The relationship between magnetism and electricity is very close. This page is dedicated to how magnetism and electricity interact in fields but before we discuss that I feel it is important to know a little bit about the electromagnetic force.

According to this physics page "[the electromagnetic force] one of the four fundamental forces the electromagnetic force manifests itself through the forces between charges (Coulomb's Law) and the magnetic force, both of which are summarized in the Lorentz force law." The lorentz force is is given by this equation:

(Images Courtesy of http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magfor.html#c2)

Below is the explanation of the variables.

Now that we know how closely related the electric and magnetic forces are it is appropriate to discuss further the interaction between electricity and magnetism in fields.

The first interaction is a magnetic force acting on a current carrying wire. This situation acts similar to the situation of a charged particle moving through a magnetic field. There is a force on the wire. It is simply a collection of charges in motion. "Hence the resultant force exerted by the field on the wire is the vector sum of the individual forces exerted on all the charged particles making up the current" (Serway). The total Magnetic force on a wire is equal to the current(I) times L(length of wire) cross B(magnetic field). If we have a closed current loop the net magnetic force acting on it is 0. The magnetic field of an electric current is shown below in a picture:

(Image courtesy of http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magcur.html#c1)

As you can see in the picture above the right hand rule is applicable. You put your thumb in the direction of the current and your curled fingers is the magnetic field.

Soon after Oersted's discovery that a compass needle is reflected by a current-carrying wire two scientist by the name of Jean-baptiste Biot and Felix Savart performed experiments dealing with the force exerted by a current on a nearby magnet (Serway). Their experiments lead to the Biot-Savart law. "The Biot-Savart Law relates magnetic fields to the currents which are their sources"(source). The Bio-Savart Law is explained in the image below.

(Images courtesy of http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/biosav.html#c1)

There are variants of magnetic fields and wires. Ampere's law can be used in many of these variants. See Ampere's law applications.

The last subject I will touch on here is Electromagnetic Fields. Electromagnetic waves (which naturally involve magnetic fields and electric fields) are predicted by Maxwell's equations. These equations are I. Gauss' law for electricity II. Gauss' law for magnetism III. Faraday's law of induction IV. Ampere's law .

The electromagnetic spectrum is given by the image below.

The reason I mentioned the electromagnetic fields is so we could discuss the energy in electric and magnetic fields. We know both fields store energy. Electric fields store energy in capacitors and magnetic fields store energy in inductors. The equations for the energy densities are shown below.

Above is energy density for electric fields

(Images Provided by http://hyperphysics.phy-astr.gsu.edu/hbase/electric/engfie.html#c1)

Below is energy density for magnetic fields

These equations, as you may have noticed, are very closely related. One will very quickly discover that magnetism and electricity are very related fields. They share many of the same rules and equations just slightly modified, and their interaction with each other is outstanding.

(Image courtesy of http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html)

The Infamous Electromagnetic Wave "Dance"!

As you can probably see magnetism and electricity are very extensive and closely related fields. I find the properties of magnets and magnetic fields to be fascinating. The integral role both areas play in our daily lives and the theory of physics is mind boggling. It really is too extensive to cover completely but the site below does an excellent job.

This page is the ultimate source on Electricity and Magnetism(http://hyperphysics.phy-astr.gsu.edu/hbase/emcon.html#emcon)