Animated Solution for Physics - Magnetic Effects of Current: Which of the following statement is (are) correct in the given figure ?

Imagine a perfectly straight, infinitely long wire piercing through the fabric of space, carrying a steady current $I_1$ directly into the page. This wire acts as the conductor of a magnetic symphony, generating a magnetic field that ripples outward in perfect, concentric clockwise circles.

Now, introduce a closed wire loop $ABCD$ into this magnetic arena, carrying its own current $I_2$. The interaction between the loop's current and the straight wire's magnetic field creates a fascinating interplay of forces and torques. Let's break down this interaction segment by segment.

A Parallel Reality

The fundamental law governing this interaction is the magnetic force equation:

This cross product holds a crucial secret: if the current element $d\vec{l}$ is parallel or anti-parallel to the magnetic field $\vec{B}$, the resulting force is absolutely zero.

Look closely at the circular arcs $AB$ and $CD$. Because they are centered exactly at the origin $O$, they perfectly trace the path of the circular magnetic field lines produced by $I_1$. The current flowing through these arcs is either perfectly parallel or perfectly anti-parallel to the magnetic field. Consequently, the magnetic force on both of these arcs vanishes entirely.

The Push and Pull

The story changes dramatically for the straight radial segments, $AD$ and $CB$. Here, the current flows radially, cutting perpendicularly across the circular magnetic field lines.

Let's apply the Right-Hand Rule to determine the direction of the forces. For the bottom segment $AD$, the current flows radially outwards ($+\hat{r}$), while the magnetic field points clockwise ($-\hat{\theta }$). The cross product $(+\hat{r}\times -\hat{\theta })$ yields a force pointing directly into the page ($-\hat{k}$).

Conversely, for the top segment $CB$, the current flows radially inwards ($-\hat{r}$), and the magnetic field still points clockwise ($-\hat{\theta }$). The cross product $(-\hat{r}\times -\hat{\theta })$ yields a force pointing directly out of the page ($+\hat{k}$).

Because the segments $AD$ and $CB$ are geometrically symmetric and sit in identical magnetic field distributions, the magnitude of the inward force on $AD$ perfectly matches the outward force on $CB$. When we sum all the forces acting on the loop, they perfectly cancel each other out.

However, the physics doesn't end there. While the net translational force is zero, these two equal and opposite forces act at different locations on the loop, creating a mechanical couple.

Imagine standing at the origin $O$ and looking down the axis $O{O}^{\prime }$. The top segment $CB$ is being pulled towards you (out of the page), while the bottom segment $AD$ is being pushed away from you (into the page). This push-and-pull dynamic generates a net torque, forcing the entire loop to rotate clockwise around the axis $O{O}^{\prime }$.

This beautiful cancellation of forces paired with the emergence of a pure torque is a classic hallmark of symmetric electromagnetic systems!