Two long, straight, parallel wires 10 cm apart carry currents in opposite directions. (a) Use the right-hand source and force rules to determine whether the forces on the wires are (1) attractive or (2) repulsive. Show your reasoning. (b) If the wires carry equal currents of 3.0 A, what is the magnetic field magnitude that each produces at the other's location? (c) use the result of part (b) to determine the magnitude of the force per unit length they exert on each other.
a) The forces on the wires are repulsive. This is because when the right hand is wrapped around the wire in the direction of the current, the thumb points away from the other wire. This means that the two currents are pushing away from each other, creating a repulsive force.
b) The magnetic field magnitude that each produces at the other's location is B = μ_0I/2πd, where μ_0 is the permeability of free space, I is the current, and d is the distance between the wires. Therefore, the magnetic field magnitude is B = (4π x 10^-7 Tm/A)(3.0 A)/(2π x 0.1 m) = 7.85 x 10^-5 T.
c) The magnitude of the force per unit length they exert on each other is F/L = B^2/μ_0. Therefore, the magnitude of the force per unit length is F/L = (7.85 x 10^-5 T)^2/(4π x 10^-7 Tm/A) = 0.62 N/m.
(a) To determine whether the forces on the wires are attractive or repulsive, we can use the right-hand source and force rules.
According to the right-hand source rule, if we point the thumb of our right hand in the direction of the current in the first wire, the fingers will curl in the direction of the magnetic field created by the first wire.
Next, using the right-hand force rule, we can determine the direction of the force on the second wire by pointing the fingers of our right hand in the direction of the magnetic field created by the first wire and the thumb in the direction of the current in the second wire.
Since the fingers on our right hand point in the opposite direction (towards each other), this indicates that the magnetic fields created by the wires are in opposite directions. Therefore, according to the right-hand force rule, the forces on the wires are repulsive.
(b) The magnetic field magnitude that each wire produces at the other's location can be determined using Ampere's Law. In this case, we have two parallel wires, and the magnetic field at a given location is given by:
B = (μ₀ * I) / (2π * r)
Where μ₀ is the permeability of free space (4π x 10^(-7) T m/A), I is the current, and r is the distance from the wire.
Since the distance between the wires is 10 cm (0.1 m), and the current in each wire is 3.0 A, we can calculate the magnetic field magnitude produced by each wire at the other's location:
B = (4π x 10^(-7) T m/A * 3.0 A) / (2π * 0.1 m)
B = 6 x 10^(-6) T
Therefore, each wire produces a magnetic field magnitude of 6 x 10^(-6) T at the other's location.
(c) The magnitude of the force per unit length (F) that the wires exert on each other can be determined using the formula:
F = μ₀ * I₁ * I₂ * L / (2π * d)
Where μ₀ is the permeability of free space, I₁ and I₂ are the currents in the wires, L is the length of the wires, and d is the distance between the wires.
Since the currents in both wires are equal (3.0 A), the length of the wires is not provided, we will assume a unit length of 1 m. Therefore, the force per unit length is:
F = (4π x 10^(-7) T m/A * 3.0 A * 3.0 A * 1 m) / (2π * 0.1 m)
F = 0.018 N/m
Therefore, the magnitude of the force per unit length that the wires exert on each other is 0.018 N/m.
To answer these questions, we can use the right-hand rule for determining the direction of the magnetic field and the right-hand force rule for determining the direction of the force between the wires.
(a) The right-hand source rule states that if you point your right thumb in the direction of the current in one wire, then the direction of the magnetic field lines produced by that wire will be in the direction your curled fingers would point. Applying this rule to the two wires, we can determine whether the forces on the wires are attractive or repulsive.
- First, let's consider the left wire. If we point our right thumb in the direction of the current flowing through it, our curled fingers would point towards us. This means that the magnetic field produced by the left wire will be pointing towards us.
- Now, let's consider the right wire. If we point our right thumb in the direction of the current flowing through it, our curled fingers would point away from us. This means that the magnetic field produced by the right wire will be pointing away from us.
According to the right-hand force rule, when two parallel wires carry currents in opposite directions, the forces between the wires are attractive. Therefore, the forces on the wires in this case are attractive.
(b) To calculate the magnetic field magnitude produced by each wire at the other wire's location, we can use the formula for the magnetic field due to a straight wire:
B = (μ₀ * I) / (2π * r)
where B is the magnetic field magnitude, μ₀ is the permeability of free space (4π × 10⁻⁷ T·m/A), I is the current in the wire, and r is the distance between the wire and the point where the field is being calculated.
Let's calculate the magnetic field produced by each wire at the other wire's location:
- Magnetic field due to the left wire at the location of the right wire:
B_left = (4π × 10⁻⁷ T·m/A * 3.0 A) / (2π * 0.10 m) = 6 × 10⁻⁵ T
- Magnetic field due to the right wire at the location of the left wire:
B_right = (4π × 10⁻⁷ T·m/A * 3.0 A) / (2π * 0.10 m) = 6 × 10⁻⁵ T
Therefore, each wire produces a magnetic field magnitude of 6 × 10⁻⁵ T at the other wire's location.
(c) The force per unit length between the wires can be calculated using the formula:
F = (μ₀ * I₁ * I₂ * L) / (2π * d)
where F is the force per unit length, μ₀ is the permeability of free space, I₁ and I₂ are the currents in the wires, L is the length of the wires, and d is the distance between the wires.
In this case, the wires are parallel, so the length L cancels out. Also, the currents in both wires are equal to 3.0 A. Using the given distance of 10 cm (0.10 m) between the wires, we can calculate the force per unit length:
F = (4π × 10⁻⁷ T·m/A * 3.0 A * 3.0 A) / (2π * 0.10 m) = 9 × 10⁻⁵ N/m
Therefore, the magnitude of the force per unit length between the wires is 9 × 10⁻⁵ N/m.