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 $\mathrm{A}$ . To the right, wire 2 carries a downward current of 4.00 $\mathrm{A}$ . Wire 3 is to be 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 from a Straight Conductor

This concept describes the magnetic field produced by a long, straight wire carrying an electric current. According to the Biot-Savart Law (or derived from Ampere’s Law), the magnetic field forms concentric circles around the wire and its magnitude decreases with distance from the wire. Understanding this is crucial for predicting how the fields from separate wires interact.

Force on a Current-Carrying Conductor

A current-carrying wire placed in a magnetic field experiences a force due to the Lorentz force law. The force per unit length, given by F/L = I × B, depends on both the magnitude of the current and the magnetic field, as well as the angle between them (usually determined using the right-hand rule). This force is central to analyzing the interactions between multiple wires.

Superposition Principle in Magnetostatics

The superposition principle states that the net magnetic field at a point due to multiple current sources is the vector sum of the fields due to each current individually. This principle is critical when determining the total magnetic interaction in a system of several wires, as it allows one to consider each wire’s effect separately before combining them.

Equilibrium Conditions in Magnetic Systems

For a system of current-carrying wires to be in equilibrium, the net magnetic forces on each wire must be zero. This involves balancing the forces from the magnetic fields generated by each current, taking into account their magnitudes and directions. Achieving such a balance often requires precise positioning of the wires and careful consideration of the direction of each current.

Transcript

00:01 Hello everyone in this problem it is given there are three long conducting wire which are suspended vertically this is the first wire this is your first wire this is the second wire and this is the third wire one wire two this carries the current of 1 .5 ampere this carries the current 4 ampere in the onward direction and the separation between the two wires 1 and 2 is 20 cm there is a third wire its position is not given and current is not given and the first part we have to find is it possible net force on each wire to be 0 to be 0 so the possible or not that we have to analyze in second part we have to find the position of third wire so that net force on each wire to be zero and in third case we have to find the current flowing through the third wire now let us see here if we see the figure first wire and second wire are repelling each other because current between them in a poor j direction so second wire apply the portion the first wire towards left if current in the third wire is such that it is also repelling that is it is applying the force to the right the resultant force may be zero that means current in third wire is same as that of second wire now we will analyze the force on second wire first and first by repelling it but third wire is attracting it to the resultant force on it may be zero similarly if we analyze the net force on third wire third and first having opposite current so first by repelling it but third and second wire if having the same current then they attract so resultant force maybe zero so answer of first is yes if third way having current in the same direction as that of second wire.

03:35 Now we will solve for the second part.

03:43 We have to find the value of the for net force 0.

03:46 D we have to find for net force on the third wire is to be 0.

04:00 It means force on third wire due to first must be equal and opposed.

04:05 Force on third due to second...

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