Animated Solution for Physics - Magnetic Effects of Current: A rectangular loop of sides 10 cm and 5 cm carrying a current I of 12 A is placed in different orientations as shown in the figures below. If there is a uniform magnetic field of 0.3 T in the positive z-direction in which orientations the loop would be in (i) stable equilibrium and (ii) unstable equilibrium?

Imagine a rectangular loop of wire, carrying a steady current, placed in a region where a uniform magnetic field permeates space. This is a classic scenario in electromagnetism that beautifully bridges the gap between moving charges and rotational dynamics. In our specific problem, we have a loop with sides $10\text{ cm}$ and $5\text{ cm}$, carrying a current $I=12\text{ A}$. The entire setup is immersed in a uniform magnetic field $\mathbf{B}=0.3\hat{k}\text{ T}$, which points straight up along the positive z-axis.

Our goal is to determine which of the four given orientations places the loop in a state of stable equilibrium and which places it in unstable equilibrium.

To tackle this, we first need to understand what equilibrium means for a current-carrying loop in a magnetic field. A loop of current acts like a tiny bar magnet, characterized by its magnetic dipole moment, denoted by $\mathbf{M}$. The magnetic moment is a vector quantity defined as $\mathbf{M}=I\mathbf{A}$, where $I$ is the current and $\mathbf{A}$ is the area vector of the loop. The direction of the area vector is determined by the right-hand rule: if you curl the fingers of your right hand in the direction of the current flow, your thumb points in the direction of $\mathbf{A}$ (and thus $\mathbf{M}$).

When this magnetic dipole is placed in an external magnetic field $\mathbf{B}$, it experiences a torque $\mathbf{\tau }$, given by the cross product:

The magnitude of this torque is $|\mathbf{\tau }|=MB\sin \theta$, where $\theta$ is the angle between $\mathbf{M}$ and $\mathbf{B}$.

For the loop to be in equilibrium, the net torque acting on it must be zero. Looking at our torque equation, $\mathbf{\tau }=0$ when $\sin \theta =0$. This occurs at two specific angles: 1. $\theta =0^{\circ }$: $\mathbf{M}$ is parallel to $\mathbf{B}$. 2. $\theta =180^{\circ }$: $\mathbf{M}$ is anti-parallel to $\mathbf{B}$.

Now, let's systematically analyze the four orientations provided in the problem to find the direction of the magnetic moment $\mathbf{M}$ for each case.

Orientation (a): The loop is positioned in the x-z plane. The current flows from the origin along the positive x-axis, then up along the positive z-axis, back along the negative x-axis, and finally down the negative z-axis. If we apply the right-hand rule and curl our fingers in this clockwise direction (when viewed from the positive y-axis), our thumb points in the negative y-direction. Therefore, ${\mathbf{M}}_a$ is along $-\hat{j}$. Since $\mathbf{B}$ is along $+\hat{k}$, the angle $\theta$ is $90^{\circ }$. The torque is maximum, and the loop is definitely not in equilibrium.

Orientation (b): Here, the loop lies flat in the x-y plane. The current flows from the origin along the positive x-axis, then along the positive y-axis, and so on. Curling our fingers in this anti-clockwise direction (viewed from above), our thumb points straight up, in the positive z-direction. Thus, ${\mathbf{M}}_b$ is along $+\hat{k}$. Since $\mathbf{B}$ is also along $+\hat{k}$, the angle $\theta$ is $0^{\circ }$. The torque is zero, meaning the loop is in equilibrium!

Orientation (c): In this case, the loop is in the y-z plane. The current flows from the origin along the positive y-axis, then up the positive z-axis. Applying the right-hand rule, our thumb points outward along the positive x-direction. So, ${\mathbf{M}}_c$ is along $+\hat{i}$. With $\mathbf{B}$ along $+\hat{k}$, the angle $\theta$ is again $90^{\circ }$. The loop experiences a torque and is not in equilibrium.

Orientation (d): Finally, the loop is back in the x-y plane, but notice the direction of the current. It flows from the origin along the positive y-axis, then along the positive x-axis. This is a clockwise flow when viewed from above. Curling our fingers accordingly, our thumb points downwards, in the negative z-direction. Therefore, ${\mathbf{M}}_d$ is along $-\hat{k}$. Since $\mathbf{B}$ is along $+\hat{k}$, the angle $\theta$ is $180^{\circ }$. The torque is zero, which means this is also a state of equilibrium!

We have identified two equilibrium states: orientation (b) at $\theta =0^{\circ }$ and orientation (d) at $\theta =180^{\circ }$. But which one is stable and which is unstable?

To answer this, we must look at the potential energy $U$ of the magnetic dipole in the magnetic field, which is given by the dot product:

Stable Equilibrium: A system is in stable equilibrium when its potential energy is at a minimum. For our loop, the minimum potential energy occurs when $\cos \theta$ is maximum, i.e., $\cos (0^{\circ })=1$.

This corresponds to orientation (b), where $\mathbf{M}$ is parallel to $\mathbf{B}$. If you slightly perturb the loop from this position, the magnetic torque will act as a restoring force, pushing it back towards alignment.

Unstable Equilibrium: Conversely, a system is in unstable equilibrium when its potential energy is at a maximum. This happens when $\cos \theta$ is minimum, i.e., $\cos (180^{\circ })=-1$.

This corresponds to orientation (d), where $\mathbf{M}$ is anti-parallel to $\mathbf{B}$. If you give the loop even the tiniest nudge from this position, the magnetic torque will pull it further away, causing it to flip $180^{\circ }$ until it reaches the stable state.

Through our rigorous analysis of magnetic moments, torque, and potential energy, we have successfully decoded the physics of the loop's orientations.

- Stable equilibrium is achieved in orientation (b). - Unstable equilibrium is achieved in orientation (d).

Matching this with our given options, the correct choice is (c), which states "(b) and (d) respectively". This problem is a beautiful demonstration of how vector mathematics and energy principles govern the rotational dynamics of electromagnetic systems.