Dimensional formula of the physical quantity, resistance is

Options

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

Answer

To find the dimensional formula of resistance, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Resistance**: According to Ohm's law, resistance (R) is defined as the ratio of the potential difference (V) across a conductor to the current (I) flowing through it. Mathematically, this is expressed as:
\[ R = \frac{V}{I} \]

2. **Finding the Dimensional Formula of Potential Difference (V)**:
- The potential difference (V) can be related to the electric field (E) and the distance (d) as:
\[ V = E \cdot d \]
- The electric field (E) can be expressed in terms of force (F) and charge (Q):
\[ E = \frac{F}{Q} \]
- Therefore, substituting for E, we have:
\[ V = \frac{F}{Q} \cdot d \]

3. **Substituting the Dimensions**:
- The dimensional formula for force (F) is given by:
\[ F = m = [M L T^{-2}] \]
- The dimensional formula for charge (Q) can be expressed as:
\[ Q = I \cdot t \quad \text{(where I is current and t is time)} \]
- Thus, the dimensional formula for charge is:
\[ [Q] = [I][T] = [A][T] \]
- Now substituting these into the expression for V:
\[ V = \frac{F \cdot d}{Q} = \frac{[M L T^{-2}] \cdot [L]}{[A][T]} = \frac{[M L^2 T^{-2}]}{[A][T]} \]
- Simplifying this gives:
\[ [V] = [M L^2 T^{-3} A^{-1}] \]

4. **Finding the Dimensional Formula for Current (I)**:
- The dimensional formula for current (I) is:
\[ [I] = [A] \]

5. **Finding the Dimensional Formula for Resistance (R)**:
- Now substituting the dimensional formulas of V and I into the formula for resistance:
\[ [R] = \frac{[V]}{[I]} = \frac{[M L^2 T^{-3} A^{-1}]}{[A]} = [M L^2 T^{-3} A^{-2}] \]

### Final Answer:
The dimensional formula for resistance is:
\[ [M L^2 T^{-3} A^{-2}] \]

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