Problem 1

Please provide a brief and clear answer to the following questions.

(1) By saying that the electrostatic filed is conservative, we do NOT mean that
(a) Its curl is identically zero.
(b) It is the opposite of the gradient of a scalar potential.
(c) The absolute potential at a point, is defined relativply to the zero location.
(d) The work done in a closed path inside the field is zero.
(e) The divergence of the electric flux density at a point is equal to the charge density at that point.

(2) Which of the following statements is NOT characteristic of a static magnetic field?
(a) It is not conservative.
(b) It is rotational.
(c) It has no sinks or sources.
(d) Magnetic flux lines are always closed.
(c) The total number of flux lines entering a given region is not equal to the total number of flux lines leaving the region.

(3) The two uniform fields shown below are near a dielectric-dielectric boundary but on two sides of it.

Which of the following configurations are NOT correct? Assume that the boundary is charge free and that ε₂ > ε₁

(4) Two electrons enter a static magnetic field (B^→) at the same time with initial velocities 2v and v, respectively. Since both velocities are perpendicular to B^→, the magnetic force becomes the centripetal force, Which of the two electrons returns to its starting point first? Why?

(5) Please explain why a complete revolution (360 degrees) around the Smith chart represents a distance of λ/2 on the transmission line

(6) Please explain how a quarter-wave transformer works to achieve load matching.

Problem 2

In free space, a uniformly charged sphere of radius α is centered at the origin. Assume that the total amount of charge within this sphere is Q.

(1) What is the volume charge density pu of this sphere?

(2) Please apply Gauss's Law to determine the electric field everywhere (i.e. 0 < r < a and r > a).

You must show details including how to construct the Gaussian surface. Merely writing down a memorized answer will reap zero points.

(3) Now assume that the sphere has a spherical hole with radius b (b < a) centered at the origin as well, which does not carry any charge any more. Determine the electric field in the region h < r < a.

(4) What is the potential difference Val, between the two spherical shells at r = a and T = b?

Problem 3

Two large fiat, metal sheets aro located at, x = 0 and x = d. and are maintanied at v(x = 0) = 0 and V(x = d) = V_0, respectively. The region in between the two sheets is charge free and is filled with two different dielectric materials with Ei and E2, respectively.

(1) List the Laplace's equations for each dielectric region, and write down the solution forms for each equation.

(2) What is the dielectric-dielectric boundary condition at the interface x = a?

(3) Determine the electric field in both dielectric regions, Hint: Since there are four different unknowns in the two Laplace's equations, we need to find four boundary conditions to solve them.

Problem 4

A conducting wire bent, as shown below lies in the xy plane and carries a current I = 10A, with L = 1m and a = 1m.

(1) If the magnetic flux density in the region is B^→ = 2a^→₂Wb/m², determine the magnetic force acting on the wire.

(2) Is your result from (1) the same as that acting on a straight wire of the length 2(L + a) which carries the same current? Why?

Problem 5

In free space, two infinitely long concentric cylindrical metal sheets have radii R₁ and R₂ (R₁ < R₂)), repectively. Each sheet caries a current I (the current directions are opposite).

(1) Calculate the magnetic field caused by these two currents everywhere, i. e. < R₁ R₁ < r < R₂ r > R₂.

(2) Calculate the magnetic flux through a given surface with length L (shown as the shaded area in the figure).

Problem 6

A 20m - long lossless transmission line is termintaed with a load having an equivalent impedance of 40 + j20 Ω at 10 MHz. The per-unit length inductance and capacitance of the line are L = 20nH/m and C = 50 pF/m, respectively.

(1) Calculate the characteristic impedance Z_o the phase constant β, and the wavelength λ.

(2) Use the Smith chart to find the input impedance at the midpoint of the line. You won't get credit if you choose to manually calculate the input impedance.

(3) Find the standing wave ratio s.