If E denotes of electric field, the dimension of a quantity `in_(0)(dE)/(dt)` are those of

Options

A. current
B. current density
C. electric potential
D. electric flux

Explanation

To solve the problem, we need to determine the dimensions of the quantity \( \epsilon_0 \frac{dE}{dt} \), where \( \epsilon_0 \) is the permittivity of free space and \( E \) is the electric field.

### Step-by-Step Solution:

1. **Identify the Dimensions of \( \epsilon_0 \)**:
The permittivity of free space \( \epsilon_0 \) has the dimensions of:
\[ [\epsilon_0] = \frac{C^2}{N \cdot m^2} = \frac{C^2}{(kg \cdot m/s^2) \cdot m^2} = \frac{C^2 \cdot s^2}{kg \cdot m^3} \]
Thus, the dimensions of \( \epsilon_0 \) are:
\[ [\epsilon_0] = \frac{M^{-1} \cdot L^{-3} \cdot T^4 \cdot I^2}{1} \]

2. **Identify the Dimensions of Electric Field \( E \)**:
The electric field \( E \) is defined as force per unit charge:
\[ [E] = \frac{N}{C} = \frac{kg \cdot m/s^2}{C} = \frac{M \cdot L}{T^2 \cdot I} \]

3. **Differentiate Electric Field with Respect to Time**:
We need to find the dimensions of \( \frac{dE}{dt} \):
\[ \left[\frac{dE}{dt}\right] = \frac{[E]}{[T]} = \frac{M \cdot L}{T^2 \cdot I \cdot T} = \frac{M \cdot L}{T^3 \cdot I} \]

4. **Combine the Dimensions**:
Now we can find the dimensions of \( \epsilon_0 \frac{dE}{dt} \):
\[ \left[\epsilon_0 \frac{dE}{dt}\right] = [\epsilon_0] \cdot \left[\frac{dE}{dt}\right] \]
Substituting the dimensions we found:
\[ \left[\epsilon_0 \frac{dE}{dt}\right] = \left(\frac{M^{-1} \cdot L^{-3} \cdot T^4 \cdot I^2}{1}\right) \cdot \left(\frac{M \cdot L}{T^3 \cdot I}\right) \]
Simplifying this:
\[ = \frac{M^{-1} \cdot L^{-3} \cdot T^4 \cdot I^2 \cdot M \cdot L}{T^3 \cdot I} = \frac{L^{-2} \cdot T \cdot I}{1} \]

5. **Final Dimensions**:
Therefore, the dimensions of \( \epsilon_0 \frac{dE}{dt} \) are:
\[ [\epsilon_0 \frac{dE}{dt}] = \frac{L^{-2} \cdot T \cdot I}{1} \]

### Conclusion:
The dimensions of the quantity \( \epsilon_0 \frac{dE}{dt} \) are those of current density, which can be expressed as \( [J] = \frac{I}{L^2} \).