A magnetic dipole moment is created by a current loop.

We will begin by finding the magnetic field of a current loop at a point along its axis, and then use our basic derivation for more complicated loops.

In reality, the loop is connected to a circuit, but we will simplify the calculation necessary to find the magnetic field it produces by idealizing its shape.

Our approach to finding the magnetic field of a current loop at a point along its axis is similar to our method for calculating the E field of a charged ring.



Here, we will sum over the magnetic field elements created by tiny current segments around the loop. We begin by defining useful quantities and noting that the x and y components of the B field cancel, by symmetry.

We use our general form for the magnetic field of a current segment, and define it in terms of the system at hand. Then it is just a matter of performing the sum over the current segments and simplifying our equation to find the net B field at a point along the z-axis through the center of the loop.



If we take the limit of this equation for z >> r, we can define the magnetic field in terms of the area of the loop. We define the magnetic dipole moment as the area vector times the current of the loop. The direction of the magnetic dipole is perpendicular to the current loop.



We can find the magnetic field at the center of a current loop with N turns by taking z = 0 and multiplying by N.

Sample question

1. What is the direction of the current in the loop? Which side of the loop is the north pole?



A. The current is clockwise, the north pole is upward.

B. The current is counterclockwise, the north pole is upward.

C. The current is clockwise, the north pole is downward.

D. The current is counterclockwise, the north pole is downward.







Recognizing that a current loop has a magnetic dipole moment gives up a powerful new understanding of how current loops behave. They can be thought of as magnets.

We saw earlier that two long straight wires exerted magnetic forces on each other. If the currents are in the same direction, the two wires attract each other. Two opposing currents cause the wires to repel.

We can see this plainly illustrated in the current loops. If the currents are in the same direction, the system is analogous to two ferromagnets with their poles parallel, attracting each other.

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.

Solenoids

A solenoid is basically just a coil of wire. We can model it as a collection of connected current loops.



The magnetic field inside a solenoid is approximately uniform. We can use the right-hand rule for a current loop contained in the solenoid to find the direction of the magnetic field. The B field outside a solenoid is very small. For an ideal solenoid, we approximate the field inside as constant and the field outside as zero.



We can use Ampere's law to derive the magnetic field inside an ideal solenoid. We take the line integral around the path enclosing one side of the loops, over a length l.



Only the part of the path inside the solenoid is nonzero, contributing to the integral.



This gives us the strength of the magnetic field inside an ideal solenoid. Here, N is the number of loops and n is N/l or the number per length.