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

Answer

To find the dimension of resistance in an electrical circuit in terms of mass (M), length (L), time (T), and current (I), we can start from Ohm's Law, which states:

\[ V = I \cdot R \]

Where:
- \( V \) is the voltage (potential difference),
- \( I \) is the current,
- \( R \) is the resistance.

### Step 1: Rearranging Ohm's Law
From Ohm's Law, we can express resistance \( R \) as:

\[ R = \frac{V}{I} \]

### Step 2: Expressing Voltage in Terms of Work and Charge
Voltage \( V \) can be defined as work done per unit charge:

\[ V = \frac{W}{Q} \]

Where:
- \( W \) is the work done,
- \( Q \) is the charge.

### Step 3: Substituting Charge in Terms of Current
Charge \( Q \) can be expressed in terms of current \( I \) and time \( t \):

\[ Q = I \cdot t \]

### Step 4: Substituting for Voltage
Now we can substitute \( Q \) back into the equation for voltage:

\[ V = \frac{W}{I \cdot t} \]

### Step 5: Substituting for Resistance
Now substituting this expression for \( V \) back into the equation for \( R \):

\[ R = \frac{W}{I \cdot t} \cdot \frac{1}{I} = \frac{W}{I^2 \cdot t} \]

### Step 6: Expressing Work in Terms of Basic Dimensions
Work \( W \) can be expressed as force multiplied by distance. The force can be expressed in terms of mass, length, and time:

\[ W = F \cdot d = (M \cdot a) \cdot L = M \cdot \frac{L}{T^2} \cdot L = M \cdot L^2 \cdot T^{-2} \]

### Step 7: Substituting Work into Resistance Equation
Now substituting \( W \) back into the equation for \( R \):

\[ R = \frac{M \cdot L^2 \cdot T^{-2}}{I^2 \cdot t} \]

### Step 8: Final Expression for Resistance
Now, substituting \( t \) in terms of \( T \):

\[ R = \frac{M \cdot L^2 \cdot T^{-2}}{I^2 \cdot T} = \frac{M \cdot L^2}{I^2 \cdot T^3} \]

### Step 9: Writing the Final Dimensions
Thus, the dimensions of resistance \( R \) in terms of \( M \), \( L \), \( T \), and \( I \) are:

\[ [R] = M^1 \cdot L^2 \cdot T^{-3} \cdot I^{-2} \]

### Final Answer
The dimension of resistance is:

\[ [R] = M^1 L^2 T^{-3} I^{-2} \]