Animated Solution for Physics - Magnetic Effects of Current: Two coaxial solenoids of different radii carry current $I$ in the same direction. Let ${\mathbf{F}}_1$ be the magnetic force on the inner solenoid due to the outer one and ${\mathbf{F}}_2$ be the magnetic force on the outer solenoid due to the inner one. Then,

Imagine you are looking at two coaxial solenoids—one nestled perfectly inside the other, like a set of Russian nesting dolls. Both are carrying a steady current $I$ in the exact same direction. The question asks us to determine the net magnetic force acting on each solenoid due to the other. Let's break this down step-by-step.

Experiencing the Outer Field

First, let's analyze the forces acting on the inner solenoid. To do this, we need to understand the environment it sits in. The outer solenoid acts like a long, current-carrying coil, which produces a highly uniform magnetic field, ${\mathbf{B}}_{outer}$, directed straight along the common central axis.

Now, look at the current flowing through the inner solenoid. The current travels in circular loops, meaning it flows tangentially (or azimuthally) to the axis. Because the current is perpendicular to the magnetic field, every small segment of the inner wire experiences a magnetic force given by the Lorentz force law:

If you apply the right-hand rule, you'll find that this force points radially outwards everywhere along the loop. It's as if the magnetic field is trying to inflate the inner solenoid like a balloon! However, because of the perfect cylindrical symmetry, for every small segment experiencing a force in one direction, there is an opposite segment experiencing an equal force in the exact opposite direction.

Mathematically, the net force is the integral over the closed loop:

Since the integral of a closed path $\oint d\mathbf{l}$ is zero, the net translational force ${\mathbf{F}}_1$ on the inner solenoid is exactly zero.

The Zero-Field Zone

What about the force on the outer solenoid? To find this, we must look at the magnetic field produced by the inner solenoid.

One of the most beautiful properties of an ideal, infinitely long solenoid is that it confines its magnetic field entirely within its core. Outside the solenoid, the magnetic field is practically zero.

Since the outer solenoid is located entirely outside the inner solenoid, it sits in a region where ${\mathbf{B}}_{inner}=0$. If there is no magnetic field, there can be no magnetic force! Therefore, the net force ${\mathbf{F}}_2$ on the outer solenoid is also zero.

The Ultimate Check

We could have saved ourselves half the work by invoking a fundamental principle of physics: Newton's Third Law of Motion. If the outer solenoid exerts a net force of zero on the inner solenoid (${\mathbf{F}}_1=0$), then the inner solenoid must exert an equal and opposite net force on the outer solenoid.

Thus, ${\mathbf{F}}_2=-{\mathbf{F}}_1=0$.

Final Conclusion: Both solenoids experience zero net magnetic force, meaning ${\mathbf{F}}_1={\mathbf{F}}_2=0$.