Potential from a distribution of charges = 1


 Elmer Quinn
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1 Lecture 7
2 Potential from a distribution of charges V = X Smooth distribution i q i r i V = X i q i r i = Z r dv Calculating the electric potential from a group of point charges is usually much simpler than calculating the electric field It s a scalar
3 iclicker: Electric Potential from Two Oppositely Charged Point Charges What s the potential and electric field in the middle between two point charges q and q separated by distance d. d q? q A : E =0,V =0 B : E = k 8q,V =0 d2 C : E = k 8q d 2,V = k q d D : E = k 8q d 2,V = k q 2d E : E =0,V = k q d 22
4 Electric Potential from Two Oppositely q V = k d/2 E = k q d/2 q (d/2) 2 + Charged Point Charges What s the potential and electric field in the middle between two point charges q and q separated by distance d. d q? q =0 q (d/2) 2 = k 8q d 2 What does V=0 mean? No energy is needed to bring a charge from rest to rest at infinity. A : E =0,V =0 B : E = k 8q,V =0 d2 C : E = k 8q d 2,V = k q d D : E = k 8q d 2,V = k q 2d E : E =0,V = k q d 22
5 d E1 q E2 q q V = k d/2 E = k q (d/2) 2 + q d/2 =0 q (d/2) 2 = k 8q d 2
6 Electric Potential from Two Oppositely Charged Point Charges The electric field lines from two oppositely charge point charges are a little more complicated The electric field lines originate on the positive charge and terminate on the negative charge The equipotential lines are always perpendicular to the electric field lines The red lines represent positive electric potential The blue lines represent negative electric potential Close to each charge, the equipotential lines resemble those from a point charge 22
7 Electric Potential from Two Identical Point Charges The electric field lines from two identical point charges are also complicated The electric field lines originate on the positive charge and terminate at infinity Again, the equipotential lines are always perpendicular to the electric field lines There are only positive potentials Close to each charge, the equipotential lines resemble those from a point charge 23
8 Example: Superposition of Electric Potential (1) Assume we have a system of three point charges: q 1 = µc q 2 = µc q 3 = µc q 1 is located at (0,a) q 2 is located at (0,0) q 3 is located at (b,0) a = 8.00 m and b = 6.00 m Question: What is the electric potential at point P located at (b,a)? 30
9 Example: Superposition of Electric Potential (2) Answer: The electric potential at point P is given by the sum of the electric potential from the three charges r 1 r 2 r 3 31
10 Conductor in Efield Charges can move freely along conductor boundary. Recall: electrostatic shielding: E=0 inside Surface of a conductor is an equipotential surface. Efield is perpendicular to conductor s surface 10
11 Electric Potential Energy for a System of Particles So far, we have discussed the electric potential energy of a point charge in a fixed electric field Now we introduce the concept of the electric potential energy of a system of point charges In the case of a fixed electric field, the point charge itself did not affect the electric field that did work on the charge Now we consider a system of point charges that produce the electric potential themselves We begin with a system of charges that are infinitely far apart This is the reference state, U = 0 To bring these charges into proximity with each other, we must do work on the charges, which changes the electric potential energy of the system 37
12 Electric Potential Energy for a Pair of Particles (1) To illustrate the concept of the electric potential energy of a system of particles we calculate the electric potential energy of a system of two point charges, q 1 and q 2. We start our calculation with the two charges at infinity We then bring in point charge q 1 Because there is no electric field and no corresponding electric force, this action requires no work to be done on the charge Keeping this charge (q 1 ) stationary, we bring the second point charge (q 2 ) in from infinity to a distance r from q 1 That requires work q 2 V 1 (r) 38
13 Electric Potential Energy for a Pair of Particles (2) So, the electric potential energy of this two charge system is where Hence the electric potential of the two charge system is If the two point charges have the same sign, then we must do positive work on the particles to bring them together from infinity (i.e., we must put energy into the system) If the two charges have opposite signs, we must do negative work on the system to bring them together from infinity (i.e., energy is released from the system) 39
14 iclicker Two large plates of area A are charged with +q and q, located distance d apart (d << size of the plates). How much electric energy is stored in the system A A +q q A : B : C : D : E : q 2 0A d q A d 2q 2 0A d 2q 2 0A d q A d d
15 iclicker Two large plates of area A are charged with +q and q, located distance d apart (d << size of the plates). How much electric energy is stored in the system A A +q q A : B : C : D : E : q 2 0A d q A d 2q 2 0A d 2q 2 0A d E = d + = potential difference between plates V = Ed = energy q A d U = q V = q2 0A d q 0A d = q 0A
16 iclicker Two large plates of area A are charged with +q and q, located distance d apart (d << size of the plates). How much electric energy is stored in the system A A +q q A : B : C : D : E : q 2 0A d q A d 2q 2 0A d 2q 2 0A d d Efield inside is constant. E = + = potential difference between plates V = Ed = energy q A d U = q V = q2 0A d q 0A d = q 0A
17 Electric Potential Energy Consider three point charges at fixed positions. Question: What is the electric potential energy U of the assembly of these charges? q1 d12 q2 d13 d23 q3 16
18 Example: Electric Potential Energy (2) Answer: The potential energy is equal to the work we must do to assemble the system, bringing in each charge from an infinite distance Let s build the system by bringing the charges in from infinity, one at a time 41
19 Example: Electric Potential Energy (3) Bringing in q 1 doesn t cost any work With q 1 in place, bring in q 2 We then bring in q 3.The work we must do to bring q 3 to its place relative to q 1 and q 2 is then: U = U 13 + U 23 = kq 3 q1 12 d 13 + q 2 d 23 U tot = U 12 + U 13 + U 23 = k q1 q 2 d 12 + q 1q 3 d 13 + q 2q 3 d 23 A sum of pair interaction energy. 42
20 Alternative way Let s calculate energy of a particle in a total potential created by all other particles. And then sum over all particles. q2 Potential created by 2 &3 at V 1 = k position of 1 is Potential created by 1 &3 at position of 2 is Potential created by 1 &2 at position of 3 is V 2 = k V 3 = k q1 q1 d 12 + q 3 d q 3 d 12 d 23 d q 3 d 23 42
21 Total energy: U tot = q 1 V 1 + q 2 V 2 + q 3 V 3 (?) U tot = q 1 V 1 + q 2 V 2 + q 3 V 3 = q2 q 1 k + q 3 + d 12 d 13 q1 q 2 k + q 3 + d 12 d 23 q1 q 3 k + q 3 d 13 d 23 q1 q 2 =2k + q 1q 3 + q 2q 3 d 12 d 13 d 23
22 Total energy: U tot = q 1 V 1 + q 2 V 2 + q 3 V 3 (?) U tot = q 1 V 1 + q 2 V 2 + q 3 V 3 = q2 q 1 k + q 3 + d 12 d 13 q1 q 2 k + q 3 + d 12 d 23 q1 q 3 k + q 3 d 13 d 23 q1 q 2 =2k + q 1q 3 + q 2q 3 d 12 d 13 d 23
23 Total energy: U tot = q 1 V 1 + q 2 V 2 + q 3 V 3 (?) U tot = q 1 V 1 + q 2 V 2 + q 3 V 3 = q2 q 1 k + q 3 + d 12 d 13 q1 q 2 k + q 3 + d 12 d 23 q1 q 3 k + q 3 d 13 d 23 q1 q 2 =2k + q 1q 3 + q 2q 3 d 12 d 13 d 23???2???
24 Total energy: U tot = q 1 V 1 + q 2 V 2 + q 3 V 3 (?) U tot = q 1 V 1 + q 2 V 2 + q 3 V 3 = q2 q 1 k + q 3 + d 12 d 13 q1 q 2 k + q 3 + d 12 d 23 q1 q 3 k + q 3 d 13 d 23 q1 q 2 =2k + q 1q 3 + q 2q 3 d 12 d 13 d 23???2??? I counted each interaction twice!
25 Total energy: U tot = q 1 V 1 + q 2 V 2 + q 3 V 3 (?) U tot = q 1 V 1 + q 2 V 2 + q 3 V 3 = q2 q 1 k + q 3 + d 12 d 13 q1 q 2 k + q 3 + d 12 d 23 q1 q 3 k + q 3 d 13 d 23 q1 q 2 =2k + q 1q 3 + q 2q 3 d 12 d 13 d 23???2??? I counted each interaction twice! U tot = q 1V 1 + q 2 V 2 + q 3 V 3 2 U tot = 1 X q i V i 2 N
26 Capacitors Capacitors are devices that store energy in an electric field. Capacitors are used in many everyday applications Heart defibrillators Camera flash units Capacitors are an essential part of electronics. Capacitors can be microsized on computer chips or supersized for high power circuits such as FM radio transmitters. 1
27 Capacitance Capacitors come in a variety of sizes and shapes. The world's first integrated circuit included one capacitor Capacitor controlling power which enters Fermilab, a particle accelerator laboratory in Batavia, IL 2
28 Capacitance Concept: A capacitor consists of two separated conductors, usually called plates, even if these conductors are not simple planes. We will define a simple geometry and generalize from there. We will start with a capacitor consisting of two parallel conducting plates, each with area A separated by a distance d. We assume that these plates are in vacuum (air is very close to a vacuum). 3
29 Parallel Plate Capacitor (1) We charge the capacitor by placing a charge +q on the top plate a charge q on the bottom plate +  Because the plates are conductors, the charge will distribute itself evenly over the surface of the conducting plates. The electric potential, V, is proportional to the amount of charge on the plates. V = de E = 0 = q 0A V = dq 0A 4
30 Parallel Plate Capacitor (2) The proportionality constant between the charge q and the electric potential difference V is the capacitance C. We will call the electric potential difference V the potential or the voltage across the plates. The capacitance of a device depends on the area of the plates and the distance between the plates, but does not depend on the voltage across the plates or the charge on the plates. The capacitance of a device tells us how much charge is required to produce a given voltage across the plates. 5
31 Definition of Capacitance The definition of capacitance is The units of capacitance are coulombs per volt. The unit of capacitance has been given the name Farad (abbreviated F) named after British physicist Michael Faraday ( ) A farad is a very large capacitance Typically we deal with µf (106 F), nf (109 F), or pf (1012 F) Pixtal/age Fotostock RF 6
32 Charging/Discharging a Capacitor (1) We can charge a capacitor by connecting the capacitor to a battery or to a DC power supply. A battery or DC power supply is designed to supply charge at a given voltage. When we connect a capacitor to a battery, charge flows from the battery until the capacitor is fully charged. If we then disconnect the battery or power supply, the capacitor will retain its charge and voltage. A reallife capacitor will leak charge slowly, but here we will assume ideal capacitors that hold their charge and voltage indefinitely. 7
33 Capacitors can be dangerous! 1 ev ~ 10000K 100 ev ~ K! 28
34 Parallel Plate Capacitor (1) Consider two parallel conducting plates separated by a distance d This arrangement is called a parallel plate capacitor. The upper plate has +q and the lower plate has q. The electric field between the plates points from the positively charged plate to the negatively charged plate. We will assume ideal parallel plate capacitors in which the electric field is constant between the plates and zero elsewhere. Reallife capacitors have fringe field near the edges. 9
35 Parallel Plate Capacitor (2) We can calculate the electric field between the plates using Gauss Law 10
36 Parallel Plate Capacitor (3) Now we calculate the electric potential across the plates of the capacitor in terms of the electric field. We define the electric potential across the capacitor to be V. We carry out the integral in the direction of the blue arrow. 11
37 Parallel Plate Capacitor (4) Remember the definition of capacitance so the capacitance of a parallel plate capacitor is Variables: A is the area of each plate d is the distance between the plates Note that the capacitance depends only on the geometrical factors and not on the amount of charge or the voltage across the capacitor. 12
38 Example: Capacitance of a Parallel Plate Capacitor We have a parallel plate capacitor constructed of two parallel plates, each with area 625 cm 2 separated by a distance of 1.00 mm. Question: What is the capacitance of this parallel plate capacitor? Answer: A parallel plate capacitor constructed out of square conducting plates 25 cm x 25 cm separated by 1 mm has a capacitance of about 0.5 nf. 13
39 iclicker Two large plates of area A are charged with +q and q, located distance d apart. How will Capacitance change if they are moved two times further A A +q q d A A +q q 2d A : C will remain the same B : C will increase, C =2C 0 C : C will decrease, C = C 0 /2
40 iclicker Two large plates of area A are charged with +q and q, located distance d apart. How will Capacitance change if they are moved two times further A A A A +q q +q q d 2d Efield inside is constant. A : C will remain the same B : C will increase, C =2C 0 C : C will decrease, C = C 0 /2
41 iclicker Two large plates of area A are charged with +q and q, located distance d apart. How will Capacitance change if they are moved two times further A A +q q d A A +q q 2d A : C will remain the same B : C will increase, C =2C 0 C : C will decrease, C = C 0 /2 q = CV = constant, V =2V 0,C= C 0 /2
42 iclicker Two large plates of area A are charged with +q and q, located distance d apart. How will Capacitance change if they are moved two times further A A A A +q q +q q d 2d Efield inside is constant. A : C will remain the same B : C will increase, C =2C 0 C : C will decrease, C = C 0 /2 q = CV = constant, V =2V 0,C= C 0 /2
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