Two long straight wires are parallel and 9.0 cm apart. They are to carry equal currents such that the magnetic field at a point halfway between them has magnitude 320 µT.
(a) Should the currents be in the same or opposite directions?
opposite
(b) How much current is needed?
a) If the currents were in the same direction in both wires, they would cancel in the middle
b) Use Ampere's law and double the field of a single wire
The current must be in the opposite direction if it has to produce some megnatic field B. B=ui/R
To determine the direction and magnitude of the current, we can use the formula for the magnetic field created by a long straight wire:
B = (μ₀ * I) / (2π * r)
Where:
- B is the magnetic field
- μ₀ is the permeability of free space (μ₀ = 4π * 10⁻⁷ Tm/A)
- I is the current
- r is the distance from the wire
Given:
- The wires are parallel and 9.0 cm apart (r = 4.5 cm = 0.045 m)
- The magnetic field at the midpoint is 320 µT (B = 320 * 10⁻⁶ T)
(a) Since the magnetic fields at the midpoint are in opposite directions, the currents in the wires must also be in opposite directions.
Therefore, the currents should be in opposite directions.
(b) Rearranging the formula for the current I:
I = (B * 2π * r) / μ₀
Substituting the given values:
I = (320 * 10⁻⁶ T * 2π * 0.045 m) / (4π * 10⁻⁷ Tm/A)
Simplifying:
I = (320 * 2π * 0.045) / 4
Calculating:
I = 9.048 A
Therefore, the currents in the wires should be opposite, and a current of approximately 9.048 A is needed.
To determine whether the currents should be in the same or opposite directions, we can apply the right-hand rule for magnetic fields.
First, let's consider part (a) - whether the currents should be in the same or opposite directions.
To apply the right-hand rule, extend your right hand with your thumb pointing in the direction of the first current. Then, curl your fingers towards the direction of the second current.
If your thumb points towards you, it means the magnetic field at the point between the wires is zero. If your thumb points away from you, it means the magnetic field is non-zero and has a specific direction.
In this case, we are given that the magnetic field at the midpoint between the wires has a magnitude of 320 µT. This implies that the magnetic field is non-zero and directed away from you. Therefore, for the magnetic fields to add up and produce a non-zero field at the midpoint, the currents in the wires should be in opposite directions.
Moving on to part (b) - determining the amount of current needed.
We can use the equation for the magnetic field produced by a straight wire to find the current. The equation is given by:
B = (μ₀ * I) / (2 * π * r)
Where:
B is the magnetic field
μ₀ is the magnetic constant (4π × 10^-7 T·m/A)
I is the current
r is the distance between the wire and the point of interest (half the separation distance between the wires in this case)
Rearranging the equation to solve for I, we have:
I = (B * 2 * π * r) / μ₀
Plugging in the given values:
B = 320 µT = 320 × 10^-6 T
r = 9.0 cm / 2 = 4.5 cm = 4.5 × 10^-2 m
μ₀ = 4π × 10^-7 T·m/A
I = (320 × 10^-6 T * 2 * π * 4.5 × 10^-2 m) / (4π × 10^-7 T·m/A)
Simplifying the equation, we find:
I = 3200 A
Therefore, the current needed in each wire to produce the given magnetic field is 3200 A (in opposite directions).