Two long, straight, parallel wires 8.0 cm apart carry currents in opposite directions.
Use the right-hand source and force rules to determine whether the forces on the wires are attractive or repulsive.
a)Repulsive
b)Attractive
If the wires carry equal currents of 4.0 A, what is the magnetic field magnitude that each produces at the other's location?
B= microT
Use the result of part C to determine the magnitude of the force per unit length they exert on each other.
F/L = microT/m
Currents in parallel wires that are in the same direction result in an attractive force between the wires. If in opposite directions, they repel.
Use Ampere's Law for the B field and F = F = B I L for the force per length L
B = (4*pi*4A)/(2*pi*0.08m)
To determine whether the forces on the wires are attractive or repulsive, we can use the right-hand source and force rules:
1. Right-hand source rule: If you point the thumb of your right hand in the direction of the current in one wire, the fingers will curl around in the direction of the magnetic field caused by that wire.
2. Right-hand force rule: If you point the fingers of your right hand in the direction of the magnetic field caused by one wire, and then point your thumb in the direction of the current in the second wire, the palm of your hand will push in the direction of the force between the wires.
Now, let's analyze the situation based on this:
Since the currents in the wires are in opposite directions, the magnetic fields created by each wire will have opposite directions as well.
Using the right-hand force rule, if we point the fingers of our right hand in the direction of the magnetic field caused by one wire (let's say Wire 1), and then point our thumb in the direction of the current in the second wire (Wire 2), the palm of our hand will push in the direction of the force between the wires.
If we follow this procedure for both wires, we will find that the forces between the wires are in the same direction, pushing them away from each other (outwards). Therefore, the forces on the wires are repulsive.
Now, to find the magnetic field magnitude that each wire produces at the other's location, we can use Ampere's Law, which states that the magnetic field produced by a long straight wire is directly proportional to the current and inversely proportional to the distance from the wire.
Using this formula, we have:
B = (μ0 * I) / (2π * R)
where μ0 is the permeability of free space (4π * 10^-7 Tm/A), I is the current in the wire, and R is the distance from the wire.
Since the wires are 8.0 cm apart and carry equal currents of 4.0 A, we can calculate the magnetic field magnitude (B) that each wire produces at the other's location:
B = (4π * 10^-7 Tm/A * 4.0 A) / (2π * 0.08 m)
Simplifying this equation, we find that B = 0.02 μT (microteslas).
Finally, to determine the magnitude of the force per unit length that the wires exert on each other, we can use the formula:
F/L = μ0 * I1 * I2 / (2π * R)
where F/L is the force per unit length, μ0 is the permeability of free space, I1 and I2 are the currents in the wires, and R is the distance between the wires.
Since the currents in the wires are equal and opposite (4.0 A), and the wires are 8.0 cm = 0.08 m apart, we can plug in the values:
F/L = (4π * 10^-7 Tm/A) * 4.0 A * 4.0 A / (2π * 0.08 m)
Simplifying this equation, we find that F/L = 0.01 μT/m (microteslas per meter).
To determine whether the forces on the wires are attractive or repulsive, we can use the right-hand source and force rules. These rules state the following:
1. Right-hand source rule: If you point your right thumb in the direction of the current in one wire, the magnetic field lines due to that current will circulate around the wire in the direction your fingers would naturally curl.
2. Right-hand force rule: If you place your right hand flat with your fingers pointing in the direction of the current in the second wire, and then curl your fingers towards the first wire, your thumb will point in the direction of the force that the second wire experiences due to the magnetic field of the first wire.
Now, let's apply these rules:
Since the currents in the two wires are in opposite directions, the magnetic fields they produce will have opposite directions at the other wire's location. According to the right-hand source rule, the magnetic field lines due to the first wire will circulate counterclockwise, while the magnetic field lines due to the second wire will circulate clockwise.
Next, let's consider the forces experienced by each wire due to the magnetic fields produced by the other wire. By using the right-hand force rule, if we curl our fingers from the direction of the magnetic field of one wire (counterclockwise in this case) towards the other wire (clockwise in this case), our thumb points away from us. This indicates that the forces the wires exert on each other are repulsive.
Moving on to the second part of the question, we need to determine the magnetic field magnitude produced by each wire at the other wire's location. To do this, we can use the formula for the magnetic field produced by a long, straight wire:
B = (μ₀ * I) / (2π * r)
where B is the magnetic field magnitude, 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 wires carry equal currents of 4.0 A, and the distance between the wires is 8.0 cm = 0.08 m, we can substitute these values into the formula:
B = (4π x 10^-7 T m/A * 4.0 A) / (2π * 0.08 m)
Simplifying the expression, we find:
B = (4π^2 x 10^-7 T m) / (2π * 0.08 m)
B = 2π x 10^-7 T
So, each wire produces a magnetic field magnitude of 2π x 10^-7 T at the other wire's location.
Finally, to determine the magnitude of the force per unit length that the wires exert on each other, we can use the formula:
F/L = μ₀ * I₁ * I₂ / (2π * d)
where F/L is the force per unit length, μ₀ is the permeability of free space, I₁ and I₂ are the currents in the wires, and d is the distance between the wires.
Since the currents in the wires are equal and opposite (4.0 A and -4.0 A), the force per unit length simplifies to:
F/L = μ₀ * (4.0 A)^2 / (2π * 0.08 m)
Substituting the values and simplifying, we get:
F/L = (4π x 10^-7 T m/A) * (4.0 A)^2 / (2π * 0.08 m)
F/L = 5 x 10^-5 N/m
Therefore, the magnitude of the force per unit length that the wires exert on each other is 5 x 10^-5 N/m.