Y E D I T E P E U N I V E R S I T Y FACULTY OF ENGINEERING AND ARCHITECTURE

Biomedica l Engineer ing Department

2012/2013-2

BME 222 Electromagnetic Fields and Waves in BME

Homework 3: Electrostatic fields.

Question 1: (20 Points) Two point charges, Q1 and Q2 are located at (1,2,0) and (2,0,0), respectively. Find the relation between Q1 and Q2 such that the total force on a test charge at point (-1,1,0) will have

a) no x – component. b) no y – component.

Question 2: (20 Points) Assuming that the electric field intensity is E = ax100x (V/m), find the total electric charge contained inside

a) a cubic volume 100 mm on a side centered symmetrically at the origin, b) a cylindrical volume around the z – axis having a radius 50 mm and a height 100 mm

centered at the origin. Question 3: (20 Points) Determine the work done in carrying a -2 μC charge from point P1 (2,1,-1) to point P2 (8,2,-1) in the electric field E = axy+ayx

a) along a parabola x = 2y2, b) along a direct path (straight line joining P1 and P2).

Note: Both paths are directed from P1 to P2. Question 4: (20 Points) A finite line charge of length L carrying a line charge density ρℓ is coincident with z – axis.

a) Determine V in the plane bisecting the line charge. b) Determine E from ρℓ using direct solution. c) Check answer in part b) with .

Question 5: (20 points) The polarization in a dielectric cube of side L centered at the origin is given by P = Po(axx+ayy+azz).

a) Determine the surface and volume bound charge densities. b) Show that the total bound charge is zero.

Question 6: (20 points) Assume that the z = 0 plane separates two lossless dielectric regions with and . If we know E1 in region 1 is ax2y-ay3x+az(5+z), what do we also know about E2 and D2 in region 2? Can we determine E2 and D2 at any point in region 2.

Question 7: (20 Points) Figure 1 shows a dielectric lens. Dielectric lenses are mostly used to collimate electromagnetic fields into a desired direction. The left surface is a circular cylinder and the right surface is a plane. If E1 at point P(ro,45°,z) in region 1 is ar5–aΦ3 , what must the dielectric constant of the lens that E3 is parallel to x – axis? (Assume that region 1 and region 3 are free space.)

y

x

P

ro

45°

1 32

Fig.1. Question 7 Question 8: (20 points) A parallel plate capacitor of width w, length L and separation d is partially filled with a dielectric medium of dielectric constant, , as shown in figure 2. A battery of Vo volts is connected between the plates.

a) Find D, E and ρs in each region. (Regions only inside the capacitor!) b) Find the total capacitance as a function of x. c) Find distance x such that, the total energy stored in both regions will be the same.

εr εo

y

x

Vo

x

L

d

Fig.2. Question 8

Due Date: 26 / 03 / 2013 Tuesday