* I1 = 2.5 A (given) * I2 = 4 A (given) * Using Kirchhoff's junction rule: I1 = I2 + I3 * So: 2.5 A = 4 A + I3 * Solving for I3: I3 = 2.5 A - 4 A = -1.5 A The value of I3 is -1.5 A. The answer is 3.

1. Electric charge
• Intrinsic property of the particles that make
up matter

2. Electric charge
• Charge can be positive or negative

3. Electric charge
• Atoms are composed of negatively-charged
electrons and positively-charged protons

4. Electric charge
• Charge is measured in Coulombs [unit: C]

5. Electric charge
• Charge is measured in Coulombs [unit: C]
• Proton and electron have equal and
opposite elementary charge = 1.6 x 10-19 C
• Charge on proton = +1.6 x 10-19 C
• Charge on electron = -1.6 x 10-19 C

6. Electric charge
• Charge cannot be created or destroyed (it is
conserved) but it can be moved around

7. Electric charge
• Charges feel electrostatic forces

8. Electric charge
• Rub a balloon on your hair and it will stick to things! Why??
• Friction moves electrons from your hair to the balloon
• The balloon therefore becomes negatively charged, so your
hair becomes positively charged (charge conservation)
• Your hair will stand on end (like charges repel), and the
balloon will stick to your hair (opposite charges attract)
• Now move the balloon near a wall. The wall’s electrons are
repelled, so the wall becomes positively charged.
• The balloon will stick to the wall! (opposite charges attract)

9. • The strength of the electrostatic force between
two charges q1 and q2 is given by Coulomb’s law
• The direction of the force is along the joining line
𝑘 = 9 × 109
𝑁 𝑚2
𝐶−2
𝐹𝑒 𝐹𝑒
Electrostatic force

10. • The electrostatic force is a vector, written Ԧ
𝐹
• Vectors have a magnitude and a direction. This
may be indicated by components Ԧ
𝐹 = (𝐹𝑥, 𝐹𝑦, 𝐹𝑧)
• The magnitude is sometimes written as Ԧ
𝐹 . It
can be evaluated as | Ԧ
𝐹| = 𝐹𝑥
2
+ 𝐹𝑦
2
+ 𝐹𝑧
2
• The direction can be indicated by a unit vector
Electrostatic force

11. Conductors and Insulators
• In metals (e.g. copper, iron) some electrons are weakly held
and can move freely through the metal, creating an electric
current. Metals are good conductors of electricity.

12. Freely moving electrons make metals good conductors of electricity and heat
Conductors and Insulators
• In metals (e.g. copper, iron) some electrons are weakly held
and can move freely through the metal, creating an electric
current. Metals are good conductors of electricity.
• In non-metals (e.g. glass, rubber, plastic) electrons are
strongly held and are not free to move. Non-metals are
poor conductors of electricity, or insulators.
• Semi-conductors (e.g. germanium, silicon) are half-way
between conductors and insulators.

13. Summary
• Matter is made up of positive and negative charges.
Electrons/protons carry the elementary charge 1.6 x 10-19 C
• Forces between charges are described by Coulomb’s Law
• Forces from multiple charges sum as vectors
• Electric field describes the force-field around charges
𝐹 =
𝑘 𝑞1 𝑞2
𝑟2
𝑘 = 9 × 109 𝑁 𝑚2 𝐶−2
𝐸 =
Ԧ
𝐹
𝑞
Ԧ
𝐹 = 𝑞 𝐸

15. Electric potential
• What is electric potential?
• How does it relate to
potential energy?
• How does it relate to
electric field?
• Some simple applications

16. Electric potential
• What does it mean
when it says “1.5 Volts”
on the battery?
• The electric potential
difference between the
ends is 1.5 Volts

17. Electric potential
1.5 V
230 V
100,000 V
So what is a volt?

18. Electric potential
The 1.5 V battery does
1.5 J of work for every
1 C of charge flowing
round the circuit
𝑊 = 𝑞 × ∆𝑉
• The electric potential difference ∆𝑉 in volts
between two points is the work in Joules needed to
move 1 C of charge between those points

19. Electric potential
• The electric potential difference ∆𝑉 in volts
between two points is the work in Joules needed to
move 1 C of charge between those points
• ∆𝑉 is measured in volts [V] : 1 V = 1 J/C
𝑊 = 𝑞 × ∆𝑉
W = work done [in J]
q = charge [in C]
∆V = potential difference [in V]

20. Potential energy
• Potential energy U is the energy stored in a system
– second example
• e.g. stretching a spring … 𝐹 = 𝑘𝑥
𝑥
𝑊 = න 𝑘𝑥 𝑑𝑥
Work = Force x Distance
= 1
2
𝑘𝑥2
→ 𝑈 = 1
2
𝑘𝑥2
𝐹
Force is varying with distance!

21. Electric potential
• e.g. moving a charge through an electric field…
𝐹
𝐸
𝐹 = −𝑞𝐸
Work = Force x Distance = −𝑞𝐸 ∆𝑥
𝑊 = 𝐹 ∆𝑥
• Potential difference ∆𝑉 is work needed to move
1C of charge: 𝑊 = 𝑞 ∆𝑉
• Equate: 𝑞 ∆𝑉 = −𝑞𝐸 ∆𝑥
𝑞
(minus sign because the
force is opposite to E)
∆𝑥
𝐸 = −
∆𝑉
∆𝑥

22. Electric potential
• Electric field is the gradient of potential 𝐸 = −
∆𝑉
∆𝑥
High V Low V
𝐸 𝑉
𝑥
𝑥
• Positive charges feel a force
from high to low potential
• Negative charges feel a force
from low to high potential

23. Two parallel plates have
equal and opposite charge.
Rank the indicated positions
from highest to lowest
electric potential.
1 2 3 4
0% 0%
0%
0%
+ + + + + + + + + + + + + + + +
- - - - - - - - - - - - -
•A
•B
•C
•D
1. A=C, B=D
2. A, B, C, D
3. C, D=B, A
4. A, B=D, C

24. Electric potential
• Analogy with gravitational potential
𝑉
𝑥
Gravitational
potential difference
exerts force on mass
Electric potential
difference exerts
force on charge
𝑞

25. Electric potential
High V Low V
𝐸
• The dashed lines are called
equipotentials (lines of
constant V)
• Electric field lines are
perpendicular to equipotentials
• It takes no work to move a
charge along an equipotential
(work done = 𝑑𝑊 = Ԧ
𝐹. 𝑑𝑥 =
𝑞𝐸. 𝑑𝑥 = 0)
• Electric field is the gradient of potential 𝐸 = −
∆𝑉
∆𝑥

26. • Summary for two plates at potential difference V
Electric potential
𝑑
𝐸 • Electric field is the potential
gradient
• Work W to move charge q
from –ve to +ve plate
𝐸 =
𝑉
𝑑
𝑊 = 𝑞 𝑉

27. • The electric potential difference ∆𝑉 between two
points is the work needed to move 1 C of charge
between those points
• This work is also equal to the potential energy
difference ∆𝑈 between those points
• Potential V = potential energy per unit charge U/q
Link to potential energy
𝑊 = 𝑞 × ∆𝑉
∆𝑈 = 𝑞 × ∆𝑉

28. Electric current
• How do we define current?
• Macroscopic and
microscopic description of
current
• Ohm’s law and resistance
• Electrical power

29. Electric current
• A flow of charge is called an electric current

30. Electric current
• A flow of charge is called an electric current
• The current (symbol I) is the amount of charge Q
[in Coulombs] flowing per unit time t [in seconds]
• The units of current are C/s or “Amperes” A
𝐼 =
𝑄
𝑡
1 𝐴 = 1 𝐶/𝑠

31. Electric current
• Current is in the direction that positive charge flows
• But in reality, current is transported by an opposite
flow of negatively-charged electrons
Sometimes described as
“conventional current”
(positive) or “electron
current” (negative)

32. Electric current
• How do we create an electric
current?
• Create an electric potential
difference between two points
• Connect those points to allow
charge to flow
• Dissipate the energy (e.g. into
light, heat)
Circuit symbols!

33. Only one terminal of the battery is
connected to the light bulb. What
happens?
1 2 3 4
0% 0%
0%
0%
1. No current flows
2. A very small current flows
3. A current flows for only a short
time
4. Current flows at half the rate it
flowed with two wires

34. Electric current
• Electrical power may be supplied as either a direct
current or an alternating current
• We will only cover direct current in this topic

35. Ohm’s Law : macroscopic version
• Ohm’s law describes the resistance of a material to
the flow of current (or its inverse – conductance)
• The greater the resistance, the less current can flow for a
given potential difference
• Resistance is measured in units of Ohms (symbol: Ω)
𝐼 =
𝑉
𝑅
𝑉 = 𝐼𝑅

36. Ohm’s Law : microscopic version
• The current density J flowing for a given electric
field E depends on the resistivity 𝜌 of the material
(or its inverse – conductivity 𝜎 = 1/𝜌)
𝐽 =
𝐸
𝜌
= 𝜎 𝐸
High resistivity 𝜌
means low current!

37. Electric current
• Current can be measured using an ammeter

38. How does the current entering
the resistor, I1, compare to the
current leaving the resistor, I2?
1 2 3
0%
0%
0%
I1
I2
CHARGE CONSERVATION: charge cannot be created or destroyed.
Energy is dissipated as current flows through a resistance, but
charge is conserved, so current in = current out.
1. I1 < I2
2. I1 > I2
3. I1 = I2

39. Electric power
• Power is the rate of use of energy : 𝑃𝑜𝑤𝑒𝑟 =
𝐸𝑛𝑒𝑟𝑔𝑦
𝑇𝑖𝑚𝑒
• How much power does an electric circuit consume?
• Moving charge Q across a potential difference V
requires work : 𝑊 = 𝑄 𝑉 (from last chapter)
• Power =
𝑊𝑜𝑟𝑘
𝑇𝑖𝑚𝑒
=
𝑄 𝑉
𝑡
= I V (in terms of current I = Q/t)
• Using Ohm’s law V = I R : Power = V I = I2R = V2/R

40. Electric power
• This power is dissipated as heat energy in the
resistance – why electrical components get hot!

41. Electric power
• Power is measured in Watts (1 W = 1 J/s)
• Your “power bill” is probably measured in “kWh” or
“kilo-Watt hours”
• This is really an “energy bill” …
• 1 kWh = 1000 J/s x 3600 s = 3.6 x 106 J = 3.6 MJ

42. Electric power
• Why do power lines operate at 100,000 V?
• P = V I : high power can be delivered using high V or high I
• Some power will be lost in heating the transmission wires
• P = I2R : low current minimizes these transmission losses

43. Electric power
Exercise: What is the resistance of a 60 W 240V light bulb?
Power P = 60 W
Voltage V = 240 V
𝑃 = 𝐼 𝑉 → 𝐼 =
𝑃
𝑉
=
60
240
= 0.25 𝐴
𝑉 = 𝐼 𝑅 → 𝑅 =
𝑉
𝐼
=
240
0.25
= 960 Ω
Exercise: What would be the power output if the bulb was
plugged into the US mains of 110 V?
𝑃 = 𝐼 𝑉 =
𝑉2
𝑅
=
1102
960
= 12 𝑊

44. Thermal runaway and fuses
“Circuit breakers” or “safety switches” either mechanical or electronic,
are now able to offer faster and more reliable protection.
If part of a circuit starts to overheat, its resistance can increase,
causing larger power dissipation, causing higher resistance etc.
For most conductors,
resistance is not completely
constant, but increases with
increasing temperature.
A fuse protects a circuit from general damage by acting as the
“weak point”; a thin wire that will physically fail (melt) if
current exceeds a safe level.

45. summary
• Electric current is the rate of flow of charge
measured in Amperes: 𝐼 = 𝑄/𝑡
• Ohm’s law relates the current to a resistance R
(𝐼 = 𝑉/𝑅) or resistivity 𝜌 (𝐽 = 𝐸/𝜌)
• Electric current dissipates power P = V I

46. Electric circuits
• Voltage and current
• Series and parallel
circuits
• Resistors and
capacitors
• Kirchoff’s rules for
analysing circuits

47. Electric circuits
• Closed loop of electrical components around which
current can flow, driven by a potential difference
• Current (in Amperes A) is the rate of flow of charge
• Potential difference (in volts V) is the work done on charge

48. Electric circuits
• May be represented by a circuit diagram.
Here is a simple case:
• R is the resistance (in Ohms Ω) to current flow

49. Electric circuits
• Same principles apply in more complicated cases!

50. Electric circuits
• How do we deal with a more complicated case?
What is the current flowing from the battery?

51. Electric circuits
• When components are connected in series, the
same electric current flows through them
• Charge conservation : current cannot disappear!
𝐼
𝐼

52. Electric circuits
• When components are connected in parallel, the
same potential difference drops across them
• Points connected by a wire are at the same voltage!
𝑉
𝑉
𝑉

53. Electric circuits
• When there is a junction in the circuit, the inward
and outward currents to the junction are the same
• Charge conservation : current cannot disappear!
𝐼1
𝐼2
𝐼3
𝐼1 = 𝐼2 + 𝐼3

54. Consider the currents I1, I2
and I3 as indicated on the
circuit diagram. If I1 = 2.5 A
and I2 = 4 A, what is the value
of I3?
1 2 3 4 5
0% 0%
0%
0%
0%
I1 I3
I2
1. 6.5 A
2. 1.5 A
3. −1.5 A
4. 0 A
5. The situation is not possible

55. Consider the currents I1, I2
and I3 as indicated on the
circuit diagram. If I1 = 2.5 A
and I2 = 4 A, what is the value
of I3?
I1 I3
I2
𝐼1 = 𝐼2 + 𝐼3
𝐼3 = 𝐼1 − 𝐼2
𝐼3 = 2.5 − 4 = −1.5 𝐴
(Negative sign means opposite direction to arrow.)
current in = current out

56. Resistors in circuits
• Resistors are the basic components of a circuit that
determine current flow : Ohm’s law I = V/R

57. Resistors in series/parallel
• If two resistors are connected in series, what is the
total resistance?
𝑅1 𝑅2
𝐼 𝐼
Potential drop 𝑉1 = 𝐼 𝑅1 Potential drop 𝑉2 = 𝐼 𝑅2
same current
Total potential drop 𝑉 = 𝑉1 + 𝑉2 = 𝐼 𝑅1 + 𝐼 𝑅2 = 𝐼 (𝑅1 + 𝑅2)

58. Resistors in series/parallel
• If two resistors are connected in series, what is the
total resistance?
• Total resistance increases in series!
𝑅𝑡𝑜𝑡𝑎𝑙
Potential drop 𝑉 = 𝐼 𝑅𝑡𝑜𝑡𝑎𝑙 = 𝐼 (𝑅1 + 𝑅2)
𝐼
𝑅𝑡𝑜𝑡𝑎𝑙 = 𝑅1 + 𝑅2

59. Resistors in series/parallel
• Total resistance increases in series!

60. Resistors in series/parallel
• If two resistors are connected in parallel, what is
the total resistance?
𝑅1
𝑅2

61. Resistors in series/parallel
• If two resistors are connected in parallel, what is
the total resistance?
𝑅1
𝑅2
𝐼1
𝐼2
𝐼 𝐼
Total current 𝐼 = 𝐼1 + 𝐼2 =
𝑉
𝑅1
+
𝑉
𝑅2
= 𝑉
1
𝑅1
+
1
𝑅2
𝑉

62. Resistors in series/parallel
• If two resistors are connected in parallel, what is
the total resistance?
• Total resistance decreases in parallel!
𝑅𝑡𝑜𝑡𝑎𝑙
𝐼
Current I =
𝑉
𝑅𝑡𝑜𝑡𝑎𝑙
= 𝑉
1
𝑅1
+
1
𝑅2
1
𝑅𝑡𝑜𝑡𝑎𝑙
=
1
𝑅1
+
1
𝑅2

63. Resistors in series/parallel
• Total resistance decreases in parallel!

64. • What’s the current flowing?
Resistors in series/parallel
(1) Combine these 2
resistors in parallel:
1
𝑅𝑝𝑎𝑖𝑟
=
1
30
+
1
50
𝑅𝑝𝑎𝑖𝑟 = 18.75 Ω
(2) Combine all the
resistors in series:
𝑅𝑡𝑜𝑡𝑎𝑙 = 20 + 18.75 + 20 = 58.75 Ω
(3) Current 𝐼 =
𝑉
𝑅𝑡𝑜𝑡𝑎𝑙
=
10
58.75
= 0.17 𝐴

65. Series vs. Parallel
String of Christmas lights – connected in series
Power outlets in house – connected in parallel
Same current through
all series elements
CURRENT Current “splits up”
through parallel branches
Voltages add to total
circuit voltage
VOLTAGE Same voltage across all
parallel branches
Adding resistance
increases total R
RESISTANCE Adding resistance
reduces total R

66. Voltage divider
The fraction of the total voltage that appears across a resistor in
series is the ratio of the given resistance to the total resistance.
Consider a circuit with several
resistors in series with a battery.
The potential difference across
one of the resistors (e.g. R1) 𝑉1 = 𝐼𝑅1
Current in circuit: 𝐼 =
𝑉
𝑅𝑡𝑜𝑡𝑎𝑙
=
𝑉
𝑅1 + 𝑅2 + 𝑅3
= 𝑉
𝑅1
𝑅1 + 𝑅2 + 𝑅3

67. What must be the
resistance R1 so that
V1 = 2.0 V?
1 2 3 4
0% 0%
0%
0%
12 V
R1
6.0 W
1. 0.80 W
2. 1.2 W
3. 6.0 W
4. 30 W
𝑉1

68. Capacitors
• A capacitor is a device in a circuit which can be used
to store charge
A capacitor consists of
two charged plates …
Electric field E
It’s charged by connecting
it to a battery …

69. Capacitors
• A capacitor is a device in a circuit which can be used
to store charge
Example : store and release energy …

70. Capacitors
• The capacitance C measures the amount of charge Q
which can be stored for given potential difference V
• Unit of capacitance is Farads [F]
+Q -Q
V
𝐶 =
𝑄
𝑉
𝑄 = 𝐶 𝑉
(Value of C depends
on geometry…)

71. Resistor-capacitor circuit
• Consider the following circuit with a resistor and a
capacitor in series
V
What happens when we
connect the circuit?

72. Resistor-capacitor circuit
• When the switch is connected, the battery charges
up the capacitor
• Move the switch to point a
• Initial current flow I=V/R
• Charge Q flows from
battery onto the capacitor
• Potential across the
capacitor VC=Q/C increases
• Potential across the
resistor VR decreases
• Current decreases to zero
V

73. Resistor-capacitor circuit
V
• When the switch is connected, the battery charges
up the capacitor

74. Resistor-capacitor circuit
• When the battery is disconnected, the capacitor
pushes charge around the circuit
• Move the switch to point b
• Initial current flow I=VC/R
• Charge flows from one
plate of capacitor to other
• Potential across the
capacitor VC=Q/C
decreases
• Current decreases to zero
V

75. Resistor-capacitor circuit
V
• When the battery is disconnected, the capacitor
pushes charge around the circuit

76. Capacitors in series/parallel
• If two capacitors are connected in series, what is
the total capacitance?
Potential drop 𝑉1 = 𝑄/𝐶1 Potential drop 𝑉2 = 𝑄/𝐶2
Total potential drop 𝑉 = 𝑉1 + 𝑉2 =
𝑄
𝐶1
+
𝑄
𝐶2
= 𝑄
1
𝐶1
+
1
𝐶2
+𝑄 +𝑄 −𝑄
−𝑄
𝐶1 𝐶2
Same charge must
be on every plate!
𝑄 = 𝐶 𝑉