If `L, R , C , and V`, respectively , represent inductance , resistance , capacitance and potential difference , then the dimensions of `L //RCV` are the same as those of

Options

A. a. Charge
B. b. `1//"Charge"`
C. c. Current
D. d. `1//"Current"`

Explanation

To find the dimensions of \( \frac{L}{R \cdot C \cdot V} \), where \( L \), \( R \), \( C \), and \( V \) represent inductance, resistance, capacitance, and potential difference respectively, we will follow these steps:

### Step 1: Write down the dimensions of each quantity.
1. **Inductance \( L \)**:
\[ [L] = M L^2 I^{-2} T^{-2} \]

2. **Resistance \( R \)**:
\[ [R] = M L^2 I^{-2} T^{-3} \]

3. **Capacitance \( C \)**:
\[ [C] = M^{-1} L^{-2} I^2 T^4 \]

4. **Potential Difference \( V \)**:
\[ [V] = M L^2 I^{-1} T^{-3} \]

### Step 2: Substitute the dimensions into the expression \( \frac{L}{R \cdot C \cdot V} \).
We need to calculate:
\[ \frac{[L]}{[R] \cdot [C] \cdot [V]} \]

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

### Step 3: Simplify the denominator.
Calculating the denominator:
1. \( [R] \cdot [C] \cdot [V] \):
\[ = (M L^2 I^{-2} T^{-3}) \cdot (M^{-1} L^{-2} I^2 T^4) \cdot (M L^2 I^{-1} T^{-3}) \]

- Combine the dimensions:
\[ = M^1 \cdot M^{-1} \cdot M^1 \cdot L^2 \cdot L^{-2} \cdot L^2 \cdot I^{-2} \cdot I^2 \cdot I^{-1} \cdot T^{-3} \cdot T^4 \cdot T^{-3} \]
- This simplifies to:
\[ = M^1 L^2 I^{-1} T^{-2} \]

### Step 4: Substitute back into the expression.
Now substituting back into the expression:
\[ \frac{M L^2 I^{-2} T^{-2}}{M L^2 I^{-1} T^{-2}} = \frac{M L^2 I^{-2} T^{-2}}{M L^2 I^{-1} T^{-2}} \]

### Step 5: Cancel out the common terms.
- Cancel \( M \), \( L^2 \), and \( T^{-2} \):
\[ = I^{-2} \cdot I^{1} = I^{-1} \]

### Step 6: Final result.
Thus, the dimensions of \( \frac{L}{R \cdot C \cdot V} \) are:
\[ = I^{-1} \]

This means the dimensions of \( \frac{L}{R \cdot C \cdot V} \) are the same as those of \( \frac{1}{\text{current}} \).

### Conclusion:
The correct answer is \( \frac{1}{\text{current}} \).

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