Magnetic Fields and Forces
Magnetic fields are produced by magnetism, the flow of electric charge, or fluctuating electric fields. In the first case, the magnetic field can be described as a region around a magnetic object, such as a magnet:
Inside the magnet, the magnetic field B is directed from the south (S) to the north pole (N). Outside the magnet, the direction of the field is represented by imaginary force lines called magnetic flux lines, which always flow in a closed loop from the north pole to the south pole. The density of these lines define the strength of the magnetic field, i.e., the more lines, the stronger the magnetic field. Magnetic field sources are always dipolar in nature, having north and south magnetic poles. But unlike electric monopoles--or charges--magnetic monopoles do not exist, i.e., an electric dipole consists of positive and negative monopoles and can be separated into isolated positive and negative charges. But separating a magnet will not yield a south and a north pole, it will yield two smaller magnets, each with a south and north pole.
A conductor through which a current is flowing will also produce a magnetic field, just as a magnet does. In this case, the magnetic field strength, H, (in units of A/m) is expressed as:
where [b.mu ] is the magnetic permeability of free space (4 x 10-7 N/A2 ). Electric and magnetic fields were first described as closely related by James Clerk Maxwell who formulated the basic principles of electromagnetic induction, which can be summarized by the following statements: Any electric field that changes over time will induce a magnetic field in the space surrounding it; any magnetic field that changes over time will also induce an electric field in the space surrounding it and the induced fields tend to have a loop pattern. These statements represent the fundamentals of electromagnetism and are mathematically expressed as Maxwell's equations. The second equation, also called Gauss's law for magnetic fields, and is expressed as:
It states that the magnetic flux normal to a closed surface is zero. This is equivalent to a statement to the effect that there are no magnetic monopoles. For a magnetic dipole, the magnetic flux directed inward toward the south pole will equal the flux outward from the north pole and the net flux will always be zero for dipole sources. The third Maxwell equation, also called Faraday's law of induction, states that the line integral of the electric field E around a closed loop is equal to the negative of the rate of change of the magnetic field B through the area enclosed by the loop:
The interaction of charge and magnetic field described by the above equation can be illustrated by a coil of wire moving in a magnetic field and generating an emf (or voltage). Faraday's law then states that an emf is induced in a coil when the magnetic field around it changes. It is expressed as: emf = -N ph/t where N is the number of turns in the coil and ph is the magnetic flux. The negative sign indicates that the emf is opposing the change that is producing it.
The fourth Maxwell equation is equivalent to Ampere's law and it quantitatively describes the relation of a magnetic field to the electric current or field inducing it. It states that for any closed loop path, the sum of the length elements (l) times the magnetic field in the direction of the length element (B)is equal to the magnetic permeability of free space ([b.mu ]) times the electric current (I) enclosed in the loop.
In the above formulation, the law can be used, for example, to calculate the magnetic field arising from the current flowing in a long, straight wire. The relationship between electric and magnetic forces is best expressed by the Lorentz force (F) which sums both forces. Mathematically, it is expressed as:
where qE, the product of the charge q and of the electric field E, is the electric force that acts in the direction of the electric field when q is positive and where the second term of the equation is the magnetic force, in that v is the velocity of the charge and B the magnetic field. The direction of the magnetic force is given by a right-hand rule, such as commonly used for right-handed coordinate systems to determine which way the cross product of two vector quantities is directed.
Just as Coulomb's law relates electric fields to the point charges that are their sources, the Biot-Savart law relates magnetic fields to the currents that induce them and it is used to calculate such induced fields for any type of circuit. The law describes the magnetic field of a current element dB and is expressed as follows:
where I is the current, dL the infinitesimal length of the conductor—or wire--carrying the current and r is the unit vector which specifies the direction of the distance vector r from the current to the field point. This law is commonly used to calculate the magnetic field resulting from an electric current distribution in straight wires or in the center or on the axis of a current loop.
Some important practical applications of electromagnetic induction are the generator and the transformer. A generator basically consists of a permanent magnet that rotates within a coil of wire. The magnet is turned by a crankshaft and as it spins, the magnetic field surrounding it changes. This induces an electric field that in turn generates a current that flows around the coil. The induced current can then be used as a power source. An example of transformer is that of the type used to convert to lower voltages and higher currents to supply electrical power to households. The coil on the input side of such transformers induces a magnetic field. Since the current is AC, the magnetic field around the input coil is also alternating and this induces an electric field which drives the current to the output coil.
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