A long, straight wire of radius R carries a current distributed u… - Vidyadip

MCQ

A long, straight wire of radius $R$ carries a current distributed uniformly over its cross section. $T$ he magnitude of the magnetic field is:

A
Maximum at the axis of the wire.
B
Minimum at the axis of the wire.
C
Maximum at the surface of the wire.
✓
Both $B$ and $C$

✓

Answer

Correct option: D.

Both $B$ and $C$

A long, straight wire of radius $R$ is carrying current $i,$ which is uniformly distributed over its cross section. According to Ampere's law,
$\oint\overrightarrow{\text{B}}.\text{d}\overrightarrow{\text{l}}=\mu_0\text{i}_\text{inside}$
At surface,
$\text{B}\times2\pi\text{R}=\mu_0\text{i}$
$\Rightarrow\text{B}_\text{surface}=\frac{\mu_0\text{i}}{2\pi\text{R}}$
Inside, $\text{B}\times2\pi\text{r}=\mu_0\text{i}$ for $\text{r}<\text{R}$
Here $i$, is the current enclosed by the amperian loop drawn inside the wire.
$B_{inside}$ will be proportional to the distance from the axis.
On the axis
$B = 0$
The magnetic fields from points on the cross section will point in opposite directions and will cancel each other at the centre.
Outside, $\text{B}\times2\pi\text{r}=\mu_0\text{i}$
$\Rightarrow\text{B}_\text{outside}=\frac{\mu_0\text{i}}{2\pi\text{r}},\ \text{r}>\text{R}$
Therefore, the magnitude of the magnetic field is maximum at the surface of the wire and minimum at the axis of the wire.

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