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Electricity & Magnetism Qualitative Problems

Here is a list of Qualitative problems which need the physics explained for each, diagrams, and equations with text are welcome.

1) The vector potential and the magnetic field inside and outside of a diamagnetic sphere in a uniform magnetic field

2) The vector potential and the magnetic field inside and outside of a uniformly magnetized sphere

3) Sketch the vector potential for at least four simple geometric cases, explain the sketches.

4) The electric potential and the electric field for (charge) monopoles, dipoles and quadrupoles

5) The vector potential and the magnetic field for (magnetic) monopoles, dipoles and quadrupoles

6) Surface polarization charges (aka, bound surface charges) versus volume polarization charges (aka, bound volume charges)

7) Surface magnetization currents (aka, bound surface currents) versus volume magnetization currents (aka, bound volume currents)

8) The relationship between the magnetic scalar potential and the magnetic field

9) The electric potential and the electric field inside capacitors containing dielectrics

10) The electric potential and the electric field inside capacitors containing ferroelectrics

11) The vector potential and the magnetic field inside inductors containing diamagnets

12) The vector potential and the magnetic field inside inductors containing ferromagnets

13) The displacement current and Maxwell's equations

14) E for a charge inside a dielectric; B for a point magnetic dipole inside a diamagnet

15) P for changing electric fields; M for changing magnetic fields

Solution Preview

Please see attached files for diagrams.

1) Sketch the vector potential for at least four simple geometric cases, explain the sketches

(a) Magnetic vector potential of an infinite wire: from equation (5.63) of section 5.4.1 of Griffiths' book it is A = (const / s) (direction of wire), where s is the distance to the wire. The sketch is in Figure 1. The field lines of the vector potential are just infinite straight lines parallel to the wire and its strength varies inversely proportional to the distance to the wire.

Figure 1: Infinite straight wire

(b) Magnetic vector potential of a rotating sphere with uniform surface charge: from equation (5.66) from example 5.11 in section 5.4.1 of Griffiths' book.

The sketch is in Figure 2. The field lines of the vector potential are circles centered on the axis of rotation.
For a fixed θ (the angle between the radius-vector r and the axis of rotation) its strength grows linearly from 0 to some value at the surface of the sphere and then falls down inversely proportional to the square of the distance to the center.
For a fixed |r|, its strength is proportional to sin θ.

As the drawing is crowded and I am not much of an artist, I drew only two circles outside of the equatorial plane.

Figure 2: Rotating sphere with uniform charge density

(c) Magnetic vector potential of an infinite uniform solenoid: from equations (5.70) and (5.71) from example 5.12 in section 5.4.1 of Griffiths' book.
The sketch is in Figure 3. The field lines of the vector potential are circles centered on the axis of rotation.
Its strength grows linearly from 0 to some value at the solenoid surface and then falls down inversely proportional to the distance to the center.

(d) Magnetic vector potential of a point magnetic dipole: from equation (5.83) in section 5.4.3 of Griffiths' book.
The sketch is in Figure 4. The field lines of the vector potential are circles centered on the axis determined by the dipole at the origin
For a fixed θ (the angle between the radius-vector r and the axis of rotation) its strength falls down inversely proportional to the square of the distance to the center.
For a fixed |r|, its strength is proportional to sin θ.

2) The electric potential and the electric field for (charge) monopoles, dipoles and quadrupoles.

Reference: sections 3.4.1 and 3.4.2 of Griffiths' book.

Formally, multipole momenta result from the expansion of a potential of some charge distribution in terms of Legendre polynomials.

For a bounded charge distribution, the potential far away from it is proportional to
1/r, 1/r2, and 1/r3 and the electric field is proportional to 1/r2, 1/r3, and 1/r4, for monopole, dipole, and quadrupole momenta, ...

Solution Summary

The detailed explanations are clear and suported by sketches as needed.

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