Question 30: Magnetism and Matter

Question

Consider a long straight wire of a circular cross-section (radius a) carrying a steady current I. The current is uniformly distributed across this cross-section. The distances from the centre of the wire’s cross-section at which the magnetic field [inside the wire, outside the wire] is half of the maximum possible magnetic field, any where due to the wire, will be :

Accepted Answer

Maximum possible magnetic field is at the surface
$$\begin{aligned}
& \mathrm{B}_{\max }=\frac{\mu_0 \mathrm{I}}{2 \pi \mathrm{a}} \
& \frac{\mathrm{~B}_{\max }}{2}=\frac{\mu_0 \mathrm{I}}{4 \pi \mathrm{a}}
\end{aligned}$$
It can be obtained inside as well as outside the wire
For inside,
$$\begin{aligned}
& \frac{\mu_0 \mathrm{I}}{4 \pi \mathrm{a}}=\frac{\mu_0 \mathrm{Ir}}{2 \pi \mathrm{a}^2} \
& \Rightarrow \mathrm{r}=\frac{\mathrm{a}}{2}
\end{aligned}$$
For outside
$$\begin{aligned}
& \frac{\mu_0 \mathrm{I}}{4 \pi \mathrm{a}}=\frac{\mu_0 \mathrm{I}}{2 \pi \mathrm{r}} \
& \Rightarrow \mathrm{r}=2 \mathrm{a}
\end{aligned}$$
Correct answer $\left[\frac{\mathrm{a}}{2}, 2 \mathrm{a}ight]$

Answer

[a/2, 3a]

Answer

[a/4, 3a/2]

Answer

[a/2, 2a]

Answer

[a/4, 2a]

Magnetism and Matter

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