Two wire A and B whose resistivity length area potential and current are `rho, L, A,V,I` and `rho, 2L, A/2, V resp. Find current in wire B

Answer

To solve the problem of finding the current in wire B, we will follow these steps:

### Step 1: Understand the parameters of wire A
- Resistivity (ρ) = ρ
- Length (L) = L
- Cross-sectional area (A) = A
- Potential difference (V) = V
- Current (I) = I

### Step 2: Calculate the resistance of wire A
Using the formula for resistance:
\[ R_A = \frac{\rho L}{A} \]

### Step 3: Apply Ohm's Law to find current in wire A
According to Ohm's Law:
\[ V = I \cdot R \]
Thus, the current \( I \) in wire A can be expressed as:
\[ I = \frac{V}{R_A} = \frac{V}{\frac{\rho L}{A}} = \frac{V A}{\rho L} \]

### Step 4: Understand the parameters of wire B
- Resistivity (ρ) = ρ (same material, so same resistivity)
- Length (L) = 2L
- Cross-sectional area (A) = A/2
- Potential difference (V) = V (same potential)

### Step 5: Calculate the resistance of wire B
Using the resistance formula again:
\[ R_B = \frac{\rho (2L)}{A/2} = \frac{2\rho L}{A/2} = \frac{4\rho L}{A} \]

### Step 6: Apply Ohm's Law to find current in wire B
Using Ohm's Law for wire B:
\[ I_B = \frac{V}{R_B} = \frac{V}{\frac{4\rho L}{A}} = \frac{V A}{4 \rho L} \]

### Step 7: Relate current in wire B to current in wire A
From Step 3, we know:
\[ I = \frac{V A}{\rho L} \]
Thus, we can express \( I_B \) as:
\[ I_B = \frac{I}{4} \]

### Conclusion
The current in wire B is:
\[ I_B = \frac{I}{4} \]