1. A line charge of uniform density ρl= 2nC/mm forms a semicircle of radius 5 cm in the upper half of xy-plane. Determine the magnitude and direction of E at the center of the semicircle.

1. A line charge of uniform density ρl= 2nC/mm forms a semicircle of radius 5 cm in the upper half of xy-plane. Determine the magnitude and direction of E at the center of the semicircle.
2. Four charge points with dimensions of; Q1 (-10,0,10), Q2 (10,0,10), Q3 (10,0,-10) and Q4 (-10,0,-10) in Cartesian system are surrounded with air. Q1= – Q4= 5 µC.
a. If Q2= Q3= 0, find Vectors E and D
b. For Q2= -Q3= 5 µC forces applied to each point from other charges.
3. A cable containing two cylindrical shape conductors are in Figure below with r1= 0.2 cm and r2= 1.2 cm filled with dielectric of εr=1.5, if ρl= 2nC/cm in the inner conductor and the outer conductor is connected to ground. Calculate the followings;
a. Surface charge density in both conductors
b. E and V in all the regions
c. The Capacitance per unit length between two conductors
d. If a 25 µC charge point is located 1meter away from the cable. What are the changes in voltage and electric field of the inner conductor?
4. 3 Sheets of conductors P1 , P2 and P3 all normal to z axis at z= 0.4×10-6,0 – and -0.4×10-6 respectively, All three sheets are in the shape of rectangle with dimensions of 2×3 mm2 lay on top of each other. The gaps between the sheets are filled with dielectric with εr= 100.
a. If a voltage of 25 V is applied to P1 and P3 is grounded (P2 is not connected to any place), Calculate; E,Q, C, and energy stored in the capacitor
b. If the voltage of a voltage of 25 V is applied to P2. P1and P3 is connected to ground. Calculate; E, Q and C
Bonus;
c. If the capacitor between P1 and P2 called C1 and the one between P2 and P3 called C2 what are the value of C in section (a) and (b) as functions of C1 and C2?
5. The volume charge density ρv =10/R2 nC/m3 exists between two concentric spheres of 3 cm<r<5 cm. Calculate total Q, Determine E and V as functions of R everywhere.
6. The cylindrical coaxial cable with the cross section shown below, has inner and outer radii of 5 mm and 15 mm respectively. The dielectric parameters are; εr=2, σ=5×10-10 S/m and µr =1. (ε0=(1/36π)x10-9 and µ0 =4πx10-7)
a. Calculate the unit length capacitance of the cable.
b. Calculate the unit length resistance between inner and outer conductors of the cable.
c. If the outer conductor voltage is 0 V and ρL=1 µC/cm, find E and V everywhere.
7. Given D = zrCos2Φaz C/m2, calculate the charge density at (1, π/4, 3) and the total charge enclosed by the cylinder of radius of 1 m and the center on z axis and the height is limited by 2 m
8. Two concentric spheres with radii of 3 and 5 cm exist. The space between them is filled with air. A 10 nC charge is uniformly distributed over a spherical shell R = 3cm and a -5 nC charge is uniformly distributed over another spherical shell R =5 Cm. find E and D the regions R <3 Cm, 3 cm< R 5 Cm.
9. Given D1 = 50ax + 80 ay -30az nC/m2 in region where x>0 with εr1 =2.1 Find D2 and E2 in region where x 0 and εr1=3 if region is where y<0 and εr2=4.5 Find;
a. D1,
2
1
1
2
b. E2 and the angle which E2 makes with y-axis,
c. The energy density in each region.
14. A lossy capacitor with the area A 0.2×0.3 cm2 and the gap d 0.1µm. The space between the two sheets are filled with a dielectric with εr=500 and σ=1×10-11 S/m.
a. Find the capacitance and resistance of the capacitor.
b. How long it takes for a 2 volts potential across the capacitor drops to 1 volt with no source connected to it?
0.1 µm Dielectric
0.3 cm Surface of the capacitor
0.3 cm
16. Two sheets of equal size of 4×4 mm2 are .1µm apart. Half of the space between the two planes filled with εr1=10 and the other half is filled with a dielectric εr2=30. Calculate the capacitance of each half and the total capacitance. What is the surface charge density of each half?
εr2=30
εr1=10 Surface of the capacitor
17. Four small spheres with equal charges of 10 nC located at the corners of a square with
2×2 cm2. What is the force applies to each sphere? What is E in the center of square? What is E at the point on the line normal to the plane of the square and crossing the plane at the center of square and is at the distance of 2?