A quantity `X` is given by `epsilon_(0) L(DeltaV)/(Deltat)`, where `epsilon_(0)` is the permittivity of free space `L` is a length `DeltaV` is a potnetial difference and `Delta` is a time internval. The dimensional forumla to `X` is the same as that of

Options

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

Answer

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

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

we will analyze each component of the expression step by step.

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

1. **Permittivity of free space (\( \epsilon_0 \))**:
The dimensional formula for permittivity of free space is given by:
\[ [\epsilon_0] = \frac{[M^{-1} L^{-3} T^4 A^2]}{[L^2]} = [M^{-1} L^{-3} T^4 A^2] \]

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

3. **Potential difference (\( \Delta V \))**:
The potential difference can be expressed in terms of electric field and distance. The dimensional formula for voltage (or 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: Combine the dimensions

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

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

Substituting the dimensions we found:

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

### Step 3: Simplify the expression

Now, we simplify the expression step by step:

1. Multiply the dimensions in the numerator:
\[ [M^{-1} L^{-3} T^4 A^2] \cdot [L] \cdot [M L^2 T^{-3} A^{-1}] = [M^{-1} L^{-3+1+2} T^{4-3} A^{2-1}] \]
This simplifies to:
\[ [M^{0} L^{0} T^{1} A^{1}] = [T A] \]

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

### Conclusion

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

\[ X = [A] \]

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

### Final Answer

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

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