Question

A long straight wire of radius a carries a steady current I. The current is uniformly distributed across its cross section. The ratio of the magnetic field at \(\frac{a}{2}\) and 2a from axis of the wire is : 

(1) 1 : 4 

(2) 4 : 1 

(3) 1 : 1 

(4) 3 : 4

Answer 1

Correct option is :  (3) 1 : 1 

Magnetic field inside wire

\( \text {Biside}=\frac{\mu_0 \mathrm{Jr}}{2}=\frac{\mu_0 \mathrm{Ir}}{2 \mathrm{A}}=\frac{\mu_0 \mathrm{Ir}}{2 \pi \mathrm{a}^2}\)

\( \mathrm{r} \rightarrow \mathrm{a} / 2 \)

\( \text {Biside}=\frac{\mu_0 \mathrm{I}}{4 \pi \mathrm{a}} \)

\( \text {Magnetic field outside }\left(B_2\right) =\frac{\mu_0 \mathrm{i}}{2 \pi \mathrm{r}} \)

\( = \frac{\mu_0 \mathrm{i}}{2 \pi(2 \mathrm{a})} \)

\(= \frac{\mu_0 \mathrm{i}}{4 \pi \mathrm{a}}\)

\( \text {Biside }=\frac{\mu_0 \mathrm{I}}{4 \pi \mathrm{a}} \)

So \( \mathrm{B}_1=\mathrm{B}_2 \Rightarrow 1: 1 \)