Animated Solution for Physics - Electromagnetic Induction: Comprehension Passage

The problem we are tackling today is a beautiful amalgamation of electromagnetic induction, magnetism, and the mechanics of work and energy. It takes us into the fascinating world of superconductors—materials that conduct electricity with absolutely zero resistance. Let's break down the physics step by step.

Imagine a closed wire loop made of a special metal that has zero electrical resistance. What happens when you try to change the magnetic flux passing through it? According to Faraday's Law of Electromagnetic Induction, a changing magnetic flux induces an electromotive force (EMF). This EMF drives an induced current.

In a normal wire, this current would quickly die out due to resistance, dissipating energy as heat. But in our zero-resistance loop, the current faces no opposition. It flows endlessly. More importantly, because the resistance is zero, the loop must maintain a strictly constant magnetic flux. If the initial flux is zero, the loop will induce whatever current is necessary to ensure the total flux remains exactly zero at all times. This is the core principle we will use to solve the first part of the problem.

We are given a magnetic dipole of moment $m$ that is brought from infinity to a distance $r$ from the center of the loop. Initially, at infinity, the magnetic field from the dipole at the loop is zero, so the initial flux is zero.

When the dipole is at a distance $r$, it creates a magnetic field. The formula for the magnetic field of a dipole on its axis is given as:

Since the distance $r$ is much greater than the radius $a$ of the loop ($r\gg a$), we can assume this magnetic field is uniform over the entire area of the loop. The external magnetic flux ${\varphi }_{\text{ext}}$ passing through the loop is simply the magnetic field multiplied by the area of the loop:

To keep the total flux at zero, the loop induces a current $i$. This current creates a self-flux ${\varphi }_{\text{self}}$ that perfectly cancels the external flux. The self-flux is proportional to the induced current, given by ${\varphi }_{\text{self}}=Li$, where $L$ is the self-inductance of the loop.

Equating the magnitudes of the external flux and the self-flux, we get:

Since $L$, ${\mu }_0$, and $a$ are constants for a given loop, we can clearly see the relationship between the induced current and the dipole parameters:

This perfectly matches our first objective, showing how the induced current scales with the dipole moment and distance.

Now, let's move to the second part of the problem: calculating the work done in bringing the dipole from infinity to the distance $r$. To find the work done, we first need to determine the force acting on the dipole.

A current-carrying loop acts like a magnetic dipole itself. The induced magnetic dipole moment $m^{\prime }$ of the loop is simply the product of the induced current and the area of the loop:

Since we already established that $i\propto \frac{m}{r^3}$, it directly follows that the induced dipole moment $m^{\prime }$ is also proportional to the same factor:

The passage provides us with a specific formula for the force between two magnetic dipoles separated by a distance $r$ on a common axis:

In our scenario, $m_1$ is the original dipole $m$, and $m_2$ is the induced dipole $m^{\prime }$. Substituting our proportionalities into this force equation, we get:

Simplifying this expression, we find how the force scales:

This force is repulsive. Why? Lenz's Law tells us that the induced current will always oppose the change that caused it. As the external dipole approaches, it increases the magnetic flux through the loop. The loop fights back by creating an opposing magnetic field, which manifests as a repulsive force pushing the dipole away.

Finally, we need to calculate the work done against this repulsive force. Work is the integral of force over distance. We are bringing the dipole from infinity to a distance $r$, so we integrate the force with respect to distance:

Substituting our proportionality for the force:

Using the standard power rule for integration, the integral of $r^{-7}$ is $\frac{r^{-6}}{-6}$. The negative sign will cancel out when we evaluate the limits from infinity to $r$, leaving us with the final proportionality:

Through a logical sequence of applying Faraday's Law, understanding dipole fields, and performing a straightforward integration, we have unraveled the mechanics of this superconducting system. The induced current scales as $m/r^3$, and the work done scales as $m^2/r^6$. This problem beautifully illustrates how macroscopic physical laws seamlessly connect to give us profound insights into the behavior of electromagnetic systems.