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 cm5 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 with εr2 =7.6. 10. A cylindrical capacitor has a radii a = 1 and b =2.5 Cm. if the space between them is filled with dielectric with εr= 2.56, Find the capacitance per meter of the capacitor. 11. Three point charges Q1= 1 mC, Q2= -2 mC, and Q3= 3 mC are respectively located at(0, 0, 4), (-2, 5, 1), and (3, -4, 6). a. Find out the potential of Vp at P(-1, 1, 2). b. Calculate the potential difference VPQ if Q at (1, 2, 3). 12. In the following figures there are two capacitors each composed of two halves with different dielectric materials. Determine the capacitance of each capacitor for d=5mm and S=30 Cm2 and εr1 =4(red) εr2 =6(blue) d 13. Given E1 = 10ax - 6ay + 12az V/m in region 1 where y>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 0.2x0.3 cm2 and the gap d 0.1µm. The space between the two sheets are filled with a dielectric with εr=500 and σ=1x10-11 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 4x4 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 2x2 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?