Due to the presence of the current I1 at the origin in the figure,
(a) the forces on AB and DC are z

Q: Why is the magnetic field circular in the yz-plane?

According to Ampere's Law, a long straight wire produces a magnetic field that wraps around it in concentric circles. Since the wire is along the x-axis, these circles lie perfectly flat in the yz-plane.

Q: Why doesn't the loop experience a force on the arcs?

The magnetic force depends on the cross product of the length element and the magnetic field. On the arcs, the current flows exactly along the circular magnetic field lines, making the angle 0 or 180 degrees. The cross product is zero!

Q: Are the forces on the straight segments AB and CD also zero?

No! On AB and CD, the current flows radially, which is perpendicular to the circular magnetic field. They experience a force along the x-axis.

Q: If AB and CD experience a force, does the loop move?

The forces on AB and CD are equal in magnitude but opposite in direction (one pushes out, one pulls in). So the net force is zero, but they do create a net torque that tries to twist the loop!

Q: What if the straight wire was along the y-axis instead?

That changes everything! The magnetic field would then be circular in the xz-plane, meaning it would pierce through the yz-plane perpendicularly. All four segments of the loop would then experience complex, non-zero forces.

Question

Due to the presence of the current $I_1$ at the origin in the figure,

Options:

(A) the forces on AB and DC are zero

(B) the forces on AD and BC are zero

(C) the magnitude of the net force on the loop is given by $\frac{\mu_0 I I_1}{4\pi} \left[ 2(b-a) + \frac{\pi}{3}(a+b) \right]$

(D) the magnitude of the net force on the loop is given by $\frac{\mu_0 I I_1}{24ab} (b-a)$

Sigma Percentile · Accepted Answer

### The 3D Visualization Challenge

When you first look at this problem, the diagram might seem a bit overwhelming. The key to unlocking it is to perfectly visualize the 3D geometry.

We are given a straight wire carrying a current $I_1$ that lies exactly along the **x-axis**. Meanwhile, the current loop $ABCD$ is placed entirely flat within the **yz-plane**. This spatial separation is the foundation of everything that follows.

### Decoding the Magnetic Field

According to Ampere's Law (or the Biot-Savart Law), a long straight wire produces a magnetic field $\vec{B}$ whose field lines form concentric circles around the wire.

Because our wire $I_1$ is along the x-axis, these circular magnetic field lines will lie perfectly flat in the yz-plane. In mathematical terms, the magnetic field in the yz-plane is purely **azimuthal** (pointing in the $\hat{	heta}$ direction).

### The Anatomy of the Loop

Now, let's dissect the loop $ABCD$ into its four distinct segments:

1. **Segments AB and CD:** These are radial segments. They point straight outward (or inward) from the origin. Their length elements $d\vec{l}$ are parallel to the radial vector $\hat{r}$.
2. **Segments AD and BC:** These are circular arcs centered at the origin. Their length elements $d\vec{l}$ perfectly follow the circular path, meaning they are parallel to the azimuthal vector $\hat{	heta}$.

### The Magic of the Cross Product

The magnetic force on any small current element is given by the elegant cross product equation:
$$ d\vec{F} = I(d\vec{l} 	imes \vec{B}) $$

Let's apply this to the circular arcs **AD and BC**. We established that the magnetic field $\vec{B}$ is azimuthal ($\hat{	heta}$). We also established that the length elements $d\vec{l}$ for these arcs are azimuthal ($\hat{	heta}$).

Because both vectors lie along the exact same line, the angle between them is either $0^\circ$ or $180^\circ$. The cross product of parallel or anti-parallel vectors is mathematically zero.

Therefore, the magnetic force acting on the arcs AD and BC is exactly zero!

### The Hidden Torque (Bonus Insight)

What about the radial segments AB and CD? Since their length elements $d\vec{l}$ are radial ($\hat{r}$) and the magnetic field $\vec{B}$ is azimuthal ($\hat{	heta}$), they are perpendicular.

They will experience a force along the x-axis ($\hat{r} 	imes \hat{	heta} = \hat{x}$). Because the currents flow in opposite radial directions, the force on AB will push out of the page, while the force on CD will pull into the page.

These forces are equal in magnitude and opposite in direction, meaning the **net force** on the entire loop is zero. However, because these forces are applied at different locations, they will create a **net torque** that attempts to twist the loop out of the yz-plane!

Animated Solution for Physics - Magnetic Effects of Current: Due to the presence of the current $I_{1}$ at the origin in the figure,

Select Answer:

Visualized Solution

$3D Geometry Setup$

$Magnetic Field of I_{1}$

$Loop Segments Analysis$

$Force on Arcs AD and BC$

$Cross Product Evaluation$

$Final Conclusion$

The Sigma Insight: Magnetic Force on Current

The 3D Visualization Challenge

Decoding the Magnetic Field

The Anatomy of the Loop

The Magic of the Cross Product

The Hidden Torque (Bonus Insight)

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Animated Solution for Physics - Magnetic Effects of Current: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Due to the presence of the current I1​ at the origin in the figure,