}] \]

2. **Permittivity (\(\epsilon\))**:

• The permittivity of a medium is from the relation \( E = \frac{D}{\epsilon} \), where \( E \) is the electric field, and \( D \) is the electric displacement field.
• The dimensional formula for electric field \( E \) is \([M L T^{-3} A^{-1}]\).
• The dimensional formula for electric displacement \( D \) is \([A T L^{-2}]\).
• Thus, the dimensional formula for permittivity \(\epsilon\) is:
  \[ \epsilon = \frac{D}{E} = \frac{[A T L^{-2}]}{[M L T^{-3} A^{-1}]} = [M^{-1} L^{-3} T^4 A^2] \]

3. **Ratio \(\frac{\mu}{\epsilon}\):**

• The dimensional formula for the ratio is:
  \[ \frac{\mu}{\epsilon} = \frac{[M L T^{-2} A^{-2}]}{[M^{-1} L^{-3} T^4 A^2]} \]
  \[ = [M^2 L^4 T^{-6} A^{-4}] \]

4. **Express this in terms of resistance (\(R\)) and time (\(T\))**:

• From Ohm’s law, resistance \( R = \frac{[M L^2 T^{-3} A^{-2}]}{[A]} = [M L^2 T^{-3} A^{-2}]\).
• The square of resistance \( R^2 = [M^2 L^4 T^{-6} A^{-4}]\).
• Hence, the dimensions of \(\frac{\mu}{\epsilon}\) in terms of resistance is:
  \[ [R^2] \]

Thus, among the given options, the correct dimension of the ratio \(\frac{\mu}{\epsilon}\) in terms of resistance \( R \) and time \( T \) is \( [R^2] \).