Two long, parallel wires hang by 4.00-cm-long cords from a...

Two long, parallel wires hang by 4.00-cm-long cords from a common axis ($\textbf{Fig. P28.65}$). The wires have a mass per unit length of 0.0125 kg/m and carry the same current in opposite directions. What is the current in each wire if the cords hang at an angle of 6.00$^\circ$ with the vertical?

Key Concepts

Magnetic Force Between Parallel Currents

This concept refers to the interaction between two current-carrying conductors, where each conductor produces a magnetic field that exerts a force on the other. The force per unit length between the currents is given by an expression involving the permeability of free space and the product of the currents divided by the distance between them. This is a fundamental aspect of electromagnetism and is essential in analyzing forces in devices like electromagnets and power transmission lines.

Lorentz Force on a Current-Carrying Conductor

The Lorentz force law states that a moving charge or a current-carrying conductor in the presence of a magnetic field experiences a force. For a current element, this force is calculated as the cross product of the current element and the magnetic field. This principle underlies the interaction between the magnetic field produced by one wire and the current flowing in another, leading to a measurable lateral force.

Equilibrium of Forces in a Suspended Conductor

When a conductor is suspended by a cord or string, it hangs in equilibrium under the influence of gravitational, magnetic, and tension forces. Analyzing the system involves resolving these forces into their components and applying the conditions for equilibrium, where the net force in any direction is zero. This approach allows one to relate the magnetic force acting horizontally to the gravitational force acting vertically.

Trigonometric Analysis in Force Balance

In many equilibrium problems involving suspended objects, trigonometry is used to resolve the tension in the supporting string into vertical and horizontal components. By relating these components to the weight of the object and the horizontal force (such as the magnetic force), one obtains equations that can be solved for unknown quantities, such as the current in the conductors. This analysis is crucial for linking the geometry of the system to the dynamical forces at play.

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