Animated Solution for Physics - Magnetic Effects of Current: A current $I$ ampere flows along an infinitely long straight thin walled tube, then the magnetic induction at any point inside the tube is

Imagine you are looking at a long, hollow pipe—a thin-walled tube. A steady current $I$ is flowing along the length of this tube. Our mission is to find the magnetic field at any point inside this hollow region.

To tackle this, we need a powerful tool that takes advantage of the cylindrical symmetry of the problem. That tool is Ampere's Circuital Law.

Ampere's Circuital Law is a beautiful principle in electromagnetism. It states that the line integral of the magnetic field $\mathbf{B}$ around any closed loop is directly proportional to the net current enclosed by that loop. Mathematically, it is written as:

Here, ${\mu }_0$ is the permeability of free space, and $I_{enc}$ is the total current passing through the area bounded by our chosen loop.

To find the magnetic field at a distance $r$ from the central axis (where $r<R$, with $R$ being the radius of the tube), we construct an imaginary circular path called an Amperian loop of radius $r$.

Now, we must ask ourselves a critical question: How much current is actually passing through the area of this Amperian loop?

Since the tube is "thin-walled," all the current $I$ is flowing exclusively on the outer surface of the cylinder. The hollow space inside contains absolutely no current. Therefore, the current enclosed by our Amperian loop is zero:

Let's substitute this crucial finding back into Ampere's Law. Because of the symmetry, the magnetic field $B$ would be constant along our circular loop, allowing us to simplify the line integral:

Since the circumference $2\pi r$ is not zero, the only way this equation holds true is if the magnetic field itself is zero:

Thus, the magnetic induction at any point inside an infinitely long, straight, thin-walled current-carrying tube is exactly zero. This is a classic result, very similar to how the electric field inside a hollow charged spherical conductor is zero!