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 difference and ` deltat ` is a time interval . The dimensional formula for `X` is the same as that of

Options

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

Explanation

To find the dimensional formula for the quantity \( X \) given by the expression

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

we will break down each component in the expression and then combine them to find the overall dimensional formula for \( X \).

### Step 1: Identify the dimensions of each component

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

2. **Length (\( L \))**:
The dimensional formula for length is:
\[ [L] = L^{1} \]

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

4. **Time interval (\( \Delta t \))**:
The dimensional formula for time is:
\[ [\Delta t] = T^{1} \]

### Step 2: Combine the dimensions

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

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

Substituting in the values we found:

\[ [X] = \frac{(M^{-1} L^{-3} T^{4} A^{2}) \cdot (L^{1}) \cdot (M L^{2} T^{-3} A^{-1})}{T^{1}} \]

### Step 3: Simplify the expression

Now we will combine the dimensions in the numerator and then divide by the dimensions in the denominator:

1. **Numerator**:
\[ M^{-1} L^{-3} T^{4} A^{2} \cdot L^{1} \cdot M L^{2} T^{-3} A^{-1} = M^{0} L^{0} T^{1} A^{1} \]

- For mass \( M \): \( -1 + 1 = 0 \)
- For length \( L \): \( -3 + 1 + 2 = 0 \)
- For time \( T \): \( 4 - 3 = 1 \)
- For current \( A \): \( 2 - 1 = 1 \)

2. **Denominator**:
\[ T^{1} \]

Now we can combine:

\[ [X] = \frac{M^{0} L^{0} T^{1} A^{1}}{T^{1}} = M^{0} L^{0} A^{1} = A^{1} \]

### Conclusion

Thus, the dimensional formula for \( X \) is:

\[ [X] = A^{1} \]

This means that the quantity \( X \) has the same dimensional formula as that of electric current.