An electric current is passed through a circuit containing two wires of the same material, connected in parallel. If the lengths and radii are in the ratio of 4/3 and 2/3, then the ratio of the current passing through the wires will be

Options

A. `8//9`
B. `1//3`
C. 3
D. 2

Explanation

To find the ratio of the current passing through two wires connected in parallel, we can follow these steps:

### Step 1: Understand the relationship between current and resistance in parallel circuits.
In a parallel circuit, the voltage across each component is the same. According to Ohm's law, the current through a resistor is given by:
\[ I = \frac{V}{R} \]
Where:
- \( I \) is the current,
- \( V \) is the voltage,
- \( R \) is the resistance.

### Step 2: Write the expressions for the resistances of the two wires.
The resistance \( R \) of a wire can be calculated using the formula:
\[ R = \frac{\rho L}{A} \]
Where:
- \( \rho \) is the resistivity of the material,
- \( L \) is the length of the wire,
- \( A \) is the cross-sectional area of the wire.

For a wire with radius \( r \), the area \( A \) is given by:
\[ A = \pi r^2 \]

Let:
- \( L_1 \) and \( r_1 \) be the length and radius of the first wire,
- \( L_2 \) and \( r_2 \) be the length and radius of the second wire.

Given:
- The ratio of lengths \( \frac{L_1}{L_2} = \frac{4}{3} \) implies \( L_2 = \frac{3}{4} L_1 \),
- The ratio of radii \( \frac{r_1}{r_2} = \frac{2}{3} \) implies \( r_2 = \frac{3}{2} r_1 \).

### Step 3: Calculate the resistances \( R_1 \) and \( R_2 \).
Using the resistance formula:
1. For wire 1:
\[ R_1 = \frac{\rho L_1}{\pi r_1^2} \]
2. For wire 2:
\[ R_2 = \frac{\rho L_2}{\pi r_2^2} \]

Substituting \( L_2 \) and \( r_2 \):
\[ R_2 = \frac{\rho \left(\frac{3}{4} L_1\right)}{\pi \left(\frac{3}{2} r_1\right)^2} = \frac{\rho \left(\frac{3}{4} L_1\right)}{\pi \left(\frac{9}{4} r_1^2\right)} = \frac{3 \rho L_1}{9 \pi r_1^2} = \frac{\rho L_1}{3 \pi r_1^2} \]

### Step 4: Find the ratio of the currents \( \frac{I_1}{I_2} \).
Using the relationship from Ohm's law:
\[ \frac{I_1}{I_2} = \frac{R_2}{R_1} \]

Substituting the values of \( R_1 \) and \( R_2 \):
\[ \frac{I_1}{I_2} = \frac{\frac{\rho L_1}{3 \pi r_1^2}}{\frac{\rho L_1}{\pi r_1^2}} = \frac{1}{3} \]

### Step 5: Conclusion
Thus, the ratio of the current passing through the two wires is:
\[ \frac{I_1}{I_2} = \frac{1}{3} \]