If C be the capacitance and V be the electric potential, then the dimensional formula of `CV^(2)` is

Options

A. `[ML^(-3)TA]`
B. `[K^0LT^(-2)A^(0)]`
C. `[ML^(1)T^(-2)A^(-1)]`
D. `[ML^(2)T^(-2)A^(0)]`

Answer

To find the dimensional formula of \( CV^2 \), where \( C \) is the capacitance and \( V \) is the electric potential, we will follow these steps:

### Step 1: Determine the Dimensional Formula of Capacitance \( C \)
The dimensional formula for capacitance \( C \) is given as:
\[ [C] = M^{-1} L^{-2} T^{-4} A^{2} \]

### Step 2: Determine the Dimensional Formula of Electric Potential \( V \)
The dimensional formula for electric potential \( V \) is given as:
\[ [V] = M L^{2} T^{-3} A^{-1} \]

### Step 3: Calculate the Dimensional Formula of \( V^2 \)
Now, we need to find the dimensional formula for \( V^2 \):
\[ [V^2] = (M L^{2} T^{-3} A^{-1})^2 = M^{2} L^{4} T^{-6} A^{-2} \]

### Step 4: Combine the Dimensional Formulas of \( C \) and \( V^2 \)
Now, we will combine the dimensional formulas of \( C \) and \( V^2 \):
\[ [CV^2] = [C] \times [V^2] = (M^{-1} L^{-2} T^{-4} A^{2}) \times (M^{2} L^{4} T^{-6} A^{-2}) \]

### Step 5: Perform the Multiplication
Now, we will multiply the dimensions together:
- For mass \( M \):
\[ M^{-1} \times M^{2} = M^{1} \]
- For length \( L \):
\[ L^{-2} \times L^{4} = L^{2} \]
- For time \( T \):
\[ T^{-4} \times T^{-6} = T^{-10} \]
- For current \( A \):
\[ A^{2} \times A^{-2} = A^{0} \]

### Step 6: Combine the Results
Now, we can combine all the results:
\[ [CV^2] = M^{1} L^{2} T^{-10} A^{0} \]

### Final Result
Thus, the dimensional formula of \( CV^2 \) is:
\[ [CV^2] = M^{1} L^{2} T^{-10} \]