Question

Let the inductance and resistance be denoted by \( L \) and \( R \) respectively. The dimensions of \( \left(\dfrac{L}{R}\right) \) are

• A. \( [L^1 M^0 T^1] \)
• B. \( [L^0 M^0 T^0] \)
• C. \( [L^0 M^1 T^0] \)
• D. \( [L^0 M^0 T^1] \)

Hint:

The quantity \( \dfrac{L}{R} \) represents a time constant in electrical circuits.

Answer 1

The Correct Option is D

Step 1: Write dimensional formula of inductance.
Inductance is defined using the relation \[ V = L \frac{dI}{dt} \] Thus, \[ [L] = \frac{[V][T]}{[I]} \] Since \[ [V] = [M L^2 T^{-3} I^{-1}] \] \[ [L] = [M L^2 T^{-2} I^{-2}] \]
Step 2: Write dimensional formula of resistance.
Resistance is given by \[ R = \frac{V}{I} \] \[ [R] = [M L^2 T^{-3} I^{-2}] \]
Step 3: Find dimensions of \( \dfrac{L}{R} \).
\[ \frac{[L]}{[R]} = \frac{[M L^2 T^{-2} I^{-2}]}{[M L^2 T^{-3} I^{-2}]} = [T] \]
Step 4: Conclusion.
The dimensional formula of \( \dfrac{L}{R} \) is \( [L^0 M^0 T^1] \).