Magnetic Force Between Parallel Currents: Videos & Practice Problems

When two parallel currents flow in the same direction, they attract each other, while opposite currents repel. Each current generates a magnetic field, described by the equation B = μ0I2l2. The force between them can be calculated using F = μ0I1I2l2πr. Understanding these principles is crucial for applications in electromagnetism and electrical engineering.

Magnetic Force Between Parallel Currents

The interaction between two parallel currents is a fundamental concept in electromagnetism, illustrating how electric currents generate magnetic fields and how these fields can exert forces on each other. When a current-carrying wire is placed in a magnetic field, it experiences a force, which can be calculated using the equation:

F = BIL \sin(\theta)

Here, F is the force, B is the magnetic field strength, I is the current, L is the length of the wire, and θ is the angle between the current and the magnetic field. The magnetic field produced by a straight current-carrying wire is given by:

B = \frac{\mu_0 I}{2 \pi r}

where μ₀ is the permeability of free space, I is the current, and r is the distance from the wire. When two parallel wires carry currents in the same direction, they attract each other, while currents in opposite directions repel each other. This behavior can be understood through the right-hand rule, which helps determine the direction of the magnetic field and the resulting force.

For two parallel wires, the force experienced by one wire due to the other can be expressed as:

F = \frac{\mu_0 I_1 I_2 L}{2 \pi r}

In this equation, I₁ and I₂ are the currents in the two wires, L is the length of the wires, and r is the distance between them. This mutual force is a result of Newton's third law, where the force exerted by wire 1 on wire 2 is equal in magnitude and opposite in direction to the force exerted by wire 2 on wire 1.

When calculating the force per unit length, the formula simplifies to:

F/L = \frac{\mu_0 I_1 I_2}{2 \pi r}

To illustrate these concepts, consider an example with two horizontal wires, each 10 meters long and separated by 0.5 meters. If the top wire carries a current of 2 A to the right and the bottom wire carries a current of 3 A to the left, the wires will repel each other due to their opposite currents. The magnitude of the force can be calculated using the earlier formula, substituting the known values:

F = \frac{(4\pi \times 10^{-7}) (2)(3)(10)}{2\pi (0.5)}

Upon calculation, this yields a force of approximately 2.4 × 10^-5 N. The direction of the force on the top wire is upward, while the force on the bottom wire is downward, confirming the repulsive interaction between the two currents.

Two very long wires of unknown lengths are a parallel distance of 2 m from each other. If both wires have 3 A of current flowing through them in the same direction, what must the force per unit length on each wire be? 
BONUS:Is the mutual force between the wires attractive or repulsive?

• A. 3.6×10⁻⁵ N/m
• B. 3.0×10⁻⁷ N/m
• C. 9.0×10⁻⁷ N/m

Answer: (C)

Here’s what students ask on this topic:

What is the magnetic force between two parallel currents?

The magnetic force between two parallel currents can be calculated using the formula:

$F=\frac{\mu 0I{1}_{}I{2}_{}l}{2\pi r}$

where $\mu 0$ is the permeability of free space, $I{1}_{}$ and $I{2}_{}$ are the currents in the wires, $l$ is the length of the wires, and $r$ is the distance between the wires. If the currents flow in the same direction, the wires attract each other; if they flow in opposite directions, the wires repel each other.

How do you use the right-hand rule to determine the direction of the magnetic field around a current-carrying wire?

To use the right-hand rule to determine the direction of the magnetic field around a current-carrying wire, follow these steps:

1. Point your thumb in the direction of the current flow.

2. Curl your fingers around the wire.

The direction in which your fingers curl represents the direction of the magnetic field lines around the wire. For example, if the current flows upward, the magnetic field will circulate around the wire in a counterclockwise direction when viewed from above.

What happens to the magnetic force between two parallel wires if the currents are in opposite directions?

If the currents in two parallel wires are in opposite directions, the wires will repel each other. This is because the magnetic fields generated by each wire interact in such a way that the forces between them push the wires apart. The magnitude of the force can be calculated using the formula:

$F=\frac{\mu 0I{1}_{}I{2}_{}l}{2\pi r}$

where $\mu 0$ is the permeability of free space, $I{1}_{}$ and $I{2}_{}$ are the currents, $l$ is the length of the wires, and $r$ is the distance between them.

How do you calculate the force per unit length between two parallel current-carrying wires?

The force per unit length between two parallel current-carrying wires can be calculated using the formula:

$\frac{F}{l}=\frac{\mu 0I{1}_{}I{2}_{}}{2\pi r}$

where $\mu 0$ is the permeability of free space, $I{1}_{}$ and $I{2}_{}$ are the currents in the wires, and $r$ is the distance between the wires. This formula gives the force experienced per unit length of the wires.

Why do parallel currents in the same direction attract each other?

Parallel currents in the same direction attract each other due to the interaction of their magnetic fields. Each current-carrying wire generates a magnetic field that circles around it. When the currents flow in the same direction, the magnetic fields interact in such a way that the forces between the wires pull them together. This attraction can be explained using the right-hand rule and the principle that magnetic fields exert forces on moving charges (currents). The force can be calculated using the formula:

$F=\frac{\mu 0I{1}_{}I{2}_{}l}{2\pi r}$

where $\mu 0$ is the permeability of free space, $I{1}_{}$ and $I{2}_{}$ are the currents, $l$ is the length of the wires, and $r$ is the distance between them.