A thick current carrying cable of radius ‘R’ carries current ‘I’ uniformly distributed across its cross-section. The variation of magnetic field B(r) due to the cable with the distance ‘r’ from the axis of the cable is represented by :

The correct graph representing the variation of the magnetic field $B\left(r\right)$ with distance $r$ from the axis of the cable is:

Step-by-Step Explanation:

Let a thick cylindrical cable of radius $R$ carry a total current $I$ uniformly distributed across its cross-sectional area $A=\pi R^2$.

To find the magnetic field $B$ at a distance $r$ from the central axis of the cable, we analyze two regions using Ampere's Circuital Law:

Case 1: Inside the cable ($r\le R$)

Consider a circular Amperian loop of radius $r$ concentric with the axis of the cable.

The current enclosed by this loop, $I_{\text{enclosed}}$, is proportional to the fraction of the cross-sectional area inside the loop:

$I_{\text{enclosed}}=I\times \frac{\pi r^2}{\pi R^2}=I\frac{r^2}{R^2}$

Applying Ampere's Circuital Law:

$\oint \overrightarrow{B}·d\overrightarrow{l}={\mu }_0{I}_{\text{enclosed}}$

$B\left(2\pi r\right)={\mu }_{0}I\frac{r^2}{R^2}$

Simplifying for $B$:

$B_{\text{in}}=\frac{{\mu }_{0}Ir}{2\pi R^2}$

Therefore, inside the cable, $B\propto r$. The magnetic field increases linearly from $B=0$ at the central axis ($r=0$) to a maximum value at the surface ($r=R$).

Case 2: Outside the cable ($r\ge R$)

For an Amperian loop outside the cable ($r\ge R$), the loop encloses the entire current $I$:

$I_{\text{enclosed}}=I$

Applying Ampere's Circuital Law:

$B\left(2\pi r\right)={\mu }_{0}I$

$B_{\text{out}}=\frac{{\mu }_{0}I}{2\pi r}$

Therefore, outside the cable, $B\propto \frac{1}{r}$. The magnetic field decreases non-linearly (hyperbolically) as $r$ increases.

Conclusion:

The magnetic field graph starts at zero at the axis, increases linearly with $r$ up to the surface $r=R$, reaches a peak, and then decreases as $\frac{1}{r}$ for $r>R$. This characteristic shape is correctly depicted in the chosen graph.