Two parallel wires in a horizontal plane carry currents I1...

Two parallel wires in a horizontal plane carry currents $I_{1}$ and $I_{2}$ to the right. The wires each have length $L$ and are separated by a distance $d$. Find the magnitudes and directions of the (a) the magnetic field due to wire 1 at the location of wire $2,$ (b) the magnetic force on wire $2,$ (c) the magnetic field due to wire 2 at the location of wire $1,$ and $(\mathrm{d})$ the magnetic force on wire $1 .$ (e) Do parallel currents in the same direction attract or repel? What about parallel currents in opposite directions? (f) Are the magnitudes and directions of the forces consistent with Newton's third law?

Two parallel wires in a horizontal plane carry currents $I_{1}$ and $I_{2}$ to the right. The wires each have length $L$ and are separated by a distance $d$. Find the magnitudes and directions of the (a) the magnetic field due to wire 1 at the location of wire $2,$ (b) the magnetic force on wire $2,$ (c) the magnetic field due to wire 2 at the location of wire $1,$ and $(\mathrm{d})$ the magnetic force on wire $1 .$ (e) Do parallel currents in the same direction attract or repel? What about parallel currents in opposite directions?
(f) Are the magnitudes and directions of the forces consistent with Newton's third law?

Key Concepts

Newton's Third Law in Electromagnetic Interactions

Newton's third law states that for every action, there is an equal and opposite reaction. This principle applies to electromagnetic forces as well, ensuring that if one current-carrying wire exerts a force on another, the second wire exerts an equal and opposite force on the first. This consistency is fundamental to the coherence of physical laws, even in electromagnetic contexts.

Right-Hand Rule

The right-hand rule is a standard tool used to determine the directions of magnetic fields and forces in electromagnetism. For the magnetic field around a current-carrying wire, you point your thumb in the direction of the current and your curled fingers indicate the direction of the magnetic field. For the force on a current element, point your fingers in the direction of the current and curl them towards the direction of the magnetic field; your thumb then points in the direction of the force.

Interaction of Parallel Currents

This concept explains how two parallel current-carrying wires interact with each other through their magnetic fields. When the currents flow in the same direction, each wire experiences a force that pulls it towards the other, and when the currents flow in opposite directions, the force is repulsive. This interaction is a direct consequence of the magnetic fields generated by each wire and the resultant Lorentz forces acting on them.

Magnetic Force on a Current-Carrying Conducting Wire

When a current-carrying wire is placed in an external magnetic field, it experiences a force. This is described by the Lorentz force law for currents, F = I L × B, where I is the current, L is the length vector of the wire, and B is the magnetic field. The direction of the force is given by the cross-product, meaning it is perpendicular to both the current direction and the magnetic field.

Magnetic Field of a Long Straight Conductor

This concept involves understanding how a current flowing in a long, straight wire generates a magnetic field around it. According to Ampère’s law, the magnetic field at a distance from the wire is given by B = ??I/(2?d), where ?? is the permeability of free space, I is the current, and d is the distance from the wire. This field forms concentric circles around the wire, with the magnitude inversely proportional to the distance from the wire.

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