1. Summary Vector Scalar 2-particle Force: (N) Potential Energy: (J)
1-particle
Electric Field: (N/C)
Potential: (V)

2. Review Current: (A) Batteries Drift velocity and Ohm’s Law:
Amount of charge per unit time past a point
Driven by difference in potential
Direction follows motion of positive charges
Batteries
Drift velocity and Ohm’s Law:

3. Review Resistance: (Ω) Measures how well charge flows
A geometric property of the resistor
Resistance (R) vs. resistivity (ρ)

4. Capacitors
A capacitor uses potential difference to store charge and energy
Example:
parallel-plate capacitor
clouds
Section 18.4

5. Capacitance Defined
Capacitance is the amount of charge that can be stored per unit potential difference
The capacitance is dependent on the geometry of the capacitor
For a parallel-plate capacitor with plate area A and separation distance d, the capacitance is
Unit is farad, F
1 F = 1 C/V
Section 18.4

6. Energy in a Capacitor, cont.
Storing charge on a capacitor requires energy
The total energy stored is equal to the energy required to move all the packets of charge from one plate to the other
Section 18.4

7. Energy in a Capacitor, cont.
The total energy corresponds to the area under the ΔV – Q graph
Energy = Area = PEcap
Q is the final charge
ΔV is the final potential difference
Section 18.4

8. Energy in a Capacitor, Final
From the definition of capacitance, the energy can be expressed in different forms
These expressions are valid for all capacitors
Section 18.4

9. Dielectrics
Most real capacitors contain two metal “plates” separated by a thin insulating region
Many times these plates are rolled into cylinders
The region between the plates typically contains a material called a dielectric
Section 18.5

10. Dielectrics, cont.
Inserting the dielectric material between the plates changes the value of the capacitance
The change is proportional to the dielectric constant, κ
Cvac is the capacitance without the dielectric and Cd is with the dielectric
κ is a dimensionless factor
Generally, κ > 1, so inserting a dielectric increases the capacitance
Section 18.5

11. Dielectrics, final
When the plates of a capacitor are charged, the electric field established extends into the dielectric material
Most good dielectrics are highly ionic and lead to a slight change in the charge in the dielectric
Since the field decreases, the potential difference decreases and the capacitance increases
Section 18.5

12. Chapter 19b Electric Circuits Circuit Diagrams?
Image taken from

13. Parallel and Series
Series—the same current passes through each component.
Parallel—the same potential change across each component
Components can be neither in series nor in parallel
Cannot be reduced to equivalence

14. Equivalence
Multiple circuit elements can be replaced by a single equivalent circuit element without affecting the rest of the circuit
Equivalence is different for resistors and capacitors
Equivalence is different for series and parallel
Reducing a circuit to equivalent components often simplifies circuit analysis
Always reduce parallel before series

15. Capacitors in Series For capacitors connected in series:
ΔVtotal = ΔV1 + Δv2 + …
The equivalent capacitance is
The equivalent capacitance is smaller in series than any of the individual capacitors
Section 19.5

16. Capacitors in Parallel
For capacitors connected in parallel
Qtotal = Q1 + Q2
The equivalent capacitance is
The equivalent capacitance is larger in parallel than any of the individual capacitors
Section 19.5

17. Resistors in Series For resistors in series
ΔVtotal = ΔV1 + Δv2 + …
The equivalent resistance is
The equivalent resistance is larger in series than any of the individual resistor
Section 19.4

18. Resistors in Parallel For resistors in series
Itotal = I1 + I2 + …
The equivalent resistance is
The equivalent resistance is smaller in parallel than any of the individual resistor
Section 19.4

19. Batteries in Series Batteries can also be connected in series
The combination of two batteries in series is equivalent to a single battery with emf of
We will not deal with batteries in parallel
Section 19.4

20. Circuits
An electric circuit is a combination of connected elements forming a complete path through which charge is able to move
Circuits are represented by circuit diagrams
a good approximation is Rwire=0
Since the resistance of the wires is much smaller than that of the resistors
Section 19.3

21. Circuit Symbols
Section 19.3

22. Circuits, cont.
Calculating the current in the circuit is called circuit analysis
Two types of circuits:
DC stands for direct current
The current is of constant magnitude and direction at the source
AC stands for alternating current (Ch. 22)
Source current changes magnitude and/or direction
The current can be viewed as the motion of the positive charges traveling through the circuit
Section 19.4

23. Circuits, cont. There must be a complete circuit for charge flow
There must be a return path from the resistor for the current to return to the voltage source
If the circuit is open, there is no current flow anywhere in the circuit
Section 19.4

24. Circuit Analysis
The main tools of circuit analysis are equivalence, Ohm’s Law, and Kirchhoff’s Rules
Loop Rule: the change in potential energy of a charge as it travels around a complete circuit loop must be zero
Junction Rule: the amount of current entering a junction much be equal to the current leaving it

25. Kirchhoff’s Loop Rule
Conservation of energy is the heart of the Loop Rule
Consider the electric potential energy of a test charge moving through the circuit
The test charge gained energy when it passed through the battery
It lost energy as it passed through the resistor
The energy is converted into heat energy inside the resistor
Section 19.4

26. Kirchhoff’s Loop Rule, cont.
Since PEelec = q V, the loop rule also means the change in the electric potential around a closed circuit path is zero
ΔV = ε – I R = 0
For the entire loop
Assumes wires have no resistance
Section 19.4

27. Power In the resistor, the energy decreases by
Power is energy per unit time
Applying Ohm’s Law, P = I² R = V² / R
The circuit converts chemical energy from the battery to heat energy in the resistors
Power is only dissipated through resistance
Section 19.4

28. Kirchhoff’s Junction Rule
Conservation of charge is the heart of the Junction Rule
The points where the currents branch are called junctions or nodes
Since charge cannot be created or destroyed, all charge entering a junction must leave the junction
Section 19.4

29. Kirchhoff’s Junction Rule, cont.
Since all charge entering a junction must leave the junction, the current entering must equal the current exiting the junction
IT = I1 + I2
Both of Kirchhoff’s Rules may be needed to solve a circuit
Section 19.4

30. Direction of Current Current directions are chosen arbitrarily
The signs of the potentials in Kirchhoff’s loop rule will depend on assumed current direction
Moving parallel to the current produces a potential drop across a resistor
The sign of the currents in Kirchhoff’s junction rule will depend on assumed current direction
Currents entering a junction are positive, currents leaving are negative
Section 19.4

31. Direction of Current, cont.
After solving the equations, the sign of the current indicates the direction
if the current is positive the assumed direction of the current is correct
If the current is negative, the direction of the current Is opposite the assumed direction
Section 19.4

32. Using Kirchhoff’s Rules
In general, if a circuit has N junctions, the junction rule can be used N – 1 times
The loop rule can be used as long as it contains at least one circuit element that is not involved in other loops
When using the loop rule, pay close attention to the sign of the voltage drop across the circuit element
You can go around the loop in any direction
Section 19.4

33. Using Kirchhoff’s Rules, cont.
Reduce all equivalences first
Apply the junction rule as many times as gives independent equations
Use the loop rule to get as many equations as unknowns in the system
Section 19.4

34. Example:

35. Real Batteries
An ideal battery always maintains a constant voltage across its terminals
The value of the voltage is the emf of the battery
A real battery is equivalent to an ideal battery in series with a resistor, Rbattery
This is the internal resistance of the battery
Section 19.4

36. Real Batteries
The current through the internal resistance and the external resistor is
The potential difference across the real battery’s terminals is
Section 19.4

37. Circuits with Capacitors
Kirchhoff’s Rules can be applied to all kinds of circuits
RL circuits
The change in the potential around the circuit is
+ε – I R – q / C = 0
Solving for I shows that I and q will be time dependent
Section 19.5

38. Charging Capacitors
When the circuit is open, there is no current in the circuit and no charge on the capacitor
When the switch is closed, current carries a positive charge to the top plate of the capacitor
When the capacitor plates are charged, there is a nonzero voltage across the capacitor
Current can no longer flow
Section 19.5

39. Charging Capacitors, cont.
The current in the circuit is described by
The voltage across the capacitor is
The charge is given by
τ = RC and is called the time constant
Section 19.5

40. Time Constant Current Voltage and charge
At the end of one time constant, the current has decreased to 37% of its original value
At the end of two time constants, the current has decreased to 14% of its original value
Voltage and charge
At the end of one time constant, the voltage and charge have increased to 63% of their asymptotic values
Section 19.5

41. Charging Capacitors, cont.
Just after the switch is closed
The charge is very small
Vcap is very small
I = ε / R
Section 19.5

42. Charging Capacitors, cont.
When t is large
The charge is very large
Vcap ≈ ε
The polarity of the capacitor opposes the battery emf
The current approaches zero
Section 19.5

43. Discharging the Capacitor
Current:
Voltage: Vcap = ε e-t/τ
Charge: q = C ε e-t/τ
Time constant: τ = RC, the same as for charging
Section 19.5

44. Uses of Capacitance

45. Ammeters An ammeter is a device that measures current
An ammeter must be connected in series with the desired circuit branch
An ideal ammeter will measure current without changing its value
Must have a very low resistance
Section 19.6

46. Voltmeters A voltmeter measures the voltage across a circuit element
It must be connected in parallel with the element
An ideal voltmeter should measure the voltage without changing its value
The voltmeter should have a very high resistance
Section 19.6

47. Temperature Dependence of Resistance
As temperature increases, the ions in a metal vibrate with larger amplitudes
This causes more frequent collisions and an increase in resistance
For many metals near room temperature,
ρ = ρo [1 + α(T – To)
α is called the temperature coefficient of the resistivity
The resistivity and resistance vary linearly with temperature
Section 19.10

48. Superconductivity
At very low temperatures, the linearity of resistance breaks down
The resistivity of metals approach a nonzero value at very low temperatures
In some metals, resistivity drops abruptly and is zero below a critical temperature
Metals for which the resistivity goes to zero are called superconductors
Section 19.10