Animated Solution for Physics - Magnetic Effects of Current: A long, straight wire of radius $a$ carries a current distributed uniformly over its cross-section. The ratio of the magnetic fields due to the wire at distance $\frac{a}{3}$ and $2a$ respectively, from the axis of the wire is

Imagine you are standing inside a massive, solid cylindrical wire carrying a steady, uniform current. The physics of electromagnetism dictates that the magnetic field you experience depends entirely on where you are relative to the central axis of this wire. This problem is a beautiful exploration of Ampere's Circuital Law, testing our understanding of how magnetic fields behave both inside and outside a current-carrying conductor.

Ampere's Circuital Law

Ampere's Law is to magnetism what Gauss's Law is to electrostatics. It states that the line integral of the magnetic field around any closed loop is proportional to the total current enclosed by that loop:

For a long, straight wire with a circular cross-section, symmetry tells us that the magnetic field lines form concentric circles around the axis. This symmetry makes evaluating the integral incredibly simple: the left side always becomes $B(2\pi r)$. The real physics lies in determining $I_{enc}$, the enclosed current.

The Linear Regime

When we are inside the wire, at a distance $r<a$, our Amperian loop does not enclose the entire current $I$. Because the current is uniformly distributed, the current density $J$ (current per unit area) is constant: $J=\frac{I}{\pi a^2}$.

The current enclosed by a loop of radius $r$ is simply the current density multiplied by the area of our loop:

Plugging this into Ampere's Law gives us the magnetic field inside the wire:

Notice that inside the wire, the magnetic field increases linearly with distance $r$. For our specific problem, we need the field at $r=\frac{a}{3}$. Let's substitute this value:

The Inverse Regime

Now, let's step outside the wire to a distance $r>a$. Here, our Amperian loop encloses the entire current $I$. The internal structure of the wire no longer matters; it behaves exactly like an infinitely thin wire located at the central axis.

Ampere's Law gives us the familiar formula:

Outside the wire, the magnetic field decreases inversely with distance $r$. We are asked to find the field at $r=2a$. Substituting this into our formula yields:

We now have the magnetic field at both locations. The final step is to find their ratio, $\frac{B_1}{B_2}$. This is where the elegance of proportionalities shines, as all the constants will perfectly cancel out.

And there we have it! The ratio of the magnetic fields is $\frac{2}{3}$. This problem serves as a fantastic reminder to always trust the fundamental laws of physics. Whether you are deep inside a conductor or far out in space, Ampere's Law will always guide you to the right answer.