Question

Two identical long straight conducting wires with a mass per unit length of \(25.0 \mathrm{g} / \mathrm{m}\) are resting parallel to each other on a table. The wires are separated by \(2.5 \mathrm{mm}\) and are carrying currents in opposite directions. (a) If the coefficient of static friction between the wires and the table is \(0.035,\) what minimum current is necessary to make the wires start to move? (b) Do the wires move closer together or farther apart?

Short answer

a) Minimum current is approximately 1.046 A. b) Wires move farther apart.

Step-by-step solution

Converting Units

First, convert the mass per unit length from grams per meter to kilograms per meter because SI units are required for calculations.Given mass per unit length: \(25.0 \text{ g/m}\)Converting to kilograms: \(25.0 \text{ g/m} = 0.025 \text{ kg/m}\)

Calculating Force of Friction

The frictional force that needs to be overcome is the product of the coefficient of static friction, mass per unit length, and gravitational acceleration.\[ F_{\text{friction}} = \mu \cdot m \cdot g \]where \(\mu = 0.035\), \(m = 0.025 \text{ kg/m}\), and \(g = 9.81 \text{ m/s}^2\).\[ F_{\text{friction}} = 0.035 \times 0.025 \times 9.81 = 0.00857975 \text{ N/m} \]

Using Ampere's Force Law

The magnetic force per unit length between two parallel currents \(I_1\) and \(I_2\) separated by a distance \(d\) is given by Ampere's force law:\[ F_{\text{mag}} = \frac{{\mu_0 \cdot I_1 \cdot I_2}}{{2\pi d}} \]For static friction to be overcome: \[ \frac{{\mu_0 \cdot I^2}}{{2\pi d}} = F_{\text{friction}} \]

Solving for Current

Using the formula from Ampere's force law and setting \(I_1 = I_2 = I\):\[ \frac{{4\pi \times 10^{-7} \cdot I^2}}{{2\pi \times 0.0025}} = 0.00857975 \]Simplifying, \(I^2\):\[ I^2 = \frac{{0.00857975 \cdot 2\pi \times 0.0025}}{{4\pi \times 10^{-7}}} \]\[ I^2 = \frac{{0.00857975 \times 0.005}}{{4\pi \times 10^{-7}}} \]Calculate \(I\):\[ I = \sqrt{\frac{{0.00857975 \times 0.005}}{{4\pi \times 10^{-7}}}} \approx 1.046 \text{ A} \]

Direction of Movement

Since the currents are in opposite directions, the force is repulsive. Once static friction is overcome, the wires will move farther apart due to this repulsion.

Key concepts

Ampere's Force Law

Ampere's Force Law is a fundamental principle in electromagnetism that describes the force per unit length between two current-carrying wires. When two parallel wires carry electric currents, they exert a magnetic force on each other. This magnetic force is attractive if the currents flow in the same direction and repulsive if they flow in opposite directions. The formula for this force per unit length, denoted by \( F_{\text{mag}} \), is: \ \[ F_{\text{mag}} = \frac{{\mu_0 \cdot I_1 \cdot I_2}}{{2\pi d}} \] \ where:

• \( \mu_0 \) is the permeability of free space, approximately \( 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A} \)
• \( I_1 \) and \( I_2 \) are the currents through the wires
• \( d \) is the distance between the wires

Understanding Ampere's Force Law helps explain how and why currents in parallel wires interact, affecting the structural integrity and behavior of circuits. In situations where precision is crucial, like in high-voltage power lines, this principle ensures stability and efficiency.

Coefficient of Static Friction

The coefficient of static friction is a measure that describes how much force is needed to start moving an object at rest relative to a surface. It represents the threshold between static and kinetic friction. Here, the coefficient of static friction is a crucial factor in determining whether the wires remain stationary or start to move. The equation for the static frictional force \( F_{\text{friction}} \) is given as: \ \[ F_{\text{friction}} = \mu \cdot m \cdot g \] \ where:

• \( \mu \) is the coefficient of static friction
• \( m \) is the mass per unit length of the wires
• \( g \) is the acceleration due to gravity, approximately \( 9.81 \text{ m/s}^2 \)

The static friction needs to be overcome for the wires to start moving. This component is vital in solving problems involving motion on surfaces, providing insights into designing and predicting the movement of objects.

Magnetic Fields

Magnetic fields are invisible forces created by moving electric charges or intrinsic magnetic moments. When current flows through a conductor, it creates a circular magnetic field around the wire. Understanding these fields is essential for explaining the interaction between current-carrying wires. The right-hand rule is a common method for visualizing magnetic fields, where your thumb points in the direction of the current and your fingers curl around in the direction of the field.
The interaction between magnetic fields from different wires results in a force, either attractive or repulsive, known as Ampere's force. Analyzing how these fields overlap and interact guides engineers and physicists in designing safe and efficient electrical systems, ensuring they harness beneficial forces while mitigating adverse effects.

Parallel Currents Interaction

Parallel currents interaction is a classic phenomenon in electromagnetism. When two parallel wires carry currents, they either attract or repel each other depending on the direction of the currents. If the currents flow in the same direction, the wires will attract; if they flow in opposite directions, they will repel. This interaction is a direct application of Ampere's Force Law, where the force is proportional to the magnitude of the currents and inversely proportional to their separation distance \( d \).
In our given problem, the currents flow in opposite directions, leading to a repulsive force. This repulsion causes the wires to move apart once the static friction is overcome. Understanding this behavior helps in practical situations such as managing space constraints in electrical systems or predicting mechanical stresses in structures like bridges that incorporate power lines. Therefore, engineers often design systems considering these forces to optimize safety and performance.