Two wires A an dB of the same material, having radii in the ratio I : 2 and carry currents in the ratio 4: I. The ratio of drift speed of electrons in A and Bis :

Options

A. `16:1`
B. `1:16`
C. `1:4`
D. `4:1`

Answer

To find the ratio of the drift speed of electrons in wires A and B, we can follow these steps:

### Step 1: Understand the given ratios
We are given:
- The ratio of the radii of the wires: \( r_A : r_B = 1 : 2 \)
- The ratio of the currents in the wires: \( I_A : I_B = 4 : 1 \)

### Step 2: Calculate the areas of the wires
The cross-sectional area \( A \) of a wire is given by the formula:
\[ A = \pi r^2 \]
Using the ratio of the radii, we can find the ratio of the areas:
\[ A_A : A_B = \pi r_A^2 : \pi r_B^2 = r_A^2 : r_B^2 = (1^2) : (2^2) = 1 : 4 \]

### Step 3: Relate drift speed to current and area
The drift speed \( v_d \) of electrons in a wire can be expressed using the formula:
\[ I = n e A v_d \]
Where:
- \( I \) is the current
- \( n \) is the number density of charge carriers (constant for the same material)
- \( e \) is the charge of an electron (constant for the same material)
- \( A \) is the cross-sectional area
- \( v_d \) is the drift speed

From this, we can derive:
\[ v_d = \frac{I}{n e A} \]
Since \( n \) and \( e \) are constants for the same material, we can write:
\[ v_d \propto \frac{I}{A} \]

### Step 4: Calculate the ratio of drift speeds
Now we can find the ratio of the drift speeds in wires A and B:
\[ \frac{v_{dA}}{v_{dB}} = \frac{I_A / A_A}{I_B / A_B} = \frac{I_A}{I_B} \cdot \frac{A_B}{A_A} \]
Substituting the known ratios:
\[ \frac{v_{dA}}{v_{dB}} = \frac{4}{1} \cdot \frac{4}{1} = 16 \]

### Final Answer:
The ratio of the drift speeds of electrons in wires A and B is:
\[ \frac{v_{dA}}{v_{dB}} = 16 : 1 \]

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