Magnetic Field Around a Wire

Magnetic Field at Different Distances (DPMT 2001)

Question: The magnetic field at 4 cm from a wire is \( 10^{-8} \, \text{T} \). What is the field at 12 cm?

(a) \( 3.33 \times 10^{-9} \, \text{T} \)

(b) \( 1.11 \times 10^{-4} \, \text{T} \)

(c) \( 3 \times 10^{-3} \, \text{T} \)

(d) \( 9 \times 10^{-2} \, \text{T} \)

Solution

Step 1: Ampère's Law
The magnetic field around a long straight wire is: \[ B = \frac{\mu_0 I}{2\pi r} \implies B \propto \frac{1}{r}. \]

Step 2: Proportionality
For distances \( r_1 = 4 \, \text{cm} \) and \( r_2 = 12 \, \text{cm} \): \[ \frac{B_2}{B_1} = \frac{r_1}{r_2} \implies B_2 = B_1 \cdot \frac{r_1}{r_2}. \] \[ B_2 = 10^{-8} \, \text{T} \cdot \frac{4}{12} = 3.33 \times 10^{-9} \, \text{T}. \]

Answer: Option (a) \( 3.33 \times 10^{-9} \, \text{T} \) is correct.