1. PHYS 1902 Electromagnetism: 2 Lecturer: Prof. Geraint F. Lewis

2. Capacitance Opposite charges, so attractive electric force
(Something must be holding them apart in a static configuration)
Potential energy is stored in this configuration

3. Capacitance Double the charge (difference)
means electric field doubles in strength
means voltage doubles

4. Capacitance

5. Capacitance Capacitance:
capacity of two conductors for holding electric potential energy
Capacitance:
Depends only on the shapes and sizes of conductors (i.e., the configuration)
and on the insulating material between them
NOT on the amount of charge or voltage

6. Capacitance of Two Parallel Plates
Plates are close enough that we can use an approx: uniform E field
What is the capacitance?

7. Spherical Capacitor – 24.3 Concentric spherical conducting shells:
inner shell: radius ra
outer shell: radius rb
Both shells are charged, the inner shell with +Q, and the outer with –Q.
What’s the capacitance?

8. Potential Energy Stored in a Capacitor
Potential energy in a capacitor C with voltage V and charge Q=CV
Work needed to “charge” the capacitor, starting from Q=0
Consider point where amount of charge q has been moved
Move small charge dq

9. Potential Energy Stored in a Capacitor
Puzzle: Alice and Bob argue about the relative “capacity” of two capacitors C and 2C to hold potential energy
Alice argues that the capacitor 2C only holds half the potential energy as C, because U=Q2/2C
Bob argues the opposite, because U=CV2/2
Which one is correct?

10. Potential Energy Stored in a Capacitor

11. Potential Energy Stored in a Capacitor

12. Electric Field Energy Capacitors store potential energy
What is the energy density between the plates?

13. Capacitors in parallel
equivalent to
Capacitors in parallel:

14. Capacitors in series
equivalent to
Capacitors in serial:

15. Capacitor problem
What is the equivalent capacitance of this combination?

16. Chapter 25
Current, Resistance, and Electromotive Force

17. Current So far, we’ve studied only static situations
Now we start considering charges in motion
Drift velocity

18. Current

19. Current During time dt, charge passing through A: dQ
Unit of current: the Ampere A = C/s

20. Current Think of current like a vector.
It ”points” in the direction of the flow of positive charge (opposite to the motion of electrons!)

21. Current density How does current relate to drift velocity?
n: concentration of particles, in m-3
Current density

22. Resistivity Ohm’s Law:
In most metals, there is a simple relationship between the electric field and the current density
Ohm’s Law:
Resistivity is independent of the electric field strength
Conductivity: inverse of resistivity

23. Resistance For an Ohmic material:
Usually more interested in voltage (rather than E) and in current (rather than J)
Resistance:
Units of Ohms

24. Closed Circuits
To maintain a steady (constant) current, we need to create a closed path – a circuit

25. Electromotive Force Electromotive force (emf) is the pump
Some other force

26. Electromotive Force Electromotive force (emf) is the pump
Change in electric potential (voltage) around a loop equals zero
(Think of the fountain)
Ideal source of emf:

27. Internal Resistance Electromotive force (emf) in an idealisation
Real devices have internal resistance
Terminal voltage, source with internal resistance

28. Internal Resistance Circuit
(current, source with internal resistance)

29. Symbols for Circuits Conductor with negligible resistance Resistor
Source of emf (longer line always represents the positive terminal)
Source of emf with internal resistance r (can be on either side)
Voltmeter (measures potential difference)
Ammeter (measures current through it)

30. Problem – 25.5
What is the reading on:
the ammeter?
the voltmeter?

31. Problem – 25.5

32. Problem
Real emf
Internal resistance
Eight flashlight batteries in series have an emf of about 12 V, similar to that of a car battery.
Could they be used to start a car with a dead battery? Why or why not?

33. Potential in a Circuit

34. Work in a Circuit
Potential energy is lost when going through a resistor
emf does work on charge, raising its potential energy
Work could be done by this current in principle

35. Power in a Circuit In time dt:
Potential energy decrease in time dt from dQ “falling” through Vab = Va – Vb
Power (rate at which energy is delivered or extracted from a circuit element):

36. Power for a Resistor Combine Ohm’s Law with our new power equation
(power delivered to a resistor)
In the case of a resistor, energy is lost as heat
Other circuit components (i.e., LEDs) use this energy in other ways

37. Power Lines
Power Station
Sydney
(power delivered to a resistor)

38. Problems
Small aircraft have 24 V electrical systems, even though their requirements are about the same as cars (with 12 V systems).
Aircraft designers say that a 24 V system weighs less, because thinner wires can be used. Why?

39. Problem – 25.9 What is rate of:
energy conversion (chemical to electrical) by the battery?
power dissipation by the battery?
power dissipation by the resistor?

40. Short circuit – 25.11
What are the meter readings on the voltmeter and ammeter?
What is the power dissipation by the battery?

41. (Sub-)Chapter 26.4
R-C Circuits

42. A Capacitor in a Circuit
Switch (open)
Ideal emf
Closing the switch initiates a non-static (time-dependent) process
What happens?
Resistor
Capacitor

43. A Capacitor in a Circuit
Switch (closed)
Ideal emf
Initially, a current will flow and charge will build up on the capacitor...
... but that capacitor builds up a voltage drop given by
Resistor
Capacitor

44. A Capacitor in a Circuit
Switch (closed)
Initially, a current will flow and charge will build up on the capacitor...
... but that capacitor builds up a voltage drop that (after a very long time) is
and the current stops
Ideal emf
Capacitor
Resistor

45. Charging a Capacitor
Use Kirchhoff’s loop rule to solve for q and i as functions of time:
Ideal emf
Switch (closed)
At t=0 (switch closed):
Resistor
Capacitor

46. Charging a Capacitor
Use Kirchhoff’s loop rule to solve for q and i as functions of time:
Ideal emf
Switch (closed)
As t -> ∞
Resistor
Capacitor

47. Charging a Capacitor
Use Kirchhoff’s loop rule to solve for q and i as functions of time:
Ideal emf
Switch (closed)
Resistor
Capacitor
Check: ?

48. Charging a Capacitor
Ideal emf
Switch (closed)
Resistor
Capacitor

49. Charging a Capacitor
Ideal emf
Switch (closed)
Resistor
Capacitor

50. Time constant τ = RC: quantifies rate of charging

51. Discharging a Capacitor
Switch (open)
What happens when the switch is closed?
Resistor
Capacitor

52. Discharging a Capacitor
Switch (closed)
Capacitor discharges and the current eventually goes to zero
Resistor
Capacitor ?

53. Discharging a Capacitor
Switch (closed)
Resistor
Capacitor

54. Discharging a Capacitor
Switch (closed)
Resistor
Capacitor