Interpret the regions

The option text appears to mean:

• for points inside the wire: $r<a$
• for points outside the wire: $r>a$

We must find how magnetic field $B$ depends on $r$ for a long straight wire carrying uniformly distributed current $I$.

Use Ampere’s law

For a circular Amperian loop of radius $r$ coaxial with the wire, $\oint \vec B\cdot d\vec l = \mu_0 I_{\text{enc}}$

By symmetry, $B$ is constant on the loop and tangent everywhere, so $B(2\pi r)=\mu_0 I_{\text{enc}}$

Case 1: Inside the wire $(r<a)$

Since current is uniformly distributed, $J=\frac{I}{\pi a^2}$

Current enclosed by radius $r$ is $I_{\text{enc}}=J\pi r^2=\frac{I}{\pi a^2}\cdot \pi r^2=I\frac{r^2}{a^2}$

Substituting into Ampere’s law: $B(2\pi r)=\mu_0 I\frac{r^2}{a^2}$

Hence, $B=\frac{\mu_0 I}{2\pi a^2}r$

Therefore, for $r<a$, $B\propto r$

Case 2: Outside the wire $(r>a)$

Entire current is enclosed, so $I_{\text{enc}}=I$

Then $B(2\pi r)=\mu_0 I$ $B=\frac{\mu_0 I}{2\pi r}$

Therefore, for $r>a$, $B\propto \frac{1}{r}$

Check the options

• A: uniform in region $r<a$ — false
• B: zero in region $r<a$ — false
• C: directly proportional to $r$ in region $r<a$ — true
• D: inversely proportional to $r$ in region $r<a$ — false

Final answer

The correct option is C.