Three long wires are parallel to a z axis, and each carries...

Three long wires are parallel to a $z$ axis, and each carries a current of 10 $\mathrm{A}$ in the positive $z$ direction. Their points of intersection with the $x y$ plane form an equilateral triangle with sides of $50 \mathrm{cm},$ as shown in Fig.29- $78 .$ A fourth wire (wire $b )$ passes through the midpoint of the base of the triangle and is parallel to the other three wires. If the net magnetic force on wire $a$ is zero, what are the (a) size and (b) direction $(+z$ or $-z)$ of the current in wire $b ?$

Three long wires are parallel to a $z$ axis, and each carries a current of
10 $\mathrm{A}$ in the positive $z$ direction. Their
points of intersection with the $x y$
plane form an equilateral triangle
with sides of $50 \mathrm{cm},$ as shown in Fig.29- $78 .$ A fourth wire (wire $b )$ passes
through the midpoint of the base of
the triangle and is parallel to the
other three wires. If the net magnetic
force on wire $a$ is zero, what are the (a) size and (b) direction $(+z$ or $-z)$ of the current in wire $b ?$

Key Concepts

Symmetry in Electromagnetic Configurations

Exploiting symmetry can simplify the analysis of electromagnetic systems by reducing the complexity of the field and force calculations. Symmetric arrangements often lead to cancellations or predictable patterns in the direction and magnitude of fields and forces, helping in setting up conditions for equilibrium or zero net force in the system.

Superposition Principle

In electromagnetic theory, the superposition principle states that the net magnetic field at any point is the vector sum of the individual magnetic fields produced by each current element. This principle allows complex arrangements of currents to be analyzed by considering the contribution of each current separately and then combining them.

Magnetic Force on a Current-Carrying Wire

When a current-carrying wire is placed in an external magnetic field, it experiences a magnetic force given by the cross product of the current element and the magnetic field. The magnitude and direction of this force depend on the current magnitude, the length of the wire segment, and the angle between the current direction and the field, following the Lorentz force law.

Magnetic Field from a Long Current-Carrying Wire

A long, straight current-carrying wire produces a magnetic field that circulates around it. The magnitude of this field at a point in space depends inversely on the distance from the wire, and its direction can be determined by the right-hand rule. This concept is fundamental in understanding how currents generate magnetic fields in space.

Right-Hand Rule

The right-hand rule is a mnemonic used in electromagnetism to determine the direction of magnetic fields and forces. For a current-carrying wire, if the right thumb points in the direction of the current, then the curled fingers indicate the direction in which the magnetic field circulates. This rule is essential for visualizing and predicting the orientation of forces and fields in electromagnetic systems.

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