Animated Solution for Physics - Electromagnetic Induction: Comprehension Passage

Imagine a point charge $Q$ moving in a circular orbit of radius $R$ in the $x$-$y$ plane. It is cruising along with an initial angular velocity $\omega$. Suddenly, a uniform magnetic field $\mathbf{B}$ is switched on along the positive $z$-axis. This magnetic field isn't static; it increases steadily from $0$ to $B$ in exactly one second.

What happens to the charge? To answer this, we must dive into the beautiful interplay between electricity and magnetism, specifically Faraday's Law of Induction.

According to Faraday's Law, a changing magnetic flux induces a circulating electric field. This isn't your regular electrostatic field created by stationary charges; its field lines form closed loops! We can express this mathematically using the integral form of Faraday's Law:

Let's calculate the magnitude of this induced electric field along the circular orbit. The line integral of $\mathbf{E}$ around the orbit is simply the magnitude $E$ multiplied by the circumference $2\pi R$. This must equal the rate of change of the magnetic flux. The flux $\Phi$ is the area of the orbit $\pi R^2$ multiplied by the magnetic field $B$.

Since the magnetic field increases from $0$ to $B$ in exactly one second, its rate of change $\frac{dB}{dt}$ is simply $B$. Substituting this into our equation, we can solve for the magnitude of the induced electric field:

This gives us the answer to the first part of our problem. The magnitude of the induced electric field is $\frac{BR}{2}$.

Now, let's move to the second part of the problem. The orbiting charge acts like a tiny current loop, creating a magnetic dipole moment $M$. We are given that this moment is proportional to its angular momentum $L$, with a proportionality constant $\gamma$:

We know that the angular momentum of a particle in a circular orbit is $L=m{R}^2\omega$. Therefore, the initial magnetic moment is:

The induced electric field we calculated earlier exerts a tangential force on our charge $Q$. This force creates a torque $\tau$ about the center of the orbit. Torque is force times radius:

Because of this torque, the charge experiences an angular acceleration $\alpha$. Using Newton's second law for rotation ($\tau =I\alpha$), where the moment of inertia $I$ is $m{R}^2$, we find:

Lenz's Law tells us that this induced effect will oppose the change that caused it. Since the magnetic field is increasing, the induced electric field will push the charge in a direction that reduces its angular velocity, thereby reducing the magnetic field it produces. So, the torque slows down the charge!

After one second, the new angular velocity ${\omega }^{\prime }$ is the initial $\omega$ minus the angular acceleration:

Finally, we can find the change in the magnetic dipole moment. Since $M$ is proportional to $\omega$, the change $\Delta M$ is proportional to the change in angular velocity:

Substituting our expression for the change in angular velocity, the mass $m$ cancels out beautifully:

This negative change represents the classical explanation for diamagnetism! When a magnetic field is applied, orbiting charges experience a torque that slightly changes their speeds, creating an induced magnetic moment that opposes the applied field.