A rectangular loop carrying a current i is situated near a long straight wire such that the wire is

Q: Why do the forces on the top and bottom segments cancel out?

Even though the magnetic field is non-uniform, at any specific distance $r$ from the straight wire, the magnetic field is the same for both the top and bottom segments. Since the currents in these segments are in opposite directions, the forces on small elements $dr$ are equal and opposite, leading to a net zero force in the vertical direction.

Q: Why do parallel currents attract?

A current-carrying wire creates a magnetic field. When another wire with parallel current is placed in this field, the Lorentz force $\vec{F} = i(\vec{L} \times \vec{B})$ acts on it. By applying the right-hand rule for the field and the left-hand rule for the force, you'll find the force is directed towards the first wire.

Q: Does the loop experience any torque?

No, in this specific orientation where the loop and the wire are coplanar, all forces act in the plane of the loop. The forces on the top and bottom segments are collinear and cancel, and the forces on the left and right segments are also in the plane, so there is no torque to rotate the loop.

Q: What if the current in the loop was counter-clockwise?

If the current was counter-clockwise, the left segment would have anti-parallel current and be repelled, while the right segment would have parallel current and be attracted. Since the repulsive force on the closer left segment would be stronger, the loop would move away from the wire.

Q: Is the magnetic field uniform across the loop?

No, the magnetic field produced by a long straight wire is inversely proportional to the distance ($B \propto 1/r$). Therefore, the field is stronger at the left segment of the loop and weaker at the right segment.

Question

A rectangular loop carrying a current $i$ is situated near a long straight wire such that the wire is parallel to one of the sides of the loop and is in the plane of the loop. If steady current $I$ is established in the wire as shown in the figure, the loop will

Options:

(A) rotate about an axis parallel to the wire

(B) move away from the wire

(C) move towards the wire

(D) remain stationary

Sigma Percentile · Accepted Answer

### The Setup: A Wire and a Loop

Imagine a long straight wire carrying a steady current $I$. Placed near it is a rectangular loop carrying a current $i$. The question asks us to determine the behavior of this loop. Will it rotate, move, or stay perfectly still?

To answer this, we need to understand the invisible magnetic interactions at play.

### The Magnetic Field of the Straight Wire

The long straight wire acts as the source of our magnetic field. According to the right-hand grip rule, if you point your thumb in the direction of the current $I$ (upwards), your fingers will curl into the page on the right side of the wire.

Therefore, the entire rectangular loop is immersed in a magnetic field $\vec{B}$ that points directly into the screen.

Crucially, this magnetic field is not uniform. Its magnitude is given by:

$$B = \frac{\mu_0 I}{2\pi r}$$

where $r$ is the distance from the wire. The field is stronger closer to the wire and weaker further away.

### Analyzing the Forces: Top and Bottom

Let's break the loop into four segments and analyze the Lorentz force $\vec{F} = i(\vec{L} 	imes \vec{B})$ on each.

For the top segment $bc$, the current flows to the right, and the magnetic field is into the page. Using Fleming's left-hand rule, the force $F_{bc}$ points upwards.

For the bottom segment $da$, the current flows to the left. The force $F_{da}$ points downwards.

Because these two segments are symmetric and lie at the same range of distances from the straight wire, the upward force perfectly cancels the downward force. The loop will not move vertically.

### Analyzing the Forces: Left and Right

Now, let's look at the vertical segments.

The left segment $ab$ carries current in the same direction as the straight wire. We know that parallel currents attract. Thus, it experiences a force $F_{ab}$ directed towards the left (towards the straight wire).

The right segment $cd$ carries current in the opposite direction. Anti-parallel currents repel. Thus, it experiences a force $F_{cd}$ directed towards the right (away from the straight wire).

### The Deciding Factor: Proximity

We have a tug-of-war! The left segment is being pulled, and the right segment is being pushed. Which force wins?

Remember our magnetic field formula? The field is stronger closer to the wire. Since segment $ab$ is closer to the straight wire than segment $cd$, the magnetic field it experiences is stronger.

Consequently, the attractive force $F_{ab}$ is significantly greater than the repulsive force $F_{cd}$.

$$F_{ab} > F_{cd}$$

**Therefore, the net force on the loop is directed towards the straight wire, and the loop will move towards it.**

Select Answer:

Visualized Solution

$System Setup$

$Magnetic Field$

$Forces on Horizontal Segments$

$Forces on Vertical Segments$

$Net Force$

$Conclusion$

The Sigma Insight: Magnetic Force on Current

The Setup

The Magnetic Field of the Straight Wire

Analyzing the Forces

Analyzing the Forces

The Deciding Factor

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A wire $A B C D E F$ (with each side of length $L$ ) bent as shown in figure and carrying a current $I$ is placed in a uniform magnetic induction $B$ parallel to the positive $y$ -direction. The force experienced by the wire is ......in the ...... direction.

Select Answer:

Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } System Setup

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Magnetic Field

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Forces on Horizontal Segments

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Forces on Vertical Segments

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Net Force

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Conclusion

The Sigma Insight: Magnetic Force on Current

The Setup

The Magnetic Field of the Straight Wire

Analyzing the Forces

Analyzing the Forces

The Deciding Factor

Similar Questions

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$System Setup$

$Magnetic Field$

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$Net Force$

$Conclusion$

A conducting loop carrying a current $I$ is placed in a uniform magnetic field pointing into the plane of the paper as shown. The loop will have a tendency to