Question

A quantity \( X \) is given by: \[ X = \frac{\epsilon_0 L \Delta V}{\Delta t} \] where:
- \( \epsilon_0 \) is the permittivity of free space,
- \( L \) is the length,
- \( \Delta V \) is the potential difference,
- \( \Delta t \) is the time interval.
The dimension of \( X \) is the same as that of:

• A. Resistance
• B. Charge
• C. Voltage
• D. Current

Hint:

For small angle approximations, remember that \( \cos(x) \approx 1 - \frac{x^2}{2} \) and \( \tan(x) \approx x \). These approximations are very useful for solving limits involving trigonometric functions as \( x \to 0 \).

Answer 1

The Correct Option is D

We need to determine the dimension of \( X \) and compare it with the dimensions of the given options. The formula for \( X \) is: \[ X = \frac{\epsilon_0 L \Delta V}{\Delta t} \]
Step 1: Analyzing the units and dimensions 1. \( \epsilon_0 \) (Permittivity of free space): The dimension of \( \epsilon_0 \) is: \[ [\epsilon_0] = \frac{\text{Coulomb}^2}{\text{Newton} \cdot \text{meter}^2} = \frac{\text{A}^2 \cdot \text{s}^4}{\text{kg} \cdot \text{m}^3} \] 2. \( L \) (Length): The dimension of length is: \[ [L] = \text{m} \] 3. \( \Delta V \) (Potential difference): The dimension of potential difference (Voltage) is: \[ [\Delta V] = \frac{\text{Joule}}{\text{Coulomb}} = \frac{\text{kg} \cdot \text{m}^2}{\text{s}^3 \cdot \text{A}} \] 4. \( \Delta t \) (Time interval): The dimension of time is: \[ [\Delta t] = \text{s} \]
Step 2: Determining the dimension of \( X \) Substitute the dimensions of each quantity into the formula for \( X \): \[ [X] = \frac{[\epsilon_0] \cdot [L] \cdot [\Delta V]}{[\Delta t]} \] \[ [X] = \frac{\left(\frac{\text{A}^2 \cdot \text{s}^4}{\text{kg} \cdot \text{m}^3}\right) \cdot \text{m} \cdot \left(\frac{\text{kg} \cdot \text{m}^2}{\text{s}^3 \cdot \text{A}}\right)}{\text{s}} \] Simplifying the expression: \[ [X] = \frac{\text{A}^2 \cdot \text{s}^4 \cdot \text{m}^3 \cdot \text{kg}}{\text{kg} \cdot \text{m}^3 \cdot \text{s}^3 \cdot \text{A} \cdot \text{s}} \] Cancel out common units: \[ [X] = \text{A} \cdot \text{s} \] Thus, the dimension of \( X \) is: \[ [X] = \text{Current} \cdot \text{Time} \]
Step 3: Conclusion The correct answer is: \[ \boxed{(D) \text{Current}} \]