# Electric Field of a uniformly Charged Thin Spherical Shell

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1 Electric Field of a uniformly Charged Thin Spherical Shell The calculation of the field outside the shell is identical to that of a point charge. The electric field inside the shell is zero. What are the differences if the shell is conducting nonconducting?

2 Electric Field of a uniformly charged large nonconducting Plane Sheet of Charge Use a cylindrical Gaussian surface The flux through the ends is EA, there is no field through the curved part of the surface. The total charge is Q = σa Note, the field is uniform. What if this surface is a conducting plane or, both sides of the nonconducting plane are uniformly charged? Section 15.9

3 Electric Field of a Nonconducting Plane Sheet of Charge, Cont. The field must be perpendicular to the sheet. The field is directed either toward or away from the sheet. Section 15.9

4 Electric Potential Energy Conservative Force Conservative force or field is a force field in which the total mechanical energy of an isolated system is conserved. Examples, Gravitation, Electrostatic, Magnetic force fields Fields or force which cause the dissipation of mechanical energy in other form heat or sound are called non-conservative. Examples: Friction, Air resistance Section 16.1

5 Work and Potential Energy There is a uniform field between the two plates. As the charge moves from A to B, work is done on it. W AB = F x Δx =q E x (x f x i ) ΔPE = - W AB = - q E x x Only for a uniform field for a particle that undergoes a displacement along a given axis SI unit of energy: J Section 16.1

6 Potential Difference The electric potential difference ΔV between points A and B is defined as the change in the potential energy (final value minus initial value) of a charge q moved from A to B divided by the quantity of the charge. ΔV = V B V A = ΔPE / q Potential difference is not the same as potential energy. Section 16.1

7 Potential Difference, Cont. ΔPE = q ΔV Both electric potential energy and potential difference are scalar quantities. Units of potential difference is J/C or Volts Section 16.1

8 Electric Potential and Charge Movements When released from rest, positive charges accelerate spontaneously from regions of high potential to low potential. When released from rest, negative charges will accelerated from regions of low potential toward region of high potential. Work must be done on the charges to make them go in the direction opposite to their spontaneous direction Section 16.1

9 Electric Potential of a Point Charge The potential created by a point charge q at any distance r from the charge is The point of zero electric potential is taken to be at an infinite distance from the charge Section 16.2

10 Electric Field and Electric Potential Depend on Distance The electric field is proportional to 1/r 2 The electric potential is proportional to 1/r Section 16.2

11 Electric Potential of Multiple Point Charges Because the electric potential or the potential energy are scalars, these quantities by a distribution of charge is given by their algebraic sum. Section 16.2

12 Electrical Potential Energy of Two V 1 is the electric potential due to q 1 at some point P The work required to bring q 2 from infinity to P without acceleration is q 2 V 1 This work is equal to the potential energy of the two particle system Charges Section 16.2

13 Positive and Negative Potential Energy If the charges have the same sign, PE is positive. Positive work must be done to force the two charges near one another. The like charges would repel. If the charges have opposite signs, PE is negative. The force would be attractive. Work must be done to hold back the unlike charges from accelerating as they are brought close together. Section 16.2

14 Problem Solving with Electric Potential or Potential Energy (Point Charges) Draw a diagram of all charges. Note the point of interest. Calculate the distance from each charge to the point of interest. Use the basic equation V = k e q/r Use PE = k e q 1 q 2 /r Include appropriate sign Section 16.2

15 Problem Example Find the potential energy of two charges 5μC and 15 μc when the distance between them is 20 cm.

16 Multiple Charges Find the electric potential at point p in the following distribution. 5 μc 4 cm 15 μc p 3 cm 10 points extra credit for solving the Electric field at P. Due next week -30 μc 25 μc

17 Problem Class Work: Graded How much work should be done on a charge (15μC) to move from point A to point B, if the potential at A is 30 V and at B is 60 V.

18 Potential and Fields on Conductors Since W = -q(v B V A ), no net work is required to move a charge between two points that are at the same electric potential. W = 0 when V A = V B All points on the surface of a charged conductor in electrostatic equilibrium are at the same potential. Therefore, the electric potential is constant everywhere on the surface of a charged conductor in electrostatic equilibrium. Section 16.3

19 Conductors in Equilibrium Review The conductor has an excess of positive charge. All of the charge resides at the surface. E = 0 inside the conductor. The electric field just outside the conductor is perpendicular to the surface. The potential is a constant everywhere on the surface of the conductor. The potential everywhere inside the conductor is constant and equal to its value at the surface. Section 16.3

20 The Electron Volt The electron volt (ev) is defined as the kinetic energy that an electron gains when accelerated through a potential difference of 1 V. Electrons in normal atoms have energies of 10 s of ev. Excited electrons have energies of 1000 s of ev. High energy gamma rays have energies of millions of ev. 1 ev = 1.6 x J Section 16.3

21 Capacitor and Capacitance A capacitor is a device used in a variety of electric circuits to store charges. It has two plates usually connected to positive or negative of a battery. The capacitance, C, of a capacitor is defined as the ratio of the magnitude of the charge on either conductor (plate) to the magnitude of the potential difference between the conductors (plates). Section 16.6

22 Capacitance, Cont. Units: Farad (F) 1 F = 1 C / V A Farad is very large Often will see µf or pf V is the potential difference across a circuit element or device. V represents the actual potential due to a given charge at a given location. What we measure in the lab is V Section 16.6

23 Parallel-Plate Capacitor, Example The capacitor consists of two parallel plates. Each has area A. They are separated by a distance d. The plates carry equal and opposite charges. When connected to the battery, charge is pulled off one plate and transferred to the other plate. The transfer stops when V cap = V battery Section 16.7

24 Parallel-Plate Capacitor The capacitance of a device depends on the geometric arrangement of the conductors. For a parallel-plate capacitor whose plates are separated by air: In a medium is replaced by ϵ Section 16.7

25 Electric Field in a Parallel-Plate Capacitor The electric field between the plates is uniform. Near the center Nonuniform near the edges The field may be taken as constant throughout the region between the plates. Section 16.7

26 Capacitors connected in Parallel When connected in parallel, both have the same potential difference, V, across them. Section 16.8

27 Capacitors in Parallel When capacitors are first connected in the circuit, electrons are transferred from the left plates through the battery to the right plate, leaving the left plate positively charged and the right plate negatively charged. The flow of charges ceases when the voltage across the capacitors equals that of the battery. The capacitors reach their maximum charge when the flow of charge ceases. Section 16.8

28 Capacitors in Parallel The potential difference across the capacitors is the same. And each is equal to the voltage of the battery The total charge, Q, is equal to the sum of the charges on the capacitors. Q = Q 1 + Q 2 C = Q/ ΔV = (Q 1 + Q 2 )/ ΔV C = C 1 + C 2 Equivalent Circuit Section 16.8

29 Capacitors in Parallel, Final C eq = C 1 + C 2 + The equivalent capacitance of a parallel combination of capacitors is greater than any of the individual capacitors. Section 16.8

30 Capacitors in Series When in series, the capacitors are connected end-to-end. The magnitude of the charge must be the same on all the plates. Section 16.8

31 Capacitors in Series When a battery is connected to the circuit, electrons are transferred from the left plate of C 1 to the right plate of C 2 through the battery. As this negative charge accumulates on the right plate of C 2, an equivalent amount of negative charge is removed from the left plate of C 2, leaving it with an excess positive charge. All of the right plates gain charges of Q and all the left plates have charges of +Q. Section 16.8

32 More About Capacitors in Series An equivalent capacitor can be found that performs the same function as the series combination. The potential differences add up to the battery voltage. Section 16.8

33 Capacitors in Series, Final The equivalent capacitance of a series combination is always less than any individual capacitor in the combination. Section 16.8

34 Problem-Solving Strategy Be careful with the choice of units. Combine capacitors following the formulas. When two or more unequal capacitors are connected in series, they carry the same charge, but the potential differences across them are not the same. The capacitances add as reciprocals and the equivalent capacitance is always less than the smallest individual capacitor. When two or more capacitors are connected in parallel, the potential differences across them are the same. The charge on each capacitor is proportional to its capacitance. The capacitors add directly to give the equivalent capacitance. Section 16.8

35 Problem-Solving Strategy, Final Redraw the circuit after every combination. Repeat the process until there is only one single equivalent capacitor. A complicated circuit can often be reduced to one equivalent capacitor. Replace capacitors in series or parallel with their equivalent. Redraw the circuit and continue. To find the charge on, or the potential difference across, one of the capacitors, start with your final equivalent capacitor and work back through the circuit reductions. Section 16.8

36 Problem-Solving Strategy, Equation Summary Use the following equations when working through the circuit diagrams: Capacitance equation: C = Q / V Capacitors in parallel: C eq = C 1 + C 2 + Capacitors in parallel all have the same voltage differences as does the equivalent capacitance. Capacitors in series: 1/C eq = 1/C 1 + 1/C 2 + Capacitors in series all have the same charge, Q, as does their equivalent capacitance. Section 16.8

37 Circuit Reduction Example Section 16.8

38 Energy Stored in a Capacitor Energy stored = ½ Q ΔV From the definition of capacitance, this can be rewritten in different forms. Section 16.9

39 Capacitors with Dielectrics A dielectric is an insulating material that, when placed between the plates of a capacitor, increases the capacitance. Dielectrics include rubber, plastic, or waxed paper. C = κc o = κε o (A/d) The capacitance is multiplied by the factor κ when the dielectric completely fills the region between the plates. κ is called the dielectric constant. Section 16.10

40 Capacitors with Dielectrics Section 16.10

41 Dielectric Strength For any given plate separation, there is a maximum electric field that can be produced in the dielectric before it breaks down and begins to conduct. This maximum electric field is called the dielectric strength. Section 16.10

42 Some Commercial Capacitor Designs Section 16.10

43 Atomic Polarization in Dielectric The presence of the positive charge on the dielectric effectively reduces some of the negative charge on the metal This allows more negative charge on the plates for a given applied voltage The capacitance increases Section 16.10

44 Problem Example Find equivalent capacitance from the above circuit

45 Classwork: Graded Find equivalent capacitance from the above circuit

46 Battery Extra Credit: 10 points Due next week Use C = 15 F And C 1 = 25 F

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