ECE 329 Tutorial 5 (2019/09/16) Problem 1 (In a Linear Dielectric): The electrostatic potential in a linear dielectric material with relative permittivity 2 ε푟 = 10.2 is 푉(풓) = 12푥푦 . Find 퐸(풓), 푃(풓) and 퐷(풓). Problem 2 (Electric Boundary Condition with Zero Surface Charge) A plane defined by 3푥 + 2푦 + 푧 = 6 (with zero surface charge) separates two dielectrics. The first dielectric, on the side of the plane containing the origin, has 푉 relative permittivity ε = 3 and uniform electric field 퐸 (풓) = 4푧̂ . The other 푟1 1 푚 dielectric has relative permittivity ε푟2 = 6. Find the electric field in the other dielectric 퐸2(풓). Problem 3 (Electric Boundary Condition with Non-zero Surface Charge): The x-y plane separates dielectric material with relative permittivity ε푟1 = 2 (for region 푧 < 0) from dielectric material with relative permittivity ε푟2 = 4 (for region 푧 > 0). In region 푧 < 0 we have uniform electric flux density 퐷1(풓) = (푥̂ + 8푦̂ + 푛퐶 푛퐶 4푧̂) . There is an uniform surface charge density ρ = 1 on the interface 푚2 푠 푚2 between the two dielectrics. Determine the electric flux density 퐷2(풓) within the dielectric material occupying the region 푧 > 0. ECE 329 Tutorial 6 (2019/09/19) Problem 1: Consider an electric field 푬(풓) present in free space, and a closed path C indicated in the figure below: Answer the following question: (a) If we have a static electric field expressed as: 푬(풓) = 2푥̂ − 3푦̂ + 5푧̂ V/m (i) What is the circulation ∮ 푬(풓) ∙ 푑풍 around the closed path C? (ii) What is the electrostatic potential change from point a to point b (푉(풂) − 푉(풃))? (b) If instead, the curl of the electric field is: ∇ × 푬(풓) = 2푥̂ − 3푦̂ + 5푧̂ V/m2 (i) What is the circulation ∮ 푬(풓) ∙ 푑풍 around the closed path C? (ii) What is the electrostatic potential change from point a to point b? Problem 2: Compute the total outward electric flux from the cube 0 ≤ 푥, 푦, 푧 ≤ 1 m , 휇퐶 containing the volumetric charge density of ρ (풓) = 16푦푧3 . 푉 푚3 Problem 3: An electric field is observed to point radially outward in cylindrical coordinate everywhere. The region < 3 m is filled with a dielectric with relative permittivity ε푟 = 5 while the region > 3 m is free space. If the electric field just outside the + 푉 = 3m interface is 푬( = 3 ) = 4̂ , and just inside the = 3m interface is 푚 푉 푬( = 3−) = 2̂ . Find the free surface charge density on the = 3m 푚 푠 interface. Problem 4: A perfect conducting spherical shell of radius b is held at a constant voltage 푉0 > 0, while a concentric perfect conducting shell of radius a (where 푎 < 푏) is grounded. The region in between of the two shells (푎 < 푟 < 푏) contains a conducting metal with conductance > 0 and relative permittivity ε푟 = 1. You make measurement of the electrical properties of the configuration, and find that there is a total charge of 푄 > 0 on one shell, while a total charge of −푄 < 0 on the other. Moreover, the electric field 푬 and volumetric charge density ρ푉 are zero for both regions 푟 < 푎 and 푟 > 푏. (a) Which shell has a positive total charge, and which has negative total charge? (b) What can you say about the voltage inside the region 푟 < 푎? (c) Applying Gauss’ law, calculate the electric field expression for 푬(푎 < 푟 < 푏). Hint: All the electric charges resides on the surface(s) of conductor (d) Applying Ohm’s law (푱 = 푬, where 푱 denotes the surface current density of the conducting current driven by electric field 푬 in a conductor with conductance ), calculate the total current 퐼 that cross a spherical shell with radius 푟0 satisfying 푎 < 푟0 < 푏, located inside the conducting metal. (e) Use the electric field 푬(푎 < 푟 < 푏) obtained from part (c), calculate the voltage different between shell with radius b and shell with radius a (in other words, calculate 푉0). .