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^(2)t)`
C. `R=V^(2)P`
D. R = Vl

Answer

To find the dimensionally correct expression for resistance \( R \) among the given options involving electric power \( P \), electric current \( I \), time \( t \), voltage \( V \), and electric energy \( E \), we will analyze each option step by step.

### Step 1: Understand the Dimensions of Each Quantity
1. **Electric Power \( P \)**:
- The dimension of power is given by:
\[ [P] = [ML^2T^{-3}] \]

2. **Electric Current \( I \)**:
- The dimension of current is:
\[ [I] = [A] \]

3. **Time \( t \)**:
- The dimension of time is:
\[ [t] = [T] \]

4. **Voltage \( V \)**:
- Voltage is defined as work done per unit charge. The dimension of work done is:
\[ [W] = [ML^2T^{-2}] \]
- The dimension of charge \( Q \) is:
\[ [Q] = [A \cdot T] \]
- Therefore, the dimension of voltage is:
\[ [V] = \frac{[W]}{[Q]} = \frac{[ML^2T^{-2}]}{[A \cdot T]} = [ML^2T^{-3}A^{-1}] \]

5. **Electric Energy \( E \)**:
- The dimension of energy is the same as work done:
\[ [E] = [ML^2T^{-2}] \]

6. **Resistance \( R \)**:
- Resistance is defined as voltage per unit current:
\[ [R] = \frac{[V]}{[I]} = \frac{[ML^2T^{-3}A^{-1}]}{[A]} = [ML^2T^{-3}A^{-2}] \]

### Step 2: Analyze Each Option
Now we will analyze each option to determine if it has the same dimensions as resistance.

#### Option 1: \( R = \sqrt{PI} \)
- Dimensions:
\[ [P] = [ML^2T^{-3}], \quad [I] = [A] \]
\[ [R] = \sqrt{[ML^2T^{-3}][A]} = \sqrt{[ML^2T^{-3}A]} = [M^{1/2}L^{1}T^{-3/2}A^{1/2}] \]
- This does not match \( [ML^2T^{-3}A^{-2}] \).

#### Option 2: \( R = \frac{E}{I^2 t} \)
- Dimensions:
\[ [E] = [ML^2T^{-2}], \quad [I^2] = [A^2], \quad [t] = [T] \]
\[ [R] = \frac{[ML^2T^{-2}]}{[A^2][T]} = \frac{[ML^2T^{-2}]}{[A^2T]} = [ML^2T^{-3}A^{-2}] \]
- This matches the dimension of resistance.

#### Option 3: \( R = \frac{V^2}{P} \)
- Dimensions:
\[ [V^2] = [ML^2T^{-3}A^{-1}]^2 = [M^2L^4T^{-6}A^{-2}] \]
\[ [R] = \frac{[M^2L^4T^{-6}A^{-2}]}{[ML^2T^{-3}]} = [ML^2T^{-3}A^{-2}] \]
- This does not match the dimensions of resistance.

#### Option 4: \( R = VI \)
- Dimensions:
\[ [R] = [V][I] = [ML^2T^{-3}A^{-1}][A] = [ML^2T^{-3}] \]
- This does not match the dimensions of resistance.

### Conclusion
The only option that matches the dimensions of resistance is:
\[ \text{Option 2: } R = \frac{E}{I^2 t} \]