Divergence and circulation

✤ A vector field is generally characterized by 1) how field lines possibly diverge away from or converge upon (point) sources plus 2) how field lines circulate, closing upon themselves.

✤ An electrostatic field diverges but has no circulation implying the force field is conservation. A test charge gains no net energy in traversing a closed circuit.

✤ This follows from Coulomb’s Law for a point charge, and, by superposition, for any collection of static charges.

✤ We will see later nonconservative electric fields may appear in nonstatic circumstances.

Surface S

Curve C

True for static fields only.

Ampere’s Law

✤ Gauss’s Law for magnetic fields says there are no magnetic monopoles.

✤ Ampere’s Law relates magnetic field circulation around any loop to the total current through the loop.

✤ In Ampere’s Law, S is an open surface with closed curve C as as its boundary and j is the electric current density (A/m2). I is the total current through any such open surface S, any because, by implicit assumption, current is conserved and if it flows through one such surface it flows through any other.

Ampere’s law applied to an infinitely long wire

✤ Integrate around a circle (a) enclosing the wire. The right-hand fingers follow the curve. The thumb gives the direction of positive current in Ampere’s Law.

✤ In (b) the orientation of the integration is reversed and the current negative.

✤ In ( c), it is easy to show the integral vanishes as it must according to Ampere’s Law.

Magnetic field outside and inside a wire

✤ We can use Ampere’s Law to DERIVE the field outside a straight wire carrying current.

✤ Assuming that the current density is uniform within the wire, we can also find the field within the wire.

✤ The field vanishes at the center.

Coaxial cable

✤ Question: An infinitely long coaxial cable carries equal and opposite currents in an inner and outer conductor as shown.

✤ Find the magnetic field everywhere in space.

Coaxial cable

✤ Applying Ampere’s law to a concentric circle,ignoring the outer conductor and outer current gives the field inside the outer conductor as for an infinite straight wire.

✤ Applying Ampere’s Law to a circular path outside the outer conductor, we find the azimuthal field vanishes as our path has vanishing net current through it.

Toroidal field

✤ The toroidal shape coil contains a magnetic field within itself and is a common “inductive” element in electronic circuits.

✤ The field strength is proportional to the number of turns N.

Magnetic field inside a solenoid

✤ A long solenoid has a strong field inside. The flux is returned over a large volume so the field strength is small outside.

✤ Applying Ampere’s Law assuming uniform field inside and vanishing field outside implies the field strength is proportional to the number of turns per unit length N/L.

Ferromagnetism

✤ The magnetic field of a “permanent magnet” is associated with the spin of charged electrons in atoms.

✤ The electrons behave like a collection of tiny permanent magnets. When these tiny magnets aligned, not randomly oriented, a net macroscopic magnetization and magnetic field is observed.

Classical orbital magnetic moment

✤ A classical electron in a circular orbit has a time averaged current proportional to speed and a magnetic moment m=IA proportional to its angular momentum L=me rxv.

✤ m = (-e/2me)L

✤ In quantum mechanics, it is found that orbital angular momentum and magnetism is quantized, Lz= n h/(2 pi ), where h is Planck’s constant. (hbar is h over 2 pi.)

Electron spin magnetic moment

✤ The electron is found to behave as if it had an intrinsic quantized spin angular momentum and an associated magnetic moment with

✤ mz = (+ or -)(-e hbar)/(2me)

✤ The magnitude is 9.27e-24 J/T.

✤ The magnetic energy U~mzB in a 1 T field is 9.27e-24 J, much smaller than 1 eV=1.602e-19 J.

Atomic magnetic moments

✤ The proton magnetic moment is about equal to the Bohr magneton with the proton mass substituted for the electron mass and ~600 times smaller. Nuclear magnetic moments are often negligible compared to electronic moments.

✤ The H atom magnetic moment is approximately the electron spin magnetic moment.

✤ In He, the two electron spins and magnetic moments cancel.

Ferromagnets

✤ In some materials, below a critical (Curie) temperature, unpaired electrons spontaneously align within small domains.

✤ Application of an external field will grow domains aligned with the field and shrink others resulting in net bulk averaged alignment and magnetism.

✤ When the external field is removed, a remnant magnetization remains. The material is “permanently” magnetized.

Curie temperature

✤ Spontaneously alignment is a “solid phase” for spin.

✤ Above a material dependent Curie temperature, the spins are randomly aligned.

✤ Other “spin phases” including antiferromagnetism are possible.

Paramagnetism and diamagnetism

✤ Paramagnetic substances have small but positive induced magnetism that results from the presence of atoms that have permanent (electron spin) magnetic moments which, when placed in an external magnetic field, tend to line up with the field. The alignment process competes with thermal motion which randomizes the moment orientations.

✤ A diamagnetic substance is one in which electron spins cancel in pairs as explained by quantum mechanics dynamics. When an external magnetic field is applied to a diamagnetic substance, a weak orbital magnetic moment is induced in the direction opposite the applied field.

✤ Diamagnetic substances are weakly repelled by a magnet.

Meissner effect

✤ A superconductor is a material which, below a critical temperature, ceases to resist current flow.

✤ When a superconductor is placed in a magnetic field, currents appear so as to EXCLUDE magnetic field from the interior, rather like, when a superconductor is placed in an electric field, charges appear so as to exclude electric field from the interior.

✤ The direction of the currents is such that the superconductor is repelled in a nonuniform field.

Force in a uniform field

✤ Recall that the magnetic forces on elements of a circuit in a uniform field result in no net force but, a net torque that depends on the orientation.

Force in nonuniform field

✤ In a nonuniform magnetic field, a net force as well as a net torque acts on a circuit.

Loudspeaker

✤ An acoustic speaker operates by energizing an electromagnet with an electrical signal. The magnetic force on a permanent magnet attached to a diaphragm causes the diaphragm to move.

✤ The inverse (motion of diaphragm causing electric current) is explained by Faraday’s Law of Induction.

Motor

✤ Can you explain this to a newbie? What about a generator? Can you explain that to someone? Or on an examination?