We have a region occupied by a homogeneous isotropic dielectricwith a relative
permeability of ?? = 1 and a relative permittivity of ?? = 4. Inthis region, the electric field has a value;
?⃑ = ?̂(-4/R^2) ?????/?????
where ?̂ is the radial unit vector, and ? is the distance fromthe origin. The current density ? is zero everywhere. The electricfield is constant (static) in time. Point A is on the z-axis atposition (? = 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 closedcircle with its center at the origin and a radius of 1 ????? in thex-y plane. Answer the following questions and state the correctunits:
(a) State (in English, no equations), what is the definition ofthe electrostatic potential (voltage)
between two points. State what units this potential is given interms of the following standard
units; meters, kilograms, seconds, Joules, and Coulombs.
(b) What is the value of the path integral of the electric fieldaround closed contour ?1?
(c) If we construct a sphere with a radius of ? = 2 ??????, andplace this sphere at the point
?(? = +1, ? = +1, ? = −1). Calculate the volume integral of thedivergence of the electric
field intensity ∫ ? ∙ ?⃑?vï® over this volume. State the correctunits.
(d) What is the total flux of the electric field ∮s1 ?⃑ ∙ ??⃑1through the closed surface ?1 surrounding this volume?
(e) Is the voltage at point C greater than, less than, or equalto zero? Write down the relevant path integral and explain yourreasoning.
(f) Where is the divergence of the electric field intensity ?⃑not equal to zero?
(g) Find the exact value of ?(?) − ?(?) by sketching the x-zcoordinate plane system, performing
the appropriate path integral, and showing the path takenstep-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 ∇ × ?⃑ atthe origin (? = 0, ? = 0, ? = 0)? Write down the relevant equationand explain your reasoning.
(i) What is the work that we would haveto perform to move a charge of ? = −3 ???????? from point C toinfinity (∞)?
(j) Where does the divergence of the magnetic flux density ?⃑⃑not to zero?
