91Ó°ÊÓ

The spring constant, often denoted as \( k \), is a measure of a spring's stiffness. It's a fundamental concept in mechanics and provides us with a clear understanding of how much force is needed to compress or extend a spring by a certain distance. This is mathematically represented by Hooke's Law:\[ F = k \times \Delta x \]where:

• \( F \) is the force applied
• \( k \) is the spring constant
• \( \Delta x \) is the change in the spring's length from its rest position

For our problem, the spring constant plays a crucial role in determining the force exerted when the spring is compressed due to the magnetic force. Given that \( k = 150 \ \text{N/m} \) and the spring is compressed by 0.02 meters, the spring's force becomes \( 3 \ \text{N} \). This allows us to relate the magnetic force on the rods to the mechanical resistance provided by the spring.

The permeability of free space, denoted as \( \mu_0 \), relates to how well it can support magnetic field formation. It's a physical constant crucial in the context of electromagnetism and plays an essential role when calculating the magnetic force between currents.For most problems involving magnetic forces between currents in air or vacuum, the value used is always \( \mu_0 = 4\pi \times 10^{-7} \ \text{Tm/A} \). It appears in the formula calculating the magnetic force between two parallel current-carrying wires: \[ F/L = \frac{\mu_0 I_1 I_2}{2\pi d} \]Here, the permeability of free space helps determine how these currents interact over a distance \( d \). Understanding \( \mu_0 \) allows us to correctly apply this formula to find the magnetic force created by the wires.

When two wires carry current in the same direction, they experience an attractive force. This fundamental principle of electromagnetism is due to the interaction between the magnetic fields generated by each current.According to Ampère's force law:- The force per unit length between two parallel conductors carrying current \( I_1 \) and \( I_2 \) is attractive and proportional to the product of the currents and inversely proportional to the distance between them.This behavior is governed by the formula:\[ F/L = \frac{\mu_0 I_1 I_2}{2\pi d} \]This formula shows how the distance \( d \) affects the magnetic attraction between currents. In the problem, the two rods carrying parallel currents of 950 A each feel this attractive force, causing the compression of the spring.

The Lorentz force describes the force on a charged particle moving through electric and magnetic fields. While it deals with individual charges, understanding this force helps grasp broader concepts like current interaction in wires.For a moving charged particle, the Lorentz force is given by:\[ \mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \]where:

• \( q \) is the electric charge
• \( \mathbf{E} \) is the electric field
• \( \mathbf{v} \) is the particle's velocity
• \( \mathbf{B} \) is the magnetic field

In the case of currents in wires, the collective motion of electrons results in a magnetic field, which then interacts with the fields from other currents. For parallel wires, this interaction leads to the magnetic forces calculated using the force per unit length formula, bringing us back full circle to the forces observed in the exercise.