Dimensions of electrical resistence are

Options

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

Answer

To find the dimensions of electrical resistance, we start with the relationship between power (P), current (I), and resistance (R). The formula we will use is:

\[ P = I^2 R \]

### Step 1: Express Power in terms of Work and Time
Power can also be defined as work done per unit time:

\[ P = \frac{W}{t} \]

Where:
- \( W \) is work done,
- \( t \) is time.

### Step 2: Substitute Work in terms of Force and Displacement
Work done can be expressed as the product of force (F) and displacement (d):

\[ W = F \cdot d \]

### Step 3: Substitute Force in terms of Mass and Acceleration
Force can be expressed using Newton's second law:

\[ F = m \cdot a \]

Where:
- \( m \) is mass,
- \( a \) is acceleration.

### Step 4: Combine the Equations
Substituting the expression for work into the power equation gives:

\[ P = \frac{F \cdot d}{t} = \frac{(m \cdot a) \cdot d}{t} \]

### Step 5: Substitute Acceleration
Acceleration can be expressed as:

\[ a = \frac{d}{t^2} \]

So, substituting this into the equation gives:

\[ P = \frac{(m \cdot \frac{d}{t^2}) \cdot d}{t} = \frac{m \cdot d^2}{t^3} \]

### Step 6: Equate the Two Expressions for Power
Now we have two expressions for power:

\[ \frac{m \cdot d^2}{t^3} = I^2 R \]

### Step 7: Rearranging for Resistance
Rearranging this equation to solve for resistance \( R \):

\[ R = \frac{m \cdot d^2}{t^3 \cdot I^2} \]

### Step 8: Substitute Dimensions
Now we substitute the dimensions for each variable:
- Mass \( [M] \) = \( m \)
- Length \( [L] \) = \( d \)
- Time \( [T] \) = \( t \)
- Current \( [I] \) = \( A \)

Thus, we have:

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

### Step 9: Final Dimensions of Resistance
This simplifies to:

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

So, the dimensions of electrical resistance are:

\[ [M][L^2][T^{-3}][A^{-2}] \]

### Conclusion
The correct option for the dimensions of electrical resistance is:

\[ [M][L^2][T^{-3}][A^{-2}] \]