Question

A metal rectangle is close to a long, straight, current-carrying wire, with two of its sides parallel to the wire. If the current in the long wire is decreasing, is the rectangle repelled by or attracted to the wire? Explain why this result is consistent with Lenz’s law.

Short answer

The rectangular loop is attracted to the wire.

Step-by-step solution

Important Concepts

Lenz’s law states that an induced current or emf always tends to oppose the change that causes it

Right-hand rule for a magnetic field around a current-carrying wire: Point your thumb of your right hand in the direction of the current. Your fingers now curl, around the wire in the direction of the magnetic field lines.

Right-hand rule for a magnetic field produced by a current in a loop;

When the fingers of your right hand curl in the direction of the current, your right thumb points in the direction of the magnetic field lines.

Two parallel conductors carrying current in the same direction attract each other. And parallel conductors carrying currents in opposite directions repel each other.

Magnetic force per unit length between two long, parallel current-carrying conductors is given by

$\frac{\mathbf{F}}{\mathbf{L}}\mathbf{=}\frac{{\mathbf{\mu }}_{\mathbf{0}}\mathbf{ll}\mathbf{\text{'}}}{\mathbf{2}\mathbf{\pi r}}$

Where r is the separation distance between the two wires.

Set Up

Let the wire is parallel to the longer sides of the rectangular loop, and the wire and the loop are in the same plane. The wire is closer to one of the longer sides of the loop and away from the other as shown in the figure below. Also, let the direction of the current in the wire is upward.

Application of right hand rule

Using the right-hand rule on the wire, we point the right thumb upward in the direction of the current, so we find that the fingers curl in the direction of the magnetic field lines. Where it is outward ( .) (out of the screen) at the rectangular loop. Since the current in the wire is decreasing, the magnetic field at the loop is also decreasing. So there is a change in the magnetic flux at the loop. Applying Lenz's law, this change in the magnetic flux at the loop induces a current in the loop to oppose this change. Thus, the magnetic field produced by the induced current is directed outward (. ) (out of the screen) to oppose the decrease in the upward magnetic field due to the current-carrying wire. Now, we use the right-hand rule on the loop; We point the right thumb outward (. ), so the fingers of the right hand curl in the direction of the induced current which is counter clockwise. This makes the direction of the current in the closer side of the loop to the wire is upward and that in the far side of the loop is downward

Conclusion

Since the current in the closer side is in the same direction of the current in the wire, they attract each other. And since the current in the far side is opposite in direction to the current in the wire, they repel each other. According to equation (1), the magnetic force between two parallel current-carrying conductors is inversely proportional to the separation distance. Thus, the magnitude of the magnetic force between the wire and the closer side is larger than that between the wire and the far side. So, the attractive force is larger than the repelling force. Therefore, the rectangular loop is attracted to the wire.

Hence,The rectangular loop is attracted to the wire.