Problem 62 - University Physics with Modern Physics

Question

A pair of long, rigid metal rods, each of length \(L,\) lie parallel to each other on a perfectly smooth table. Their ends are connected by identical, very light conducting springs of force constant \(k\) (Fig. 28.55 ) and negligible unstretched length. If a current \(I\) runs through this circuit, the springs will stretch. At what separation will the rods remain at rest? Assume that \(k\) is large enough so that the separation of the mods will be much less than \(L\) .

Step-by-Step Solution

Answer

The separation is \( d = \sqrt{\frac{\mu_0 I^2 L}{2\pi k}} \)."

Step 1: Understanding the Forces Involved

In this system, there are electromagnetic forces due to the currents in the rods and mechanical forces exerted by the springs. The rods repel each other due to the magnetic force generated by the currents, while the springs try to keep them together due to their restoring force.

Step 2: Applying Magnetic Force Formula

The force between two parallel currents is given by the formula: \( F_m = \frac{\mu_0 I^2 L}{2\pi d} \), where \( F_m \) is the magnetic force, \( \mu_0 \) is the permeability of free space, \( I \) is the current, \( L \) is the length of the rods, and \( d \) is the separation between the rods.

Step 3: Considering the Spring Force

The spring force exerted by the springs follows Hooke's Law: \( F_s = kd \), where \( F_s \) is the spring force, \( k \) is the spring constant, and \( d \) is the separation distance over which the spring is stretched.

Step 4: Equating the Forces for Equilibrium

In equilibrium, the magnetic force is balanced by the spring force. Therefore, equate the magnetic force to the spring force: \( \frac{\mu_0 I^2 L}{2\pi d} = kd \).

Step 5: Solving for the Separation Distance

Rearrange the equation \( \frac{\mu_0 I^2 L}{2\pi d} = kd \) to solve for \( d \). Multiply both sides by \( d \) to get \( \frac{\mu_0 I^2 L}{2\pi} = kd^2 \). Then solve for \( d \): \( d = \sqrt{\frac{\mu_0 I^2 L}{2\pi k}} \).

Key Concepts

Parallel Currents

When electric currents flow through two parallel wires, a fascinating interaction occurs: they exert magnetic forces on each other. These forces arise because moving electric charges, like those in a current, create magnetic fields. Here's how it works:

• If two currents flow in the same direction in parallel wires, they attract each other.
• If the currents flow in opposite directions, they repel each other.

This phenomenon is due to Ampère's law, which describes how currents interacting in magnetic fields generate these forces. In the original exercise, the rods with currents are repelling each other because they likely have currents in opposite directions. This repulsion tends to push the rods apart, a central factor in solving the problem.

Hooke's Law

Hooke's Law is a fundamental principle in physics that describes how springs behave under tension or compression. It states that the force exerted by a spring is proportional to its displacement from its original position, and can be mathematically expressed as:\[ F_s = kx \]where:

• \( F_s \) is the force exerted by the spring.
• \( k \) is the spring constant, a measure of the spring's stiffness.
• \( x \) is the displacement from the spring's unstretched position.

In the context of this exercise, Hooke's Law is used to calculate the force the springs exert when the rods are magnetically repelling each other. The balance of forces between the magnetic repulsion and the spring force keeps the rods at a stable separation when they are at rest.

Spring Constant

The spring constant \( k \) is a crucial figure that describes a spring's rigidity. A higher \( k \) value implies a stiffer spring, providing more resistance to being stretched or compressed.In the given exercise, the springs used to connect the rods are described as having a very large spring constant. This means they are quite stiff, which is important because:

• A large \( k \) ensures that the rods will not move significantly apart, even under strong magnetic forces.
• The springs effectively oppose the magnetic repulsion between the rods.

Understanding how \( k \) affects the system is key to solving the exercise, as the balance between the magnetic force and the spring's resistance must be maintained for the rods to stay at rest. The relationship is clarified by the equilibrium equation, which helps to find the separation distance \( d \).