Dimension of resistance in an elecatrical circuit, in terms of dimension of mass `M`, of length `L`, of time `T`, and of current `I`, would be

Options

A. `[ML^(2)T^(-3)l^(-1)]`
B. `[ML^(2)T^(-2)]`
C. `[ML^(2)T^(-1)l^(-1)]`
D. `[ML^(2)T^(-3)l^(-2)]`

Explanation

To find the dimension of resistance in an electrical circuit in terms of mass (M), length (L), time (T), and current (I), we can follow these steps:

### Step 1: Understand the formula for resistance
Resistance (R) is defined as the ratio of voltage (V) to current (I):
\[ R = \frac{V}{I} \]

### Step 2: Express voltage in terms of work done and charge
Voltage (V) can be defined as work done (W) per unit charge (Q):
\[ V = \frac{W}{Q} \]

### Step 3: Substitute the expression for voltage into the resistance formula
Substituting the expression for voltage into the resistance formula gives:
\[ R = \frac{W/Q}{I} \]
This can be rearranged to:
\[ R = \frac{W}{Q \cdot I} \]

### Step 4: Write the dimensions of work and charge
1. **Work (W)** has the dimension of energy, which is given by:
\[ W = \text{Force} \times \text{Distance} = (M \cdot L \cdot T^{-2}) \cdot L = M L^2 T^{-2} \]
So, the dimension of work is:
\[ [W] = M^1 L^2 T^{-2} \]

2. **Charge (Q)** can be expressed in terms of current (I) and time (T):
\[ Q = I \cdot T \]
Thus, the dimension of charge is:
\[ [Q] = I^1 T^1 \]

### Step 5: Substitute the dimensions into the resistance formula
Now we can substitute the dimensions of work and charge into the resistance formula:
\[ R = \frac{M^1 L^2 T^{-2}}{I^1 T^1 \cdot I^1} \]
This simplifies to:
\[ R = \frac{M^1 L^2 T^{-2}}{I^2 T^1} \]

### Step 6: Simplify the expression
Now, we can simplify the expression:
\[ R = M^1 L^2 T^{-2} \cdot T^{-1} \cdot I^{-2} \]
\[ R = M^1 L^2 T^{-3} I^{-2} \]

### Final Result
Thus, the dimension of resistance (R) in terms of mass (M), length (L), time (T), and current (I) is:
\[ [R] = M^1 L^2 T^{-3} I^{-2} \]