Indiana University Physics P331: Theory of Electromagnetism Review Problems #3

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1 Indiana University Physics P331: Theory of Electromagnetism Review Problems #3 Note: The final exam (Friday 1/14 8:00-10:00 AM will be comprehensive, covering lecture and homework material pertaining to HW #1-1. The following review problems cover material not included in Exams 1 and. 1. The boundary between two linear dielectric materials with different permittivities is given by the z = 0 plane with ɛ 1 above and ɛ below the plane. Identify the correct formula(s satisfied by the normal derivative of V at the boundary (that is, the derivative of V in the direction normal to the boundary. Recall E = V, and V 1 refers to the potential in region 1 and V refers to the potential in region. (a V 1 V = σ free (b V 1 z (c ɛ 1 V 1 z (d ɛ 1 V 1 z z=0 V = σtot ɛ 0 V z=0 ɛ z = σ free V z=0 ɛ = σ bound ɛ 0. Which of the following is a statement of charge conservation? (a ρ t = J (b ρ t = S J d a (c ρ t = V J dτ (d None of the above. 3. Why can t we use a scalar potential to find the (static magnetic field, as we have done with the (static electric field, i.e., why can t we use B( r = V B ( r (a Because the divergence of B is always zero (no magnetic monopoles. (b Because only either E or B can be described with a scalar potential, not both. (c Because B can have a non-zero curl according to Ampere s law: B = µ 0 J. (d None of the above. 1

2 4. A laterally large plane of charge with surface charge density σ f sits between two (large neutral, linear dielectric wafers (dielectric constant ɛ r, as shown. +"+"+"+"+"+"+"+"+"+"+"+"+"+"+"+"+"+"+"+"+"+"+"+"+"+"+"+"+"+"+"! f (a Determine D and E everywhere in space. (b Find the polarization density P inside the wafers. (c Determine any discontinuities in D and E at all boundaries. 5. A coaxial cable consists of a copper wire of radius R 1, surrounded by a concentric copper tube of inner radius R and outer radius R 3. The space between is filled with material of dielectric constant ɛ r. Find the capacitance per unit length of the cable. 6. A spherical conductor of radius R 1, is uniformly charged, with surface charge density σ 0. It is coated with a neutral, linear dielectric of susceptibility χ e from r = R 1 to r = R. (a What is the electric field, E, outside the coating, for r > R? (b What is the electric field, E, within the bulk of the coating, for R 1 < r < R? (c What are the bound surface charge densities at r = R 1 and r = R? What is the bound volume charge density in the region R 1 < r < R? (d What boundary conditions does E satisfy at r = R 1 and r = R? Show that the solutions in parts (a and (b are consistent with these boundary conditions. (e Find the bound surface and volume charge densities, σ b and ρ b, respectively. 7. A segment of wire is bent into an arc of radius R and subtended angle θ. Point P is at the center of the circular segment. The wire carries constant current, I. (a What is the magnetic field at P? (b An electron is located at P, moving at v = v 0ˆx. What is the magnetic force on the electron?

3 8. A thick slab of nonmagnetic material carries a uniform volume current density, J = J ˆx as shown in the figure. The slab is infinite in the x and y directions and extends from z = a to z = +a. (a Find the magnetic field inside the slab. (b Find the magnetic field outside the slab. 9. Consider a long wire of radius a, and suppose that the current density within the wire has the form: J(s = I 0 s π a 4, in units of Amps per square meter, where s is the radial distance from the axis of the wire, and the direction of J is along the wire. (a What is the total current I running through the wire? (b What is the magnitude of the magnetic field at points outside the wire (i.e., for s > a? (c What is the magnitude of the magnetic field at points inside the wire (i.e., for s < a? 10. The polarizability, α, of a methane molecule is given cm 3 : recall p = α E. The atomic polarizability of a carbon atom is cm 3, and that of hyrdogen is cm 3. Comment on the effect of binding of atoms into a molecule on the electronic structure. Recall that methane is CH Consider two water molecules, each with dipole moment p = Cm. Determine the dipole-dipole interaction force between them (responsible for hydrogen bonding when they are a distance r = 1 nm apart for the special cases where (a ˆp 1 = ˆp and ˆp 1 ˆr = ˆp ˆr = 0. (b ˆp 1 = ˆp = ˆr Recall that the potential energy of a dipole in an electric field is given by U = p E and the electric field of a dipole oriented in the z direction is E = 1 p ( cos θ ˆr + sin θ 4πɛ 0 r ˆθ 3 3

4 1. A long, thick cylindrical shell of inner radius a and outer radius b is made of dielectric material with frozen-in polarization given by P = k s ŝ (a Locate all the bound surface and volume charge densities. (b Use Gauss law to find the electric field in all three regions 0 < s a, a s b and s b. 4

5 Formula Sheet E(r = 1 ρ(r r r ( r r r r dτ, (Coulomb s Law V = U/q, (Electric Potential Difference ˆ r V (r = E dl, E = V, (Def n of Electric Potential O ˆˆ O E da = 1 ε 0 q enclosed, E = 1 ε 0 ρ, (Gauss s Law E dl = 0, E = 0, (Path Indep. of Elec. Potential V (r = 1 ˆˆ O F da = ρ(r r r dτ, (Coulomb s Law for Potential [V = 0] ( F dτ (Divergence Theorem ˆˆ F dl = ( F da, (Stokes Theorem ˆ b ( ψ dl = ψ(b ψ(a, a (Fundamental Theorem for Gradients U = 1 ρ(rv (r dτ = ε 0 E(r dτ, (Electrostatic Potential Energy C = Q/ V, (Capacitance 5

6 More possibly useful formulas, etc Poisson s Equation [Laplace s Equation] V = 1 ε 0 ρ [ V = 0] General solution to Laplace s Equation in spherical polar coordinates Boundary Conditions (so far V (r, θ = l=0 ( A l r l 1 + B l P r l+1 n (cos θ, E E 1 = σ/ɛ 0, E 1 = E V 1 = V. V ( r = 1 n=0 1 r n+1 ρ( r (r n P n (cos θ dτ, (Multipole expansion V ( r = 1 p ˆr r, (Dipole potential E( r = 1 p ( cos θˆr + sin θˆθ, (Dipole in electric field 4πɛ 0 r 3 p = i q ir i, p = ρ( r r dτ, (Dipole moment P 0 (x = 1, P 1 (x = x, P (x = (3x 1/, P 3 (x = (5x 3 3x/, (Legendre Polynomials cosh u = eu + e u, sinh u = eu e u, (Hyperbolic Sine/Cosine D ε 0 E+P, (Def n. of Electric Displacement Vector D = ρ free, σ b = P ˆn, ρ b = P, (Gauss s Law in mat l, bound charge densities 6

7 P = χ e ε 0 E, ε ε 0 (1 + χ e, (Linear Dielectrics J = ρ t, (Continuity Equation ˆ I = J da (Current and Current Density B(r = µ 0 4π C ˆ I(r dl (r r r r 3, (Biot-Savart Law B dl = µ 0 I enclosed, B = µ 0 J (Integral/Differential Forms of Ampere s Law E = 1 ρ, ɛ 0 E = 0, B = 0, B = µ 0 J (Maxwell s equations for static charge and current distributions in vacuum P 0 (x = 1, P 1 (x = x, P (x = (3x 1/, P 3 (x = (5x 3 3x/, (Legendre Polynomials 7

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