a quantity `X` is given by `epsilon_(0)L(DeltaV)/(Deltat)` where `in_(0)` is the permittivity of the free space, L is a length, `DeltaV` is a potential difference and `Deltat` is a time interval. The dimensinal formula for `X` is the same 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 equation:

\[ X = \frac{\epsilon_0 L \Delta V}{\Delta t} \]

where:
- \( \epsilon_0 \) 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 \( \epsilon_0 \)**:
The dimensional formula for \( \epsilon_0 \) can be derived from the formula for capacitance:
\[ C = \frac{\epsilon_0 A}{d} \]
where \( A \) is the area and \( d \) is the distance. The dimensions of capacitance \( C \) are \( [C] = [M^{-1} L^{-2} T^4 A^2] \).

Since \( A \) has dimensions \( [L^2] \) and \( d \) has dimensions \( [L] \), we can rearrange:
\[ \epsilon_0 = \frac{C \cdot d}{A} = \frac{[M^{-1} L^{-2} T^4 A^2] \cdot [L]}{[L^2]} = [M^{-1} L^{-1} T^4 A^2] \]

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

3. **Potential difference \( \Delta V \)**:
The dimensional formula for voltage (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 equation for \( X \).

Now substituting the dimensional formulas into the equation for \( X \):
\[ X = \frac{\epsilon_0 L \Delta V}{\Delta t} \]

Substituting the dimensions:
\[ X = \frac{[M^{-1} L^{-1} T^4 A^2] \cdot [L] \cdot [M L^2 T^{-3} A^{-1}]}{[T]} \]

### Step 3: Simplify the expression.

Now we simplify:
\[ X = \frac{[M^{-1} L^{-1} T^4 A^2] \cdot [L^3] \cdot [M T^{-3} A^{-1}]}{[T]} \]
\[ = \frac{[M^0 L^3 T^{4-3} A^{2-1}]}{[T]} = [L^3 T^{1} A^{1}] \]

### Step 4: Identify the dimensions.

The dimensional formula we derived is:
\[ X = [L^3 T^{1} A^{1}] \]

### Step 5: Compare with known quantities.

Now we compare this with the known quantities:
- **Resistance**: \( [M L^2 T^{-3} A^{-2}] \)
- **Charge**: \( [A T] \)
- **Voltage**: \( [M L^2 T^{-3} A^{-1}] \)
- **Current**: \( [A] \)

The dimensional formula \( [L^3 T^{1} A^{1}] \) does not match any of the standard quantities directly, but we can see that it relates to the current when considering charge per unit time.

### Conclusion:

Thus, the dimensional formula for \( X \) is the same as that of **current**.