Question

A long current carrying wire produces a magnetic field of \(1\ \text{T}\) at a distance \(r\). The magnetic field at
\[ (a)\ \frac{r}{2}, \qquad (b)\ 2r, \qquad (c)\ 3r \] is

• A. \((a)\ 2\text{T},\ (b)\ \dfrac{1}{2}\text{T},\ (c)\ \dfrac{1}{3}\text{T}\)
• B. \((a)\ 3\text{T},\ (b)\ \dfrac{1}{3}\text{T},\ (c)\ \dfrac{1}{6}\text{T}\)
• C. \((a)\ \dfrac{3}{2}\text{T},\ (b)\ \dfrac{1}{4}\text{T},\ (c)\ \dfrac{1}{8}\text{T}\)
• D. \((a)\ \dfrac{5}{2}\text{T},\ (b)\ \dfrac{1}{2}\text{T},\ (c)\ \dfrac{1}{3}\text{T}\)

Hint:

For a long straight current carrying wire: \[ B=\frac{\mu_0 I}{2\pi r} \] The magnetic field decreases inversely with distance from the wire.

Answer 1

The Correct Option is A

Step 1: Recall the inverse proportionality of magnetic field with distance.
For a long straight current-carrying wire, the magnetic field at distance $ r $ is \[ B = \frac{\mu_0 I}{2\pi r} \implies B \propto \frac{1}{r} \] This means if you double the distance, the field halves; if you halve the distance, the field doubles.
Step 2: Establish the reference value.
We are given that at distance $ r $, the field is $ B_0 = 1\text{ T} $. We use this as our reference to scale the answer for each new distance.
Step 3: Find B at distance r/2.
Halving the distance from $ r $ to $ r/2 $ doubles the field: \[ B_{r/2} = B_0 \times \frac{r}{r/2} = 1 \times 2 = 2\text{ T} \] This makes intuitive sense: moving closer to the wire increases the field.
Step 4: Find B at distance 2r.
Doubling the distance from $ r $ to $ 2r $ halves the field: \[ B_{2r} = B_0 \times \frac{r}{2r} = 1 \times \frac{1}{2} = \frac{1}{2}\text{ T} \]
Step 5: Find B at distance 3r.
Tripling the distance from $ r $ to $ 3r $ reduces the field to one-third: \[ B_{3r} = B_0 \times \frac{r}{3r} = 1 \times \frac{1}{3} = \frac{1}{3}\text{ T} \]
Step 6: Summarize the results.
\[ (a)\ B_{r/2} = 2\text{ T},\quad (b)\ B_{2r} = \frac{1}{2}\text{ T},\quad (c)\ B_{3r} = \frac{1}{3}\text{ T} \] \[ \boxed{(a)\ 2\text{ T},\ (b)\ \tfrac{1}{2}\text{ T},\ (c)\ \tfrac{1}{3}\text{ T}} \]