Two long wires hang vertically. Wire 1 carries an upward...

Two long wires hang vertically. Wire 1 carries an upward current of 1.50 A. Wire 2, 20.0 cm to the right of wire 1, carries a downward current of 4.00 A. A third wire, wire 3, is to be hung vertically and located such that when it carries a certain current, each wire experiences no net force. (a) Is this situation possible? Is it possible in more than one way? Describe (b) the position of wire 3 and (c) the magnitude and direction of the current in wire 3.

Two long wires hang vertically. Wire 1 carries an upward current of 1.50 A. Wire 2, 20.0 cm to the right of wire 1, carries a downward current of 4.00 A. A third wire, wire 3, is to be hung vertically and located such that when it carries a certain current, each wire experiences no net force. (a) Is this situation possible? Is it possible in more than one way? Describe (b) the position of wire 3 and
(c) the magnitude and direction of the current in wire 3.

Key Concepts

Magnetic Field of a Long Straight Conductor

A long straight conductor carrying an electric current produces a magnetic field that circulates around it. According to Ampère's law, the magnitude of this magnetic field at a distance from the wire is given by B = ??I/(2?r), where I is the current and r is the distance from the wire. This fundamental principle is key to understanding the interactions between current-carrying wires.

Magnetic Force Between Currents

Current-carrying wires exert forces on each other due to their magnetic fields. Specifically, if the currents flow in the same direction, the wires attract; if they flow in opposite directions, they repel. The magnitude of this force per unit length is given by F/L = ??I?I?/(2?r), which shows that the force depends on both currents and their separation. This concept is central when analyzing equilibrium conditions in wire systems.

Right-Hand Rule

The right-hand rule is an essential tool for determining the direction of the magnetic field created by a current, as well as the direction of the magnetic force experienced by a conductor in a magnetic field. By aligning the thumb in the direction of the current, the curled fingers indicate the circular direction of the magnetic field, which is critical for predicting the interactions between different conductors.

Superposition Principle in Electromagnetism

The superposition principle states that the net magnetic field at any point in space is the vector sum of the individual fields produced by each current-carrying conductor. Similarly, the net force on a conductor is the sum of the forces due to each external magnetic field. This principle allows for complex configurations, such as the equilibrium of forces on multiple wires, to be analyzed by summing their individual contributions.

Equilibrium Conditions in Magnetic Systems

Achieving zero net force in a system involving magnetic interactions requires a careful balance between the magnetic forces resulting from different currents. By adjusting parameters such as the magnitude, direction, and position of currents, it is possible to create conditions where the attractive and repulsive forces cancel out. This concept is fundamental for understanding how a system can remain in a state of equilibrium despite the presence of multiple magnetic interactions.

Transcript

00:01 In this problem, we have two wires hanging vertically, and we want to figure out if we had a third wire, it is possible to set up a situation where each of the wires experiences no net force.

00:12 All right, so when we look at this problem, we know that before wire three is placed, wires one and two are going to repel each other because their currents are going in opposite directions.

00:23 If wire 3 is between wires 1 and 2, then no matter what the direction of the current in wire 3, either wire 1 or wire 2 are going to be repelled, right? since no matter what direction it goes, one of the other wires will be pointing in the opposite direction and they'll repel.

00:40 If wire 3 is located to the right of wire 2, then because wire 2 carries a greater current than wire 1, the force that's going to be exerted on wire 2, the force that's going to be exerted by wire 2 on wire 3 is going to be greater than the force exerted by wire 1 on wire 3.

01:02 So, in that scenario, wire 3 is not going to be an equilibrium.

01:06 One of the wires is going to pull it more strongly than the other.

01:09 So finally, if wire 3 is located to the left of wire 1 and the current in it is downward, it's going to repel wire 1, but it's going to attract wire 2.

01:21 And that means that it's not going to experience a net force.

01:24 It's going to be pulled equally in all directions.

01:26 So the situation is possible, but only in one specific setup.

01:31 Now, the next part of the problem is asking us to solve for this equilibrium distance.

01:38 And we start once again by setting up our equilibrium forces...

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