Phys 742 - EM I In-class test 2 - 4 April 1995

1. Solve the boundary-value problem for a (type I) superconducting sphere of

radius a in an external magnetic field B_{0}. What is the magnetization M in the sphere? What is the induced dipole moment m (in magnitude and direction)?

The magnetization is equivalent to a surface current K, which in this case is a real persistent surface current. Find the components of K as a function of suitably chosen angles θ and ϕ.

Find a vector potential A that describes the total B outside the sphere.

2. Two large parallel metal plates, separated by a distance d, are lowered verti-

cally into a liquid dielectric of density ρ. If the liquid rises a distance h between the plates when a potential difference V is established between them, what is the dielectric constant ϵ of the liquid?

Supply “typical” values of ϵ, ρ, V, d and estimate the corresponding value of h.

3. A long hollow cylinder of inner radius a, outer radius b, and of magnetic

permeability μ, is placed in a region of initial magnetic-flux density B_{0}, at right angles to the field. Describe qualitatively the H and B fields at all points in space, for large positive μ.

For extra credit, if time permits: Determine explicitly the screening factor, defined as the ratio between the field

inside the cavity and B_{0}.

4. What is the torque exerted by a uniform magnetic field B on a circular loop

of radius a, carrying a current I? In particular, what is the torque if the plane of the loop is (a) perpendicular to the field, (b) parallel to the field?

—————————— Formulas for ∇ × A ——————————

e_{ρ} (\frac{1}{ρ} \frac{∂A_{z}}{∂ϕ} − \frac{∂A_{ϕ}}{∂z} ) + e_{ϕ} (\frac{∂A_{ρ}}{∂z} − \frac{∂A_{z}}{∂ρ} ) + \frac{e_{z}}{ρ} (\frac{∂ (ρA_{ϕ})}{∂ρ} − \frac{∂A_{ρ}}{∂ϕ} )

\frac{e_{r}}{r sin θ} (\frac{∂ (A_{ϕ} sin θ)}{∂θ} − \frac{∂A_{θ}}{∂ϕ} ) + \frac{e_{θ}}{r} (\frac{1}{sin θ} \frac{∂A_{r}}{∂ϕ} − \frac{∂ (rA_{ϕ})}{∂r}) + \frac{e_{ϕ}}{r} (\frac{∂ (rA_{θ})}{∂r} − \frac{∂A_{r}}{∂θ} )