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Electromagnetic Field

Electromagnetic Field

Superposition of fields

Electric fields

The dipole field

The field of a line of charge

Magnetic fields

Electromagnetic fields

Maxwells equations

An electromagnetic field exists in any volume of space in which electric and magnetic forces are interacting. It can arise from electric charges (such as those borne by electrons and protons) in motion; it can also be created by a changing magnetic field or changing electric field. Since a changing electric field can generate an electric field and vice versa, a self-sustaining pair of electric and magnetic fields can move through space, the two fields rapidly exchanging energy. These electromagnetic rays permeate the universe; depending on the frequency with which they oscillate, they form radio waves, infrared rays, visible light, x rays, and other forms of electromagnetic radiation.

An electromagnetic field is best understood as a mathematical function or property of space time, but

may be represented as a group of vectors, quantities (usually drawn as arrows) with specific strength and direction. A stationary charge produces an electric field, while a moving charge additionally produces a magnetic field. Since velocity is a relative concept dependent on ones choice of reference frame, magnetism and electricity are not independent, but linked together, hence the term electromagnetism.

Superposition of fields

Since all charges, moving or not, have fields associated with them, we must have a way to describe the total field due to all randomly distributed charges that would be felt by a positive charge, such as proton or positron, at any position and time. The total field is the sum of the fields produced by the individual particles. This idea is called the principle of superposition.

Let us look at arrow or vector representations of the fields from some particular charge distributions. There is an infinite number of possibilities, but we will consider only a few simple cases.

Electric fields

According to Coulombs law, the strength of the electric field from a nonmoving point charge depends directly on the charge value q and is inversely proportional to the distance from the charge. That is, farther from the source charge will be subject to the same strength of force regardless of whether it is above, below, or to the side of the source, as long as the distance is the same. A surface of the same radius all around the source will have the same field strength. This is called a surface of equipotential. For a point source charge, the surface of equipotential is a sphere, and the force F will push a positive charge radially outward. A test charge of mass m and positive charge q will feel a push away from the positive source charge with an acceleration (a) directly proportional to the field strength and inversely proportional to the mass of the test charge.

If a charge does not move because it is acted upon by the electromagnetic force equally from all directions, it is in a position of stable equilibrium.

The dipole field

Now let us consider the field from two charges, one positive and one negative, a distance d apart. We call this combination of charges a dipole. Remember, opposite charges attract, so this is not an unusual situation. A hydrogen atom, for example, consisting of an electron (negative charge) and a proton (positive charge) is a very small dipole, as these particles do not sit right on top of each other. According to the superposition principle mentioned above, we can just add the fields from each individual charge and get a rather complicated field. If we only consider the field at a position very far from the dipole, we can simplify the field equation so that the field is proportional to the product of the charge value and the separation of the two charges. There is also dependence on the distance along the dipole axis as well as radial distance from the axis.

The field of a line of charge

Next, we consider the field due to a group of positive charges evenly distributed along an infinite straight line, defined to be infinite because we want to neglect the effect of the endpoints as an unnecessary complication here. Just as the field of a point charge is directed radially outward in a sphere, the field of a line of charge is directed radially outward, but at any specific radius the surface of equipotential will be a cylinder.

Magnetic fields

Recall that the force of a stationary charge is F = qE, but if the charge is moving the force is F = qE +qv× B. A steady (unchanging in time) current in a wire, generates a magnetic field. Electric current is essentially charges in motion. In an electrical conductor like copper wire, electrons move, while positive charges remain steady. The positive charge cancels the electric charge so the overall charge looks like zero when viewed from outside the wire, so no electric field will exist outside the wire, but the moving charges create a magnetic field from F = qv× B where B is the magnetic field vector. The cross product results in magnetic field lines circling the wire. Because of this effect, solenoids (a current-carrying coil of wire that acts as a magnet) can be made by wrapping wire in a tight spiral around a metallic tube, so that the magnetic field inside the tube is linear in direction.

Relating this idea to Newtons first law of motion, which states that for every action there is an equal and opposite reaction, we see that an external magnetic field (from a bar magnet, for example) can exert a force on a current-carrying wire, which will be the sum of the forces on all the individual moving charges in the wire.

Electromagnetic fields

A simple example of a combination of electric and magnetic fields is the field from a single point charge, say a proton, traveling through space at a constant speed in a straight line. In this case, the field vectors pointing radially outward would have to be added to the spiral magnetic field lines (circles extend into spirals because an individual charge is moving) to get the total field caused by the charge.

For a static electric fieldmeaning an unchanging electric field, which can only be generated by charges at restthe force F on a test charge is F = qE, where q is the value of the test charge and E is the vector electric field. For a static magnetic field (caused by moving charge inside an overall neutral group of charges, or a bar magnet, for example) the force is given by F = qv× B, where v is the charge velocity, B is the vector magnetic field, and the× indicates a cross-product of vectors.

Maxwells equations

A description of the field from a current which changes in time is much more complicated, but is calculable owing to James Clerk Maxwell (1831 1879). His equations, which have unified the laws of electricity and magnetism, are called Maxwells equations. They are differential equations, which completely describe the combined effects of electricity and magnetism, and are considered to be one of the crowning achievements of the nineteenth century. Maxwells formulation of the theory of electromagnetic radiation allows us to understand the entire electromagnetic spectrum, from radio waves through visible light to gamma rays.

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