Question

We have a region occupied by a homogeneous isotropic dielectric with a relative

permeability of ?? = 1 and a relative permittivity of ?? = 4. In this region, the electric field has a value;

?⃑ = ?̂(-4/R^2) ?????/?????

where ?̂ is the radial unit vector, and ? is the distance from the origin. The current density ? is zero everywhere. The electric field is constant (static) in time. Point A is on the z-axis at position (? = 0, ? = 0, ? = −2 ??????); point B is on the z-axis at (? = 0, ? = 0, ? = +4 ??????) and point C is at (? = +1 ?????, ? = +1 ?????, ? = +1 ?????). The voltage at infinity is set to zero. Finally, you are given a closed contour ?1 which forms a closed circle with its center at the origin and a radius of 1 ????? in the x-y plane. Answer the following questions and state the correct units:

(a) State (in English, no equations), what is the definition of the electrostatic potential (voltage)

between two points. State what units this potential is given in terms of the following standard

units; meters, kilograms, seconds, Joules, and Coulombs.

(b) What is the value of the path integral of the electric field around closed contour ?1?

(c) If we construct a sphere with a radius of ? = 2 ??????, and place this sphere at the point

?(? = +1, ? = +1, ? = −1). Calculate the volume integral of the divergence of the electric

field intensity ∫ ? ∙ ?⃑?v over this volume. State the correct units.

(d) What is the total flux of the electric field ∮s1 ?⃑ ∙ ??⃑1 through the closed surface ?1 surrounding this volume?

(e) Is the voltage at point C greater than, less than, or equal to zero? Write down the relevant path integral and explain your reasoning.

(f) Where is the divergence of the electric field intensity ?⃑ not equal to zero?

(g) Find the exact value of ?(?) − ?(?) by sketching the x-z coordinate plane system, performing

the appropriate path integral, and showing the path taken step-by-step. No credit will be given

for any other technique of obtaining an answer.

(h) What is the curl of the magnetic field intensity ∇ × ?⃑ at the origin (? = 0, ? = 0, ? = 0)? Write down the relevant equation and explain your reasoning.

(i) What is the work that we would have to perform to move a charge of ? = −3 ???????? from point C to infinity (∞)?

(j) Where does the divergence of the magnetic flux density ?⃑⃑ not to zero?

Answer 1