Three long wires (wire 1, wire 2, and wire 3 ) hang...

Three long wires (wire $1,$ wire $2,$ and wire 3 ) hang vertically. The distance between wire 1 and wire 2 is $20.0 \mathrm{cm} .$ On the left, wire 1 carries an upward current of 1.50 A. To the right, wire 2 carries a downward current of $4.00 \mathrm{A}$. Wire 3 is located such that when it carries a certain current, each wire experiences no net force. Find (a) the position of wire $3,$ and (b) the magnitude and direction of the current in wire 3.

Key Concepts

Superposition Principle

The superposition principle states that the net magnetic field or force at any point due to multiple sources is the vector sum of the fields or forces produced individually by each source. In systems with several current-carrying wires, this principle allows for the calculation of overall effects by considering each contribution separately before summing them.

Magnetic Field from a Current-Carrying Conductor

A long, straight conductor carrying an electric current produces a magnetic field that circles around the wire. The strength of this field at a distance from the wire is described by Ampere's law, and it depends inversely on the distance from the wire. This concept is fundamental for understanding how currents interact via magnetic forces.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field relative to the direction of current flow. By pointing the thumb in the direction of the current, the curled fingers show the looping direction of the magnetic field lines. This rule helps in visualizing and predicting the directional behavior of magnetic effects in current-carrying conductors.

Magnetic Force on a Current-Carrying Wire

A wire carrying an electric current placed in an external magnetic field experiences a force, as described by the Lorentz force law. The magnitude and direction of this force depend on the current, the magnetic field strength, and the angle between the current direction and the magnetic field. This concept is critical in analyzing interactions among multiple current-carrying wires.

Equilibrium Conditions

Equilibrium conditions in electromagnetic systems occur when the net force acting on an object or system is zero. In the context of multiple current-carrying wires, achieving equilibrium means arranging the currents and positions so that the attractive and repulsive forces balance. Understanding these conditions is essential for designing systems with stable configurations.

Transcript

00:02 All right, we have got, oh yeah, three long wires.

00:06 Wire 1, 2, 3 hang vertically like so.

00:13 There we go.

00:15 And the distance between wire 1 and wire 2 is 20 centimeters.

00:22 And wire 1 carries an upward current of 1 .5 amps.

00:31 And wire 2 carries a downward current of 4 amps.

00:40 Wire 3 is somewhere so that each wire experiences no net force.

00:48 Magnitude, where is it, magnitude, and direction of the current on wire 3.

00:53 All right, so if we assume for the moment that wire 3 is over here on the right, then force is calculated by mu 0.

01:03 I -1, i -2 over 2 pi and the distance between the wires.

01:12 And if we take a look at the separate pairs, the first pair of wires will cause opposite, so they repel.

01:29 And if we assume the third wire, the amp is going up, the current's going up, then the same, so the first and the third wire would attract, and then opposite, so the second and third wire would repel.

01:46 So that's good because all the arrows are different, so when you add them together, you can get zero, so that's good.

01:54 All right, if we take the first pair of wires, wire 1 and wire 2 and we calculate so we get mu 0 times 1 .5 times 4 divided by 2 pi and 0 .2 minus.

02:14 Okay, so i'm doing the first wire, first wire and the third wire.

02:18 So mu 0 and 1 .5 times whatever the current is in the third wire.

02:28 So i just called it a for some reason, so 2 pi.

02:32 And then the distance between them would be 0 .2 plus x.

02:38 So x, there go.

02:40 So the distance between 1 and 2 is 0 .2 meters.

02:44 And the distance between 2 and 3, we are calling x for these purposes here...