Point P is midway between two long, straight, parallel wires that run north-south in a horizontal plane. The distance between the wires is 1cm. Each wire carries a current of 1A toward the north.
A) Find the magnitude and direction of the magnetic field at point P.
B) Repeat the question if the current in the wire on the east side runs toward the south instead.
SOLUTION This solution is FREE courtesy of BrainMass!
Please see the attachments for the solution.
First, let's calculate the field of an infinite wire carrying a current I.
From symmetry we can conclude that the magnitude of the magnetic field is constant on a concentric circular path of radius r around the wire. The direction of B is always perpendicular to the current and the radius (Biot-Savart law) -hence it is tangential to the loop.
If we use Ampere's law that states that the path integral of the magnetic field along a closed loop is proportional to the current that goes through this loop we get:
This turns to (the magnetic field is tangential to dl):
And then magnetic field magnitude is constant on the loop:
But so we get for the field of an infinite wire:
The direction of the field is determined by the right-hand-rule.
Now to our question.
The first is to find the magnetic field midway between two infinite wires that carry current in the same direction.
From pure symmetry we see that the magnetic fields will have the same magnitude, but from the right-hand-rule we can see that their directions will be exactly opposite to one another. So in this case the two magnetic fields will cancel each other and the total magnetic field midpoint between the wires is zero.
Now, when we switch the direction of the current in one of the wires, one of the magnetic fields changes direction. Their magnitude is still the same, but now they do not cancel each other:
In this cae, the total magnetic field midway between the wires will be double the magnetic field contribution from a single wire, namely:
All that is left is to plug in the numbers (MKS units):
The magnetic field midway between the wires is 80 micro-tesla.© BrainMass Inc. brainmass.com December 24, 2021, 8:01 pm ad1c9bdddf>