The dimensionally correct expression for the resistance R among the following is
[P = electric power, l = electric current, t = time, V = volatge and E = electric energy]

Options

A. `R=sqrt(Pl)`
B. `R=E/(l^2t)`
C. `R=V^2P`
D. R=Vl

Answer

To find the dimensionally correct expression for resistance \( R \), we need to analyze the dimensions of the given quantities: electric power \( P \), electric current \( I \), time \( t \), voltage \( V \), and electric energy \( E \).

### Step-by-Step Solution:

1. **Understand the dimensions of resistance**:
The resistance \( R \) has the dimensions given by:
\[ [R] = [M^1 L^2 T^{-3} I^2] \]
where \( M \) is mass, \( L \) is length, \( T \) is time, and \( I \) is electric current.

2. **Identify the dimensions of electric power \( P \)**:
Electric power is defined as:
\[ P = VI \]
The dimensions of voltage \( V \) can be expressed as:
\[ [V] = [M^1 L^2 T^{-3} I^{-1}] \]
Therefore, the dimensions of power \( P \) become:
\[ [P] = [V][I] = [M^1 L^2 T^{-3} I^{-1}][I] = [M^1 L^2 T^{-3}] \]

3. **Identify the dimensions of electric energy \( E \)**:
Electric energy is defined as:
\[ E = Pt \]
Thus, the dimensions of energy \( E \) are:
\[ [E] = [P][T] = [M^1 L^2 T^{-3}][T] = [M^1 L^2 T^{-2}] \]

4. **Check the dimensions of the given expressions**:
- **Option 1**: \( R = \sqrt{PI} \)
- Dimensions:
\[ [R] = [\sqrt{[P][I]}] = [\sqrt{[M^1 L^2 T^{-3}][I]}] = [\sqrt{M^1 L^2 T^{-3} I^1}] = [M^{1/2} L^{1} T^{-3/2} I^{1/2}] \]
This does not match the dimensions of resistance.

- **Option 2**: \( R = \frac{E}{I^2 t} \)
- Dimensions:
\[ [R] = \frac{[E]}{[I^2][t]} = \frac{[M^1 L^2 T^{-2}]}{[I^2][T^1]} = \frac{[M^1 L^2 T^{-2}]}{[I^2][T^1]} = [M^1 L^2 T^{-3} I^2] \]
This matches the dimensions of resistance.

5. **Conclusion**:
The dimensionally correct expression for resistance \( R \) is given by:
\[ R = \frac{E}{I^2 t} \]
Therefore, the correct answer is **Option 2**.