A loop of flexible wire of irregular shape carrying current is placed in an external magnetic field.

Q: Why does the magnetic force point outward on every segment?

This is a direct consequence of the right-hand rule for the cross product $d\vec{l} \times \vec{B}$. If you assume the magnetic field points into the page and the current flows clockwise, crossing the tangential current vector with the inward field vector always yields a force vector pointing radially away from the center of the loop.

Q: Why does the loop form a circle and not a square or a rectangle?

The outward magnetic forces push the flexible wire to enclose the maximum possible area. According to the isoperimetric inequality in mathematics, for a given fixed perimeter (the length of the wire), the shape that encloses the maximum area is a perfect circle.

Q: What would happen if the current direction was reversed?

If the current direction is reversed, the cross product $d\vec{l} \times \vec{B}$ would flip, meaning the magnetic forces would point radially inward. Instead of expanding into a circle, the flexible loop would crumple and collapse inward.

Q: Why must the plane of the loop be normal to the magnetic field?

A current loop acts as a magnetic dipole. A dipole is in stable equilibrium when its magnetic moment vector $\vec{M}$ is parallel to the external magnetic field $\vec{B}$. Since $\vec{M}$ is perpendicular to the plane of the loop, the plane itself must be normal (perpendicular) to the magnetic field.

Q: Does the magnetic field do any work in expanding the loop?

Yes. As the loop expands to a circular shape, its area increases. This increases the magnetic flux passing through the loop. The magnetic force does positive work during this expansion until the loop reaches its maximum area (a circle).

Question

A loop of flexible wire of irregular shape carrying current is placed in an external magnetic field. Identify the effect of the field on the wire.

Options:

(A) Loop assumes circular shape with its plane normal to the field.

(B) Loop assumes circular shape with its plane parallel to the field.

(C) Wire gets stretched to become straight.

(D) Shape of the loop remains unchanged.

Sigma Percentile · Accepted Answer

### The Setup: A Flexible Wire in a Magnetic Field

Imagine you are holding a flexible, irregularly shaped loop of wire. Now, let a steady current $I$ flow through it, and place this entire setup into a uniform external magnetic field $\vec{B}$. What happens next is a beautiful demonstration of electromagnetic forces and geometric optimization.

To understand the behavior of the loop, we must zoom in and look at the microscopic forces acting on the wire.

### The Invisible Hand: Magnetic Force on a Current Element

We know from the Biot-Savart Law and Lorentz force principles that any current-carrying element placed in a magnetic field experiences a magnetic force. For a tiny segment of the wire represented by the vector $d\vec{l}$, the magnetic force $d\vec{F}$ is given by the cross product:

$d\vec{F} = I(d\vec{l} 	imes \vec{B})$

Let's apply the right-hand rule to this cross product. If you point your fingers in the direction of the current $d\vec{l}$ and curl them towards the magnetic field $\vec{B}$ (which we can assume points into the screen for visualization), your thumb points in the direction of the force $d\vec{F}$.

If you trace this around the entire irregular loop, you will notice a fascinating pattern: the force on every single small segment points **radially outward**. Because the wire is flexible, these outward forces act like an invisible hand, pulling and stretching the loop in all directions.

### The Geometry of Equilibrium: Why a Circle?

As the outward forces stretch the flexible wire, the loop tries to expand. However, the total length (perimeter) of the wire is fixed. The loop will continue to change shape until the outward magnetic forces are perfectly balanced by the internal tension of the wire.

Geometrically, for a given fixed perimeter, the shape that encloses the maximum possible area is a **circle**. Therefore, the irregular loop will stretch and smooth out its kinks until it forms a perfect circular shape.

### The Final Alignment: Stable Equilibrium

But we are not done yet! We must also consider the orientation of this newly formed circular loop. A current-carrying loop acts as a magnetic dipole, with a magnetic dipole moment $\vec{M}$ given by:

$\vec{M} = I\vec{A}$

where $\vec{A}$ is the area vector of the loop. In physics, a magnetic dipole placed in an external magnetic field $\vec{B}$ experiences a torque $\vec{	au} = \vec{M} 	imes \vec{B}$. This torque attempts to rotate the loop until it reaches a state of **stable equilibrium**, which occurs when the dipole moment $\vec{M}$ is perfectly parallel to the external field $\vec{B}$.

Since the area vector $\vec{A}$ is perpendicular to the plane of the loop, aligning $\vec{M}$ with $\vec{B}$ means that the plane of the circular loop must become perfectly **normal (perpendicular)** to the magnetic field.

Thus, the flexible wire not only becomes a circle but also aligns its plane normal to the magnetic field, making option (a) the correct answer.

Animated Solution for Physics - Magnetic Effects of Current: A loop of flexible wire of irregular shape carrying current is placed in an external magnetic field. Identify the effect of the field on the wire.

Select Answer:

Visualized Solution

Visualizing the Setup

Magnetic Force on a Current Element

Direction of Magnetic Force

Equilibrium Shape

Orientation of the Loop

Conclusion

The Sigma Insight: Magnetic Force on Current

The Setup

The Invisible Hand

The Geometry of Equilibrium

The Final Alignment

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