A current-carrying wire produces a magnetic field, concentric about the wire.
We will first define the magnetic field produced by a moving charged particle,
then use that field to find the magnetic field of by a current-carrying wire.
The magnetic field has units of teslas, equal to Newtons per Amp-meter. The symbol μ0 is called the permeability constant.
A moving charged particle feels a force from a magnetic field, the Lorentz force.
A moving charged particle also creates a magnetic field, given by the equation above. This equation is called the Biot Savart law.
Sample question
Consider a proton at the origin with constant vx = 1.0 x 107 m/s.
Use q = 1.60 x 10-19 C.
A. Find the magnitude and direction of the magnetic field caused by the motion of this charged particle at the position
(x, y, z) = (1.0 mm, 0, 0).
B. Find the magnitude and direction of the magnetic field caused by the motion of this charged particle at the position
(x, y, z) = (0, 1.0 mm, 0).
Magnetic field of a current-carrying wire
Replacing q Δv with I Δs gives us the equation for a short segment of a current carrying wire.
To find the formula for the magnetic field at a point P, a distance d away from a long current-carrying wire, we need to integrate over current segments.
We define the useful quantities sin θ and r for our system at hand and substitute into the segment equation.
Integrating over all of the segments, we approximate a long wire by setting our limits for an infinitely long wire.
Evaluating at these limits gives us a relatively simple equation for the magnetic field of a current-carrying wire.
The diagram above shows the B field of a current-carrying wire.
The direction of the positive current is into the page, marked by an "x."
The magnetic field of the wire is in the shape of concentric circles about the wire.
Notice the circles get farther apart as the distance from the wire increases,
indicating that the strength of the B field decreases with distance.
This diagram shows the current-carrying wire at an angle. Here, the direction of the B field at particular
points about the wire is shown as arrows. Not that the arrows are longer in the inner circles, closer to the wire.
The right-hand rule for a current-carrying wire is illustrated above.
The hand grasps the wire with the thumb in the direction of the current.
The fingers wrap around the wire, giving the direction of the B field in the concentric circles.
Sample questions
1. What is the direction of the magnetic field at point P, due to this current-carrying wire?
A. up
B. down
C. right
D. into the page
E. out of the page
2. Where is the magnetic field stronger, at A or B??
A. A
B. B
C. they are the same
Magnetic forces on current-carrying wires
We can easily derive the formula for the magnetic force on a length l of a current-carrying wire from the Biot Savart law
using the drift velocity over a length of wire. The length vector has the same direction as the positive current flow.
Sample questions
1. What is the direction of the magnetic force on wire B from wire A?
A. up
B. down
C. into the page
D. out of the page
E. the force is zero
2. What is the direction of the magnetic force on wire B from wire A?
A. up
B. down
C. right
D. left
E. the force is zero
3. What is the direction of the net torque on wire A from wire B?
A. up
B. down
C. into the page
D. out of the page
E. the net torque is zero
4. Consider two long parallel wires carrying current as shown.
At what point or points on the x-axis does the magnetic field equal zero?
Ampere's law
Ampere's law employs a strategy for finding current inside a bounding loop, similar to the way Gauss's
law employs Gaussian surfaces to find the charge enclosed.
The closed line integral around the current-carrying wire traverses an Amperian loop of circumference 2πr.
Notice the important result that the radius of the circle cancels out.
The integrated magnetic field around any Amperian loop equals the current enclosed by the loop.
This result holds true for any loop, not necessarily circular in shape.
