`A` quantity` X` is given by `epsilon_(p) L(delta V)/(delta t)` , where `epsilon_(p)` is the permitivity of free space ,L is a length , `delta V` is a potential diffrence and ` deltat ` is a time interval . The dimensional formula for `X` is the seme as that of

Options

A. resistance
B. charge
C. voltage
D. current

Explanation

To solve the problem, we need to find the dimensional formula for the quantity \( X \) given by the expression:

\[ X = \frac{\epsilon_p L \Delta V}{\Delta t} \]

where:
- \( \epsilon_p \) is the permittivity of free space,
- \( L \) is a length,
- \( \Delta V \) is a potential difference,
- \( \Delta t \) is a time interval.

### Step 1: Determine the dimensional formula for each component

1. **Permittivity of free space (\( \epsilon_p \))**:
The permittivity of free space has the dimensional formula:
\[ [\epsilon_p] = \frac{[M^{-1} L^{-3} T^4 A^2]}{[L^2]} = [M^{-1} L^{-3} T^4 A^2] \]

2. **Length (\( L \))**:
The dimensional formula for length is:
\[ [L] = [L] \]

3. **Potential difference (\( \Delta V \))**:
The dimensional formula for electric potential difference is:
\[ [\Delta V] = [M L^2 T^{-3} A^{-1}] \]

4. **Time interval (\( \Delta t \))**:
The dimensional formula for time is:
\[ [\Delta t] = [T] \]

### Step 2: Substitute the dimensional formulas into the expression for \( X \)

Now we substitute these dimensional formulas into the expression for \( X \):

\[ X = \frac{[\epsilon_p] \cdot [L] \cdot [\Delta V]}{[\Delta t]} = \frac{[M^{-1} L^{-3} T^4 A^2] \cdot [L] \cdot [M L^2 T^{-3} A^{-1}]}{[T]} \]

### Step 3: Simplify the expression

Now we can simplify the expression step by step:

1. Combine the dimensions in the numerator:
\[ [\epsilon_p] \cdot [L] \cdot [\Delta V] = [M^{-1} L^{-3} T^4 A^2] \cdot [L] \cdot [M L^2 T^{-3} A^{-1}] \]

This results in:
\[ = [M^{-1} L^{-3} T^4 A^2] \cdot [M] \cdot [L^2] \cdot [T^{-3}] \cdot [A^{-1}] \]
\[ = [M^{0} L^{-3 + 1 + 2} T^{4 - 3} A^{2 - 1}] = [L^{0} T^{1} A^{1}] = [T A] \]

2. Now divide by the dimension of time:
\[ X = \frac{[T A]}{[T]} = [A] \]

### Conclusion

The dimensional formula for \( X \) is:
\[ [X] = [A] \]

This indicates that the dimensional formula for \( X \) is the same as that of electric current.

### Final Answer

The dimensional formula for \( X \) is the same as that of current \( I \).

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