An electric current is the flow of charge past a point in a circuit.

 

 

It is measured in a unit called an ampere, or amp, which is equivalent to a flow rate of one coulomb or charge each second. When discussing this rate at which charges flow, we need to mention that once an electric field is established across a wire, each individual charge actually drifts at a very slow (less than 1 mm/sec) speed along the wire. It is the very high number of charges moving at any one time that creates the "larger current".

 

1 amp = 6.25 x 10¹⁸ electrons/sec

 

Within a conductor, it is often useful to define current in terms of its current density, or j. The units on j are amps/m². Current is sometimes referred to as "the flux of current density" over an cross-sectional area.

 

 

 

 

No only do charged particles moving through an external magnetic field experience a magnetic force according to

 

 

they also create magnetic fields. Gauss' Law stated that the net flux from a closed surface was equal to the ratio of the enclosed charge divided by the permittivity of free space, =8.85 x 10^-12 C²/Nm².

 

 

This statement reflects that positive charges are "sources" of flux lines and negative charges are "sinks" for flux lines allowing an unequal number of field lines to either originate or terminate inside of a closed surface. Gauss' Law for Magnetism

 

 

states that the net magnetic flux through a closed surface is zero as there are no such things as monopoles. Since each magnet has both a north and a south pole, any magnet enclosed within a closed surface would have the same number of flux lines both exiting and entering the surface - the net flux would be zero.

 

Ampere's Law says that if we replace the closed surface integral with a closed line integral then the magnetic field multiplied by the length of the curve will equal the sum of the enclosed currents times the permeability of free space, µ_o = 4pi x 10^-7 N/A².

 

 

Remember that a dot product () reflects that we must look for the component of B along the path; that is, we are interested in choosing an Amperian path that flows in parallel to the magnetic field  B so that theta equals either 0º or 180º.

 

 

 

Now let's use Ampere's Law to determine the strength of the magnetic field a distance r from a wire carrying a current I.

 

image courtesy of John Hopkins University
Physics Lecture Demonstrations

 

            

 

Suppose we needed to investigate the strength of the magnetic field both inside and outside of this wire.

 

 

Let's assume that the wire has a radius of a and a uniform current density j (shown in the diagram as a solid grey area with the current flowing in the +z direction), allowing us to express the current as

 

 

If we once again examine the magnetic field outside of the wire at a distance r > a, then we get the expression

 

 

If we now examine the magnetic field within the wire at a distance r < a, then we get the expression

 

 

Combining these two equations graphically shows us 

 

           

 

two general "shape" combinations in this graph that are reminiscent to those for an electric field of a uniformly charged insulator as well as a gravitational field of a solid sphere having a uniform density (except that the later two are inverse square relationships when r > a).

 

 

 

Recall that an ideal solenoid is a tightly turned coil of wire which produces a uniform magnetic field down its center. We will now use Ampere's Law to calculate the magnitude of this central magnetic field.

 

image courtesy of John Hopkins University Physics Lecture Demonstrations

 

 

Beginning on the left side we will use the Amperian path ABCDA.

 

 

where B represents the strength of the central magnetic field and represents the length of the line CD.

 

Completing our solution, we now have

 

 

where the ratio  represents the total number of loops in the solenoid divided by the length of the solenoid. This value is a constant represented by the variable n.

 

 

 

Now let's wrap a solenoid into a circle to form a toroid and examine its magnetic field.

 

image courtesy of Oberlin College
Demonstration Website

 

Once again, we will use Ampere's Law to calculate the magnetic field. To begin we will place an Amperian loop along the center of the toroid's coils so that its radius, r, lies between the inner and outer radii of the torus, a < r < b.

 

 

For r < a and r > b, B = 0. In the case of r < a, no currents are enclosed within the Amperian loop. For r > b, the currents flowing "in" (-z) and the currents flowing "out" (+z) cancel, giving a net enclosed current of zero.