Chapter 16
Electric Current and Circuits
1
Electric Current
Electron flow:
Conventional current (is the direction of current (I) and is not
necessarily the same as the direction of
moving charges)
_
+
higher potential
Electron flow
•
•
the free electrons in metals cause the conduction of electricity
in the presence of potential difference, electrons will move from
negative end to positive end with higher potential
Electric current is the motion of charge (positive) from one point to another
and is defined as:
dQ
I=
dt
net charge passing through an area per unit time
Unit of I : Ampere or A
(current is not a vector)
2
The moving charges in metals are always electrons (negative charge).
In an ionized gas/solution, the moving charges include electrons
(negative) and charged ions (positive).
Same current is produced by the +ve and –ve
charges if number of charges and drifting velocity
are the same
I
I
direction of v
same as E
-
v
v
+
E
E
+
v
v
-
direction of v
opposite to E
v : drift velocity of the charged particle
battery
_ +
I
I
+
3
Ohm’s Law: the electric current passing through a fixed conductor at
constant temperature is proportional to the potential
difference between the start and end points of the conductor.
V = IR
Where,
V = potential difference between the start and end points of the
conductor [volt or V];
I = current passing through the conductor [Ampere or A]; and
R = the resistance of the conductor [ohm or Ω].
Current is not a vector. However, current moves in one direction along a conductor.
When a direction is defined as +ve, the opposite direction is –ve or clockwise
around a circuit is +ve and counter-clockwise is –ve.
4
Resistance of a conductor (R):
R=
ρL
A
Where,
R = resistance [Ω];
ρ = resistivity of the metal of which the conductor is made of (a constant
for a particular metal) and is a function of temperature (T);
A = cross-sectional area of the metal wire [m2]; and
L = length of the metal wire [m].
ρ = ρ (T )
ρT = ρ 0 (1 + αT )
Where,
neglecting the thermal
expansion of the metal
RT = R0 (1 + αT )
The reciprocal of resistivity is conductivity (unit: Ω-1 m-1)
ρ0 = resistivity of the metal at temperature = 0oC [Ωm];
ρT = resistivity of the metal at temperature = ToC [Ωm]; and
α = temperature coefficient of the metal [(Co)-1] .
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Resistance in series:
I
A
R
R
1
2
V1
V2
B
The current passing
through each resistor is
the same
V = V1 + V2 = IR1 + IR2
R = R1 + R2
6
Resistance in parallel:
I1
R
1
I
I2
A
R
B
The potential difference
between A and B must be
the same
2
V
I = I1 + I 2
V
I1 =
R1
V
and I 2 =
R2
1 I
1
1
= = +
R V R1 R2
7
Electric circuit: the conducting path forming a closed loop where the
charge moves.
Direct current (DC) circuit:
a circuit with the direction of current not
changing with time.
Alternative current (AC) circuit: a circuit with the direction of current
oscillating back and forth.
Electromotive force (emf): the influence causing the current flow moving
from lower potential to higher potential, e.g.
water pump, cells
ideal emf : V12 = emf or E
E = V12 = IR {unit: volt]
R
R1
R2
R3
E1
E2
E3
+
R = R1 + R2 + R3
E = E1 + E2 + E3
E
Single cell
cell connected in series
8
Examples of sources of emf:
•
•
•
batteries
electric generators
solar cells, etc.
Through the connection of a source of emf, electrical potential energy is
transferred into the electric circuit.
An ideal source of emf has a constant potential difference between its
terminals and is independent of the current passing through it.
emf can be defined as the magnitude of the potential difference.
Vab = IR = ε
Vab = IR = ε − Ir
Ideal source of emf
Source of emf with internal resistance
I : current; r: internal resistance of the emf
9
Kirchhoff’s Rules:
Junction rule:
•
the algebraic sum of the currents passing through any junction is zero
∑I = 0
At any junction
conservation of electric charge
Loop rule:
•
the algebraic sum of the potential differences in any loop is zero. Both
emf and resistors can be included in the loop.
∑V = 0
For any closed loop
electrostatic force is conservative
10
Junction rule:
∑I = 0
junction
I1 + I 2 = I 3
Loop rule:
R1
ΣV = 0
+
I1
Traveling in counter-clockwise direction:
R3
E1 − I1 R1 + I 2 R3 + I 2 R2 − E2 = 0
E1 − E2 = I1 R1 − I 2 R3 − I 2 R2
E1
I2
R2
+
E2
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Sign convention:
• travel thru a source from – to +, emf is +ve
• travel thru a source from + to -, emf is –ve
• travel thru a resistor in the same direction as the current, IR term
is –ve (decreasing potential)
• travel thru a resistor in the opposite direction as the current, IR term
is +ve (increasing potential)
R
-E
+
emf
R
+E
-
+
-
E
E
traveling direction
- IR
resistor
I
+ R
+ IR
-
traveling direction
I
- R
+
12
Example 16.1: Find the emf’s ε1 and ε2 in the circuit as shown in the figure, and
find the potential difference of point b relative to point a.
1 Ω 20V
+
ε1
1.0A
4Ω
a
2.0A
6Ω
1Ω
+
b
ε2
1Ω +
2Ω
Solution: From the given currents in the diagram, the current through the middle
branch of the circuit must be 1.00 A (the difference between 2.00 A and 1.00 A).
We now use Kirchhoff’s Rules, passing counterclockwise around the top loop:
20.0V − (1.00 A)( 6.00 Ω + 1.00 Ω) + (1.00 A)( 4.00 Ω + 1.00 Ω) − ε 1 = 0
⇒ ε 1 = 18.0V .
Traveling around the external loop of the circuit, the emf ε2 and Vab are:
20.0V − (1.00 A)(6.00 Ω + 1.00 Ω) − (2.00 A)(1.00 Ω + 2.00 Ω) − ε 2 = 0
⇒ ε 2 = 7.0V .
And Vab = 18 V – (1.00 A)(4.00 Ω + 1.00 Ω) = +13.0 V.
13
Energy and Power in Electric Circuits
dW = V12 dQ = V12 Idt
dW
p=
= V12 I
dt
dQ
QI =
dt
V1
V2
I
I
1
2
where,
W = work done
Q = charge
I = current
t = time
p = power or time rate of energy transfer (Watt or W)
V12 = V1 – V2 = potential difference between point 1 and point 2.
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