INTRODUCTION: Electric potential can be stated as potential energy per unit charge, just as electric field is defined as force per unit charge. The concept of electric potential can be derived from work done in electric systems. If a test charge q is introduced in the space where an electric field exists, it will experience an electric force. Work is done if the test charge is moved from one point to another in the presence of electric field. The electric potential at a point is defined as the amount of work required to bring a unit positive charge from infinity to the point under consideration. The electric potential V at a distance r from a charge q is,
V = kq/r .................................. (1)
where, k = 8.99x10^{9} N.m^{2}/C^{2}. Electric potential is a scalar quantity. The unit of potential is called volt.
The principle of superposition applies to the electric potential. When more than one charge is present, the electric potential at a point is the algebraic sum of the potentials due to each of the charges present. If N number of charges are present, then the resultant electric potential V at a point is given by,
V = V_{1} + V_{2} + V_{3} + ........... + V_{N} = S(q_{i}/r_{i}) ............... (2)
where, V_{1}, V_{2}, ........................... V_{N} are the potentials due to N different charges and, ri is the distance of the charge q_{i} in general. It can be shown that the potential due to a uniformly charged sphere at points on the surface or outside the surface is,
V = kq/r ................................................ (3)
where q is the total charge on the sphere and r is the distance from the center of the sphere. The result is the same whether the total charge q is uniformly distributed or placed at the center.
ELECTRIC POTENTIAL ENERGY
The electric potential can also be defined as the electric potential energy per unit charge. Hence,
V = U/q = W/q ..................... (4)
where the magnitude of electric potential energy U is the same as the work done W. A line joining all points having the same potential is called Equipotential line. A surface that has the same potential everywhere on it, is called Equipotential surface.
POTENTIAL DIFFERENCE
When a charge q is brought from infinity to a point where the electric potential is V, work of the amount qV is said to be performed. The difference in electric potential between two points is called potential difference. So, the work done W when a charge q is moved from a point A to a point B is,
W = q(V_{B}  V_{A}) ................................. (5)
where VA is the potential at A and VB is the potential at B. Electron volt (eV) is the unit of electric energy.
1 eV = 1.602x10^{19} joules
CAPACITANCE
The ability or capacity to store electric charges in conductors and hence electric energy, is related to the capacitance of the conductors. A capacitor or condenser is made of two conductors separated by a dielectric or insulating medium. The capacitance C of a capacitor is defined as,
C = Charge on either conductor/Potential difference between the two conductors
That is,
C = q/V .............................................. (6)
The unit of capacitance is called Farad (F). The capacitance of a parallel plate capacitor, made of two parallel plate conductors of plate area A, facing each other and separated by a distance d is,
C = e(A/d) ................................... (7)
where e is the electric permittivity of the medium between the two conductors. The electric permittivity of the medium is given by
e = e_{o}K .......................................... (8)
where K is called dielectric constant and eo = 8.85x10^{12} F/m is the electric permittivity in vacuum. K = 1 in vacuum.
e_{o} = 8.85x10^{12} C^{2}/N.m^{2}
The potential difference V between the two plates of a parallel plate capacitor is,
V = Ed .............................................. (9)
where E is the electric field inside the two plates.
ENERGY STORED IN A CAPACITOR
The energy W stored in a capacitor that has a potential difference V and q amount of charge on each plate is,
W = (1/2)qV ................................... (10)
Using equation (6), we may write
W = (1/2)CV^{2} = (1/2)q^{2}/C .............................. (11)
CAPACITOR CONNECTED IN SERIES OR IN PARALLEL
Equivalent Capacitance of two or more capacitors connected in series is equal to the sum of individual capacitances. If two or more capacitors are connected in parallel, then sum of the reciprocal of the individual capacitances is equal to the reciprocal of the equivalent capacitance. That is, the equivalent capacitance for N capacitors connected in parallel is,
C_{eq} = C_{1} +C_{2} + C_{3} + C_{4} + .................. + C_{N} ...................... (12)
If N number of capacitors are connected in series, then
1/C_{eq} = 1/C_{1} + 1/C_{2} + 1/C_{3} + 1/C_{4} + ............ + 1/C_{N} ................(13)
TEST YOURSELF:
1. Find the electric potential at a distance of 5 m from a point charge of 15 nC.
2. What are the electric potentials at distances r=25 cm, 45 cm and r=75 cm from a point charge of 5 mC?
3. Find the electric potential at the center of a square as shown in the figure 21, if four point charges are placed at the four corners of the square 50 cm on each side, when
(a) each of the four charges is +5mC
(b) two charges are +5mC and two are 5mC.
Figure 21
4. Find the electric potential at x=0 if three point charges are placed along xaxis as follows:
(a) +5 nC at x=15 cm, +3 nC at x=20 cm and +4 nC at x=25 cm
(b) +7 nC at x=25 cm, 6 nC at x=34 cm and +5 nC at x=15 cm.
5. Find the electric potential at a point A due to the presence of 4 point charges q_{1}, q_{2}, q_{3} and q_{4} at distances r_{1}, r_{2}, r_{3} and r_{4} respectively, as shown in the diagram below.
q_{1}= +5 nC at r_{1}=0.45 m, q_{2} = 3 nC at r_{2}= 0.15 m, q_{3} = +7 nC at r_{3} = 0.20 m and q_{4} = 4 nC at
r_{4} =0.35 m.
6. Two point charges q_{1} and q_{2} are of same magnitude and opposite sign Are placed in xy plane. The charges are separated by a distance d as in the diagram below. What are the points at which the electric potential is zero?
Figure for Problem number 6.
7. An electron and a proton are placed at x=1.5x10^{10} m and x=+1.5x10^{10} m, respectively along xaxis(an arrangement of charges known as electric dipole). Find the electric potential at,
(a) x=1.0x10^{6} m
(b) at y=1.0x10^{6} m.
8. What will be the work done if 1 C of charge is moved from a positive terminal to the negative terminal of a 12 V battery?
9. An electron starting from rest falls through potential difference of 100 V moving from higher to lower potential. Find the speed of the electron.
10. Two point charges +4 mC at x=20 cm and 5 mC at x= 75 cm are placed along xaxis. Find the work done if +3 mC of point charge is moved from x=25 cm to x=70 cm.
11. Two large parallel metal plates are separated by a distance of 1.0 cm and, are connected to a 120 V battery as shown below.
Find the
(a) electric field between the plates
(b) force experienced by a point charge +3 mC placed between the plates
12. A parallel plate capacitor has a plate area of 8.0x10^{3} m^{3} and a plate separation distance of 1.5 mm in the air. Find the capacitance and the charge on each plate if connected to a 6.0 V battery.
13. A capacitor that uses paraffin (K=2.2) as dielectric is connected to a 24 V battery. Its capacitance value in air is 25 mF. Calculate the potential energy stored in the capacitor.
14. A capacitor with a capacitance of 25 mF is connected to a 9.0 V battery. What is the magnitude of the charge stored in each plate?
15. A parallel plate capacitor of capacitance 5 mF filled with air has a plate area of 0.0025 m^{2}. What is the distance of plate separation in this capacitor?
16. A parallel plate capacitor has a capacitance of 10 mF. What will be its capacitance if the plate separation is doubled?
17. A parallel plate capacitor filled with air has a capacitance of 25 mF. It is filled with a material of dielectric constant K=7.5 and charged with a 12 V battery as shown below.
What is the
(a) new capacitance of the capacitor?
(b) magnitude of charge stored in each plate?
18. Find the equivalent capacitance of two capacitors with capacitance values of 10 mF and 20 mF if connected in
(a) series
(b) parallel.
19. Find the equivalent capacitance and the total energy stored by the group of capacitors in the diagram below.
20. Two uncharged capacitors in series are connected with a source power supply V_{o}=500V as shown. C_{1} =10.0 mF and C_{2}=20.0 mF
Find the magnitude of the charge stored and the potential difference in each of the two capacitors.
21. Two capacitors C_{1} =10.0 mF and C_{2}=20.0 mF are connected in parallel to V_{o}=1000 V power supply as shown.
Find the magnitude of charge stored in each of the two capacitors.
22. A 4.5 mF capacitor is charged to 2.5 kV. It is then connected to an identical uncharged capacitor. Find the new charge and energy stored in each capacitor.
23. Two capacitors C_{1}=5.0 mF and C_{1}=7.0 mF are each separately charged with a 15 V battery. The two capacitors are then connected together such that the positive plate of one is joined with the negative plate of the other. What is the final charge of each capacitor.
24. A 5.0 mF capacitor is charged to 50 V, and a 25 mF capacitor is charged to 25 V. The positive plates of the two are joined together and the negative plates of the two are joined together. What are the new potential difference and charge stored in the capacitors?
25. An air filled parallel plate capacitor has a plate area of 25 cm^{2}. A break down in air occurs (air becomes conductive) if the electric field exceeds 2.5x10^{6} V/m. Calculate the amount of charges that can be stored in the capacitor.
CHECK YOURSELF: Solution of the problems will soon be avilable in the Crumb Library Reserve Folder #301.
GRADE YOURSELF
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Updated 10/19/99