EXCEL REVIEW CENTER

ELECTRICAL ENGINEERINGElectrostatic/CapacitanceEXCEL REVIEW CENTERElectrostatics/CapacitanceElectrostatics is the branch of science that deals with the phenomena and properties of stationary or slow-moving (without acceleration ) electric charges. (Wikipedia)

Laws of Electrostatics:

Like charges repel each other, unlike charges attract each other. The force exerted between two charges is directly proportional to the product of their strengths and is inversely proportional to the square of the distance between the two charges.

Coulombs Law:The fundamental equation of electrostatics is Coulomb's law, which describes the force between two point charges. The magnitude of the electrostatic force between two point electric charges Q1 and Q2 is directly proportional to the product of the magnitudes of each charge and inversely proportional to the surface area of a sphere whose radius is equal to the distance between the charges: (Wikipedia)

F = k

k =9 x 109

F force in Newton Q1, Q2 respective charges on the particles in Coulombs

- absolute permittivity

= 8.85410-12 Farad/meter

- relative permittivity or dielectric constant

= 1 - free space d - distance between the charged particles in meters

In CGS:F force in DyneQ1, Q2 respective charges on the particles in Statcoulombd - distance between the charged particles in centimetersCapacitance is the ability of a body to hold an electrical charge.

Electric Intensity: E or (Field Intensity)

E+QdE

E - Electric Intensity in Newton/CoulombQ charge in Coulombs d distance in meters

VPotential at a point: V

V potential at a point in volts Q charge in Coulombs d distance in meters

Electric field outside AN isolated sphere in free space: E

E = 9109 E electric field outside an isolated sphere in free space in volts/meter Q the total charge in Coulombs d the distance from the center of an isolated sphere located in free space in meters.

Electric flux density: D

DD

D - electric flux density in Coulombs /sq. meter

- electric flux in CoulombsA - area in sq. meters

CAPACITANCE: C

C = is the property of the capacitor which delays any change of voltage across it.

C capacitance in FaradQ charge in CoulombsV voltage in Volts

ElasTANCE: S

the reciprocal of capacitance

S = S elastance in DarafQ charge in CoulombsV voltage in Volts

CAPACITANCE of two parallel plate capacitor:

C =Vd

C capacitance in FaradA area of each plate sq. metersd - thickness of the dielectric material in meters

CAPACITANCE of n parallel plates Capacitor with the same dielectric material and thickness of insulation:

drddrrC =

C capacitance in Faradn the number of platesCAPACITANCE of several parallel plates Capacitor with different dielectric material and thickness of insulation:

d1r1d2d3r2r3

C =

Properties of Various Insulating (Dielectric) Materials

Vacuum1

Air1.0006

Asbestos2

Bakelite5

Cellulose film5.8

Marble7

Mica6

Paper (Dry)2.2

Paper (Treated)3.2

Glass6

Porcelain5.7

Pressboard6.2

Quartz, fused3.5

Rubber2.6

Silica, fused3.6

Water70

Wax, paraffin2.2

Major Types of Capacitors:

1Electrolytics

2Dielectrics

3Plastic film types

4Metallized plastic types

5Glass and ceramics

6Mica and mica/paper types

7Air/vacuum types

CAPACITANCE of an isolated sphere:

C = 4 o r rr+Q+++++++r

Capacitance of a Spherical Capacitor:

r1+rr2++++Q+-------

C = 4or

Capacitance of Coaxial cable:

C = (Farad/meter)r1r2

capacitors in series:

=++

capacitors in Parallel:

CT = C1 + C2 + Cn

energy stored in a capacitor:

W = CV2 = = VQ

W stored energy in (Joules)C capacitance in (Farads)V voltage across a capacitor in (Volts)Q charge in (Coulombs)

Example1: Calculate the force between two charged particles placed 10-13 m. apart from each other. The charge of each particle is 3.510-19 C. (Ans. ________ )

Example2: Two identical charge particles 1.5 meters apart placed under water experience a force of 2 Newton. If the dielectric constant of water is 80, calculate each charge. (Ans. ________ )

Example3: Q1 and Q2 are placed 50 cm apart in an x-axis. In between the two charges another charge Q3 is placed 20cm from Q1. If Q1= 3C, Q2= 8C and Q3=-5C, calculate the net force on Q3.(Ans. ________ )

Example4: Three point charges are placed at the corners of an equilateral triangle as shown. Calculate the resultant force exerted on the charge 3Q where Q = 2C. (Ans. ________ )

Example5: Two charges are placed 40 cm apart on the x-axis. Q1=3C is placed at 0 while Q2= 5C is at 40cm. Calculate the distance from Q1 a new negatively charge Q3 be placed if the force on it is zero.(Ans. ________ )

Example6: Calculate the electric field E in air at a distance of 30 cm from a point charge Q1=5.0x10-9 C.(Ans. ________ )

Example7:. Two spheres having charges of Q1=5C and Q2=3C are placed 20cm apart. The spheres are made to touch each other and then again separated at a distance of 30cm. Calculate the force after this condition.(Ans. ________ )

Example8: Two positive point charges of Q1=12x10-10C and Q2=8x10-10C are placed 10 cm apart. Find the work done in bringing the two charges 4 cm closer.(Ans. ________ )

Example9: A capacitor consists of 15 aluminum sheets connected in parallel separated by plates of glass in between having dielectric constant of 6 and a thickness of 2 mm. If the area of each sheet is 5 cm by 10 cm, calculate the charge on the capacitor when 24 volts is applied to it.(Ans. ________ )

Example10: Two metallic plates separated by 1-cm are connected across a 12-V battery. A certain plastic material is inserted completely filling the space between the plates and the charge on each plate is observed to double. What is the dielectric constant of the plastic material? (Ans. ________ )

Example11. An isolated charge sphere having a radius of 0.1cm is place in a medium having a dielectric of 2.2. Determine its capacitance.(Ans. ________ )

Example12: Two conductors having a diameter each of 1.5-cm are placed 45-cm apart from each other. What is the capacitance of the line in Farad per meter. (Ans. ________ )Example13: The series combination of two capacitors is connected to a voltage source of 1000 volts. If C1= 3pF and C2= 6pF, calculate (a) its equivalent capacitance Ceq (b) the charges on each capacitors: Q1, Q2 (c) the voltage across each capacitors: V1, V2 (d) the energy stored in each capacitor: W1, W2.(Ans. ________ )

Example14: The parallel combination of two capacitors is connected across a voltage source of 120 volts. If C1=2pF and C2=6pF, calculate (a) its equivalent capacitance: Ceq (b) its equivalent elastance: St. (Ans. ________ )

Example15 Two capacitors 3F and 4F are individually charged across a dc voltage source of 6.0 volts. After being disconnected from the dc source, the capacitors are connected with the negative plate of one end attached to the positive plate of the other. Calculate the final charge on each capacitor. (Ans. ________ )

Example16: Two capacitors are connected in series and the series combination is connected in parallel with a third capacitor. If the three capacitors are of the same value, i.e. 150F, what is the total capacitance? (Ans. ________ )

Example17: A length of a co-axial cable is 300 km long. Its conductor has a diameter of 0.5cm and its insulation covering is 0.4cm thick. Calculate the capacitance of the cable if the insulation covering has a relative permittivity of 4.5.

Problems to Solve

Problem1. Determine the voltage across a 4F capacitor when charged with 5mC.A. 2.25 kVB. 6.55 kVC. 3.25 kVD. 1.25 kV *

Problem2.Two identical charges are place 1.5 meters apart. If the force on each charge is about 2 Newton, calculate how large is the charge of each?A.4x10-6CB. 7x10-9CC.5x10-9CD. 2x10-5C *

Problem3. A direct current of 4A flows into a previously uncharged 20F capacitor for 3ms. Determine the voltage between the plates.A. 220 VB. 600 V *C. 410 VD. 650 V

Problem4.Two parallel rectangular plates measuring 20cm by 40cm carry an electric charge of 0.2C. Calculate the electric flux density.A.2.5 C/m2* B. 3.5 C/m3C.6.5 C/m5D. 2.7 C/m2

Problem5.The flux density between two plates separated by mica of relative permittivity 5 is 2 C/m2 . Find the voltage gradient between the plates.A. 15.7 kV/mB. 47.1 kV/mC. 45.2 kV/m *D. 32.9 kV/m

Problem6. A ceramic capacitor has an effective plate area of 4cm 2 separated by 0.1mm of ceramic of relative permittivity 100. Calculate the capacitance of the capacitor in picofarads.A. 5340 pFB. 4760 pFC. 2580 pFD. 3540 pF *

Problem7. A waxed paper capacitor has two parallel plates, each of effective area 800cm2. If the capacitance of the capacitor is 4425pF, determine the effective thickness of the paper ifits relative permittivity is 2.5.A. 0.62 mmB. 0.4 mm *C. 1.2 mmD. 0.95 mm

Problem8. A parallel plate capacitor has nineteen interleaved plates each 75mm by 75mm separatedby mica sheets 0.2mm thick. Assuming the relative permittivity of the mica is 5, calculate the capacitance of the capacitor.A.22.4 nF *B. 22.4 nFC.22.4 nFD. 22.4 nFProblem10. Calculate the equivalent capacitance of two capacitors of 6F and 4F connected in parallel.A.15 FB. 21 FC.10 F *D. 9 F

Problem11. Calculate the equivalent capacitance of two capacitors of 6F and 4F connected in series.A.2.4 F *B. 1.9 FC.3.2 FD. 2.6 F

Problem12.What capacitance must be connected in series with a 30F capacitor for the equivalentcapacitance to be 12F?A. 42 FB. 20 F *C. 36 FD. 29 F

Problem13.Capacitances of 1F, 3F, 5F and 6F are connected in parallel to a direct voltagesupply of 100V. Determine the total charge. A.0.3 mCB. 2.1 mCC.1.5 mC *D. 1.3 mC

Problem14. Capacitances of 3F, 6F and 12F are connected in series across a 350 V supply.Calculate the voltage across capacitor 1.A. 120 VB. 210 VC. 350 VD. 200 V *

Problem15. Determine the energy stored in a 3F capacitor when charged to 400 V. Find also the average power developed if this energy is dissipated in a time of 10s.A.26 kWB. 32 kWC.24 kW *D. 29 kW

Problem16. A 12F capacitor is required to store 4J of energy. Find the voltage to which thecapacitor must be charged.A. 726.8 VB. 527.3 VC. 941.5 VD. 816.5 V *

Problem17. A capacitor has a capacitance of 8F with air between its plates. Determine its capacitance when a dielectric of value 6 is placed between its plates.A. 31FB. 48F *C. 59FD. 72F

Problem18. The potential difference between metal plates is 120 volts. If the distance between theses plates is 3.0mm, calculate the electric field between them.A. 31kV/mB. 40kV/m *C. 64kV/mD. 42kV/m

Problem19. A charge Q1of 0.20C is 30cm from another charge Q2 of 3.0F in vacuum. Calculate the work needed to bring Q1 18 cm closer to Q2.A. 0.132JB. 0.232JC. 0.079JD. 0.027J *

Problem20. A charge Q1 of 2.0F is place at point (0,0) while a second charge Q2 is place at point (100,0) cm. Determine at what point in the positive x-axis will the absolute potential be zeroA. 63cmB. 40cm *C. 50cmD. 32cm

Problem21. Three point charges are located at the corners of a square of side L equal to 30cm. The charge Q1 of -4C is at (0,0), charge Q2 of 8C is at (L,0), charge Q3 of -5C is at (L,L), Calculate the electric field strength at (L,0)A. 2.45NB. 0.79NC. 1.48N *D. 1.95N

Problem23. Three point charges are placed at the corners of an equilateral triangle. The charge Q1 of 2C is at (0,0), charge Q2 of 3C is at (20,0), and charge Q3 of 4C is placed at the 1st quadrant. Calculate the force on Q3.A. 3.9 N*B. 2.5 NC. 4.2 ND. 3.1 N

Problem24. Determine the electric field strength E at a distance 30cm away from a point charge of 5nF.A. 0.62 kN/CB. 0.36 kN/C C. 0.49 kN/CD. 0.50 kN/C *

Problem25. In a concentric cable capacitor the diameters of the inner and outer cylinders are 3 and 10 mm respectivelv. If the relative permittivity for insulation is 3 find its capacitance per metre. A. 241.7pFB. 138.8pF *C. 337.1pFD. 291.5pF

Problem26. The conductors of a two-wire transmission line (4 km long) are spaced 45cmbetween centre. If each conductor has a diameter of 1.5 cm, calculate the capacitance of the line. A. 31.6 nFB. 27.2 nF *C. 25.3 nFD. 34.9 nF

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