# physics 111N electric potential and capacitance

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1 physics 111N electric potential and capacitance

2 electric potential energy consider a uniform electric field (e.g. from parallel plates) note the analogy to gravitational force near the surface of the Earth physics 111N 2

3 potential between parallel plates potential energy define a quantity that depends only upon the field and not the value of the test charge the potential measured in Volts, V = J/C actually only differences of potential are meaningful, we can add a constant to V if we like physics 111N 3

4 potential between parallel plates 9.00 V 6.75 V 4.50 V 2.25 V 0 V equipotential lines lines of equal value of potential arbitrarily choose V=0 at the right-hand plate 9 V 0 V physics 111N 4

5 a capacitor 9.00 V 6.75 V 4.50 V 2.25 V 0 V physics 111N 5

6 conservation of energy using potential A 9 V battery is connected across two large parallel plates that are separated by 9.0 mm of air, creating a potential difference of 9.0 V. An electron is released from rest at the negative plate - how fast is it moving just before it hits the positive plate? physics 111N 6

7 potential energy between point charges Q Q physics 111N 7

8 potential from a point charge depend only on distance from the charge Q arbitrarily choose V=0 infinitely far from the charge physics 111N 8

9 a set of point charges q1 q3 P q2 at the point P there is an electric field E and an electric potential V physics 111N 9

10 electric field from a set of point charges q1 q3 q2 physics 111N 10

11 electric field from a set of point charges q1 q3 q2 physics 111N 11

12 electric field from a set of point charges q1 q3 q2 have to do vector addition - difficult physics 111N 12

13 electric potential from a set of point charges q1 q3 q2 just scalar addition - easy physics 111N 13

14 for example find the electric potential 10.0 pc 4.00 mm 10.0 mm -8.0 pc 5.00 mm 20.0 pc what potential energy would a charge of 2.0 nc have at this position? physics 111N 14

15 equipotential diagrams equipotentials are defined as the surfaces on which the potential takes a constant value - hence different equipotentials never intersect usually draw them with equal potential separations physics 111N 19

16 equipotentials for a point charge 110 V 90 V 70 V 50 V 30 V physics 111N 20

17 equipotentials from a dipole 0 V -20 V 20 V physics 111N 21

18 equipotentials from a dipole -20 V 20 V notice that the field lines are always perpendicular to the equipotentials physics 111N 22

19 equipotentials from two equal point charges notice that the field lines are always perpendicular to the equipotentials physics 111N 23

20 a capacitor 9.00 V 6.75 V 4.50 V 2.25 V 0 V notice that the field lines are always perpendicular to the equipotentials physics 111N 24

21 equipotentials and field lines we can use some logical deduction to see that electric fields must be perpendicular to equipotentials we can move a test charge along an equipotential without changing potential hence the potential energy does not change thus no work is done if the E-field had a component parallel to the equipotential we would do work hence there can be no component of E parallel to an equipotential physics 111N 25

22 electric field as the gradient of the potential Δs consider two adjacent equipotential surfaces separated by a small distance, Δs potential difference between the surfaces is ΔV VΔV V for a small distance, the E-field is approximately constant, so the work done per unit charge in moving from one surface to the other is E Δs this equals the change in potential, -ΔV Δs hence we can express the E-field as a potential gradient VΔV V electric fields point downhill physics 111N 27

23 electric field as the gradient of the potential physics 111N 28

24 topological maps physics 111N 30

25 electric fields at the surface of conductors electric fields meet the surface of conductors at right angles the electric field in a conductor is zero means the potential can t have a gradient potential in a conductor is constant physics 111N 34

26 capacitors & capacitance consider two conductors, separated in space, carrying equal and opposite charge this is a capacitor electric fields will fill the space between the conductors a potential difference will be set up between the conductors electrostatic energy is stored in the fields the potential difference between a and b is proportional to the charge Q the constant of proportionality that tells us how much charge do I need per unit potential is called the capacitance, C units are farads physics 111N 35

27 parallel plate capacitors two parallel metal plates, of area A, separated by a distance d we can show that the electric field between large plates is uniform and of magnitude physics 111N 36

28 what s this ε0 thing? it s a property of the vacuum of empty space that tell us how strong electric fields should be it was in Coulomb s law, but we hid it in the constant k physics 111N 37

29 parallel plate capacitors two parallel metal plates, of area A, separated by a distance d we can show that the electric field between large plates is uniform and of magnitude since it s uniform the potential difference must be hence so the capacitance is which depends only on the geometry of the capacitor physics 111N 38

30 parallel plate capacitors physics 111N 41

31 circuit diagrams and rules wires are treated as being resistance-less, they act as equipotentials 24 V 24 V potential changes occur when electrical components are attached to the wires e.g. a battery - keeps two wires at fixed potential difference 24 V 0 V e.g. a capacitor - potential drop 24 V 0 V so we can build a legitimate circuit out of these two components and wires 24 V 24 V 0 V 0 V physics 111N 42

32 circuit diagrams and rules net charge doesn t accumulate, a circuit starts with total charge of zero and always has total charge of zero Q 24 V 0 V Q -Q 24 V 0 V positive charge build up on one plate pushes positive charge off the other plate, leaving negative charge behind Q -Q Q 24 V 0 V positive charges pushed off end up on the next plate Q -Q Q -Q 24 V 0 V same `push-off occurs here physics 111N 43

33 circuit diagrams and rules net charge doesn t accumulate, a circuit starts with total charge of zero and always has total charge of zero Q -Q Q -Q 24 V 0 V but where do the pushed-off positive charges go? remember we have to make a circuit Q -Q Q -Q 24 V 0 V 24 V 0 V they moved all the way around the circuit and formed the first set of positive charges physics 111N 44

34 capacitors in series imagine removing the `inner plates and the wire joining them : V Q -Q Q -Q 0 V Q -Q 0 C1 C2 = C V 0 V 0 physics 111N 45

35 capacitors in series we can derive a formula for C in terms of C1 and C2 : Q -Q Q -Q V V - V1 V - V1 - V2 = 0 C1 C2 V1 V2 V 0 V Q -Q C 0 V 0 physics 111N 46

36 capacitors in series - using the formula many students use this formula wrongly it is NOT the same as C = C1 C2 e.g. C1 = 1 F, C2 = 1 F, then C1 C2 = 2 F but so and hence physics 111N 47

37 capacitors in parallel Q1 -Q1 V 0 V C1 0 Q2 -Q2 V 0 C2 total charge on the left-hand plates V 0 physics 111N 50

38 capacitors in parallel Q1 -Q1 V 0 suppose we joined the plates together V C1 0 V 0 Q2 -Q2 V 0 = Q1Q2 -(Q1Q2) C2 V 0 V 0 V Q -Q 0 = C V 0 physics 111N 51

39 capacitors in parallel Q1 -Q1 V 0 V C1 Q2 -Q2 0 total charge on the left-hand plates V 0 C2 V 0 V Q -Q 0 = C V 0 physics 111N 52

40 combining capacitors Two capacitors, one with capacitance 12.0 nf and the other of 6.0 nf are connected to a potential difference of 18 V. Find the equivalent capacitance and find the charge and potential differences for each capacitor when the two capacitors are connected in (a) series (b) parallel physics 111N 54

41 stored energy in a capacitor getting the charges in place on the plates requires work - this work ends up as energy stored in the electric fields consider charging up a capacitor from zero charge to a charge Q if at some time the charge is q, the potential is v = q/c to add another small amount of charge Δq, will need to do work of ΔW = v Δq the extra work needed is the area of the rectangle physics 111N 57

42 stored energy in a capacitor getting the charges in place on the plates requires work - this work ends up as energy stored in the electric fields consider charging up a capacitor from zero charge to a charge Q if at some time the charge is q, the potential is v = q/c to add another small amount of charge Δq, will need to do work of ΔW = v Δq the total work required is the area under the curve physics 111N 58

43 energy stored in electric fields capacitors store energy - this can be thought of as residing in the field between the plates define energy density as the energy per unit volume for a parallel plate capacitor this formula turns out to be true for all electric field configurations physics 111N 60

44 dielectrics so far we ve assumed that the gap between the plates is filled with vacuum (or air) it doesn t have to be - suppose we place some nonconducting material in there physics 111N 61

45 adding a dielectric physics 111N 62

46 dielectrics the voltage changes - reflects a change in the capacitance capacitance with dielectric capacitance without dielectric dielectric constant if the voltage goes down (for fixed charge) when a dielectric is added, what can we say about K? 1. K is negative 2. K is less than 1 3. K is greater than 1 4. K is zero physics 111N 63

47 dielectrics the voltage drop corresponds to a reduction of the electric field in the gap the reason is induced charges on the surface of the dielectric physics 111N 64

48 dielectrics the voltage drop corresponds to a reduction of the electric field in the gap the reason is induced charges on the surface of the dielectric physics 111N 65

49 dielectrics the induced charge is caused by the polarization of electric dipoles in the dielectric physics 111N 66

50 dielectrics the induced charge is caused by the polarization of electric dipoles in the dielectric cancellation of charges in the bulk surface charge remains physics 111N 67

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