A 0.85 kg copper rod rests on two horizontal rails 1.0 m apart and carries a current of 65 A f

step 1 For this problem on the topic of magnetic fields, we are told that a copper rod with a mass of 1

A 0.85 kg copper rod rests on two horizontal rails 1.0 m...

A $0.85 \mathrm{~kg}$ copper rod rests on two horizontal rails $1.0 \mathrm{~m}$ apart and carries a current of 65 A from one rail to the other. The coefficient of static friction between rod and rails is $0.50$. What are the (a) magnitude and (b) angle (relative to the vertical) of the smallest magnetic field that puts the rod on the verge of sliding?

Key Concepts

Vector Decomposition of Forces

Vector decomposition involves breaking a force into its orthogonal components, typically along the horizontal and vertical directions. This approach simplifies the analysis of forces acting at angles, enabling the separate consideration of their effects in different directions and facilitating the application of equilibrium conditions to solve for unknowns such as magnitudes and directions.

Force Equilibrium

Force equilibrium occurs when the total forces acting on an object cancel out, resulting in no net acceleration. In problems involving the onset of motion, setting up equilibrium conditions allows one to relate different forces—such as gravitational, frictional, and applied forces—to determine critical values like the minimum force required to initiate movement.

Lorentz Force

The Lorentz force describes the force exerted on a current-carrying conductor when it is placed in a magnetic field. It is mathematically given by F = I(L × B), where I is the current, L is the length vector in the direction of the current, and B is the magnetic field. This concept is central to understanding how magnetic fields act on moving charges and currents, and it provides the basis for many electromagnetic applications.

Static Friction

Static friction is the force that resists the initiation of sliding motion between two contacting surfaces. Its maximum value is determined by the product of the coefficient of static friction and the normal force acting on the object. Understanding static friction is important in analyzing situations where an applied force must overcome this threshold to start motion.

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