Electricity and Magnetism, Part 5

Question 1: Discuss Gauss’s law for magnetism.

Answer 1: Gauss’s law for magnetism solidifies the assertion that magnetic monopoles cannot exist. It does so by making statements about magnetic flux, which is defined in much the same way as electric flux, Fm = B·A, where A is the surface’s normal vector. Magnetic flux has SI units of webers (Wb). If the surface is taken to be an enclosing surface, or a Gaussian surface, Gauss’s law states that the total flux through the surface will equal zero. Unlike the case with electric flux, which allows for contained net charge, there can be no net magnetic poles inside of a closed surface. For every north pole, there must be a south pole of equal magnitude. If, for instance, we envision our Gaussian surface as enclosing one end of a short solenoid (which will set up a magnetic field resembling that of a magnetic dipole), we will notice that the magnetic field B will enter the Gaussian surface inside the solenoid (at the north pole) and will leave it outside the solenoid. No lines will begin or end in the interior of the surface, so the total flux for the surface is zero.

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Question 2: Discuss the Biot-Savart law and Ampere’s law. Discuss displacement current.

Answer 2: Any current-carrying wire will produce a magnetic field that circles the wire in the direction indicated by the right-hand rule (with the thumb pointing in the direction of the current). The Biot-Savart law gives the magnitude and direction of the magnetic field created by each infinitesimal element of current. The equation form of this law is given as dB = (µ0I/4r3pi)*dL×r, where µ0 is a physical constant called the permeability of free space, dL is an infinitesimal vector pointing along the length of the wire in the direction of current, and r is the vector pointing from the wire element to the location where the field is to be calculated. Integrating this expression along the full length of wire gives the total resultant field. Ampere’s law states this concept in the form of a closed-loop line integral:?·?=?0?For transient cases, such as RC circuits, it is necessary to take into account what is known as the displacement current. Displacement current is found by the equation Id = e0(?Fe/?t), where e0 is the permittivity of free space and ?Fe/?t is the instantaneous rate of change of electric flux. It is not actually a current, but for use in Ampere’s law, it is treated like one. Thus, for transient cases, the I in Ampere’s law is calculated as I = Iactual + Id.

Question 3: Discuss the application of Ampere’s law to a straight wire, a loop of wire, and a solenoid.

Answer 3: Ampere’s law is described by the closed-loop line integral:.For a straight length of wire, the magnetic field simply encircles the wire in the direction indicated by the right-hand rule (with the thumb pointed in the direction of current and the fingers curling to indicate the magnetic field). For a very long wire where end effects may be ignored, the magnetic field is given as B = µ0I/(2pi*r), where r is the distance from the wire.For a loop of current-carrying wire, the magnetic field direction can still be determined from the right-hand rule. However, the field magnitude at most points around and inside the loop is difficult to determine. The field magnitude at the center of the loop is found to be B = µ0I/2r.For a solenoid, a long coil of current-carrying wire that is sometimes wrapped around a metal core, the magnetic field direction may again be determined by the right-hand rule. At all points inside the solenoid, the field has magnitude B = µ0nI/h, where n is the number of loops in the solenoid and h is its height. For a long solenoid, it can be shown that the field magnitude outside the solenoid is negligible.

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