(a) An infinitely thin and infinitely long wire is placed on the z-axis. The wire is
carrying a current I in the +z direction. A point magnetic dipole of magnitute m
and direction +y is placed on the x-axis at x=a. Find the force on the dipole due
to the magnetic field generated by the wire.
(b) Find the torque on the dipole.
(c) Find the force on the wire due to the magnetic field generated by the dipole.
(d) The wire described above is now replaced with another wire with finite radius R, but
still infinitely long and coaxial with the z-axis. This wire too carries a current I
(uniformly distributed) in the +z direction. Find the force acting on the dipole
when r<a and r>a.
A cylindrical permanent magnet with uniform magnetization M parallel to its axis has radius R
and length L. It is placed with its axis on the z-axis as shown in the figure.
(a) Find the "magnetic charge density" associated with the scalar magnetic potential for the H-field.
(b) Use this density to find the H-field along the axis of the magnet (i.e. the z-axis).
(You will have to evaluate a limit carefully near the circular surfaces. Check that
your expressions are consistent with the R→∞ [infinite charged plane] limit.)
(c) Find the B-field along the axis of the magnet.
(d) Discuss the dependence of the magnetic field B at the origin on R and L.
Valentine's Day special: For extra credit (not much), analyze the love story presented in the performance of the "Magnet and the Churn" in the Gilbert and Sullivan opera Patience and explain the physics behind the broken heart.
A very long straight wire carries a current I(t) = (1 Ampere) cos[2π t/(20ms)]. A setup to measure this currrent is to be designed so that a rectangular loop of dimensions b by c is placed a distance a from the wire as shown in the figure. It is desired to be able to measure a sinusoidal potential difference V with amplitude 1Volt across the loop for the current value given above. Design a loop (i.e. determine the lengths a, b and c) which achieves this. (If you find that the size of the loop is somewhat impractical you can use a loop with N turns.)
A point charge Q is moving along the z-axis with constant velocity so that its
z-coordinate at time t may be taken as z'=vt. Note that we have both a
o scalar source [charge density ρ = Q δ(z'-vt) δ(x') δ(y') ]
o and a vector source [current density J = Qv δ(z'-vt) δ(x') δ(y') in the +z direction]
in this problem.
First, use the charge density and calculate
(a) the flux of the electric field (at time t) through a circular area
of radius R on the x-y plane, centered at the origin,
(b) the magnitude and the direction of the induced magnetic field on
the x-y plane, a distance R from the origin.
Now, use the current density to calculate
(c) the magnetic field on the x-y plane, a distance R from the origin
using the Biot-Savart law.
In class, we showed that an electromagnetic wave with the fields
E = Eo x cos [k(z - vt)]
B = Bo y cos [k(z - vt)]
satisfied the four differential conditions associated with the Maxwell Equations.
This is called a "plane wave" because the fields depend only on the z-coordinate,
and therefore are constant on any plane perpendicular to the z-axis.
A different type of wave is the "spherical wave", in which the wave magnitudes are
proportional to cos[k(r - vt)] /r far away from the origin. (r is the radial spherical
coordinate.) Take the electric field as E = θ Eo cos[k(r - vt)] /r and find the
corresponding magnetic field. Show that this form also satisfies Maxwell Equations
if you neglect terms which decay faster than 1/r. What are the relations between
these fields and v?
Consider a plane electromagnetic wave, moving at a 45o angle to the z-axis. A perfect conductor is placed on the x-y plane. The wave is incident on the plane (from the -z side) with its electric field perpendicular to the plane of incidence. (a) Find the electric and magnetic fields at all points in space. (b) Find the average Poynting vector associated with the total field.
Solve part (a) of Homework Problem 7 again, but this time with the space
for z>0 filled with a material with properties ε, μ and conductivity σ.
A very short current pulse travels along a wire on the z-axis with speed of light c
so that the current density is given as J(r,t) = z Io δ(x)δ(y)δ(z-ct).
Note that since there is no net charge at any time, the scalar potential is always zero.
(Actually, there is an electric dipole traveling along with the current
due to the continuity condition, but neglect its effect for this problem.)
Find the vector potential, and obtain the electric and magnetic fields.
(Hint: Without loss of generality, you can find the potentials and the fields
for a point on the x-axis, say at (r,0,0).
Do the z' integral first, which will be trivial, then do the t' integral, carefully!)