Animated Solution for Physics - Magnetic Effects of Current: Figures A and B shown two long straight wires of circular cross-section ($a$ and $b$ with $a<b$), carrying current $I$ which is uniformly distributed across the cross-section. The magnitude of magnetic field $B$ varies with radius $r$ and can be represented as

Imagine you are exploring the magnetic landscape around two long, straight wires. Wire A is thin with a radius $a$, while Wire B is thicker with a radius $b$. Both wires carry the exact same total current $I$, uniformly distributed across their cross-sections. Our mission is to visualize how the magnetic field $B$ changes as we move away from the center of these wires.

The Linear Ascent

Let's dive inside the wire first. Using Ampere's Circuital Law, we draw an imaginary circular path—an Amperian loop—of radius $r$ ($r<R$) inside the wire. The current enclosed by this loop isn't the total current $I$, but a fraction of it proportional to the area: $I_{enc}=I\left(\frac{\pi r^2}{\pi R^2}\right)$.

Applying Ampere's Law, $\oint \mathbf{B}\cdot d\mathbf{l}={\mu }_0{I}_{enc}$, we get:

Notice the beautiful simplicity here: all terms are constant except $r$. This means that inside the wire, the magnetic field is directly proportional to the distance from the center ($B\propto r$). If we graph this, it's a perfect straight line starting from the origin.

The Hyperbolic Descent

Now, let's step outside the wire ($r>R$). Our Amperian loop now encloses the entire current $I$. Ampere's Law simplifies to:

Outside the wire, the magnetic field is inversely proportional to the distance ($B\propto \frac{1}{r}$). The graph transitions into a smooth, decreasing curve known as a rectangular hyperbola.

Comparing the Peaks and the Merge

The magnetic field reaches its absolute maximum right at the surface of the wire, where $r=R$. The peak value is:

This is the crucial insight! Because $R$ is in the denominator, a thinner wire will have a higher peak magnetic field at its surface. Since Wire A is thinner ($a<b$), its graph will peak higher at $r=a$ than Wire B's graph at $r=b$.

But here is the most fascinating part: what happens when we are far away, at a distance $r>b$? To an observer out there, both wires look identical. They both enclose the exact same total current $I$. Therefore, they must produce the exact same magnetic field $B=\frac{{\mu }_{0}I}{2\pi r}$.

This means the two graphs must perfectly merge into the same $\frac{1}{r}$ curve for all distances greater than $b$. Looking at the given options, only option (c) captures this physical reality: a higher peak for the thinner wire, a lower peak for the thicker wire, and a flawless merging of their paths in the outside region.