HOMEWORK SET 2 - Due: Tuesday, Feb. 16
(a) Calculate the magnetic field along the axis of a current ring.
(b) Now consider two rings, separated by a distance d. Calculate the magnetic field along z.
(c) Show that, if d = a, then at z = 0 the magnetic field varies, to lowest order, as (z/a)4. Hint: Taylor expand and show that the first 3 derivatives are zero. You can grind it out, or plot B versus z and use symmetry arguments for first and third order terms.
(d) Explain why a Helmholtz coil may be useful.
(a) Approximate the magnetic field inside a torus of radius a and diameter b that has N winds of a wire carrying current I. Assume that a >> b.
(b) Approximate the magnetic field if I = 1 A, a = 5 cm, b = 0.5 cm, and there is a permeable material inside with mr = 1000. Compare to the Earth's field (~0.5 Gauss = 5x10-5 T)
Near the poles of the earth, the magnetic field is ~0.5 G (5x10-5 T).
(a) What is the magnetic dipole moment of the earth (RE = 6378 km)?
(b) Suppose the cross-sectional area of the earth's central core is ~106 km2. Estimate how much total current is flowing inside the earth.
(c) The magnetic field from the sun (at the Earth's orbit) is ~10 nT. Estimate how far from earth must one go to detect the sun's magnetic field (e.g. Bsun ~ Bearth). Solve the problem at the equator, express your answer in units of RE (Earth Radius).
Consider a conducting plane with thickness d carrying a current density (J A/m2) in the y direction. There is a magnetic field in the x direction. The magnetic field will deflect the electrons carrying the current to the +z direction creating surface charge, and therefore an electric field.
(a) Calculate the "Hall" electric field and surface charge. (Hint: J = nev, where n is the density of charge carriers and v is their velocity).
(b) If one measures the potential between the top and bottom, can the density of charge carriers be determined?
