Physics 2112
Unit 9: Electric Current
Today’s Concept:
Electric Current
Electricity & Magnetism Lecture 9, Slide 1
A Big Idea Review

kq1q2
F1, 2  2 rˆ1, 2
r1, 2
Coulomb’s Law
Force law between
point charges
Electric Field
Force per unit charge
Gauss’ Law
Flux through closed
surface is always
proportional to charge
enclosed
Electric Potential
Potential energy per
unit charge
Capacitance
Relates charge and
potential for two
conductor system

 F
E
q
0
q2
Gauss’ Law
Can be used to
determine E field
b 

U ab

   E  dl
q
a
Q
C
V
r1,2
Electric Field
Property of Space
Created by Charges
Superposition
  Qenc
 E  dA 
Vab
q1
F1,2
Spheres
Cylinders
Infinite Planes
Electric Potential
Scalar Function that can
be used to determine E


E  V
Electricity & Magnetism Lecture 9, Slide 2
Applications of Big Ideas
Conductors
Charges free to move
Field Lines &
Equipotentials
What Determines
How They Move?
Spheres
Cylinders
Infinite Planes
They move until
E0!
Gauss’
Law
E  0 in conductor
determines charge
densities on surfaces
Capacitor Networks
Work Done By E Field
  b  
  F  dl   qE  dl
b
Wab
a
a
Change in Potential Energy

U a b  Wa b    qE  dl
b 
a
Series:
(1/C23)  (1/C2) + (1/C3)
Parallel
C123  C1 + C23
Electricity & Magnetism Lecture 9, Slide 3
Key Concepts:
1) How resistance depends on A, L, s, r
2) How to combine resistors in series and parallel
3) Understanding resistors in circuits
Today’s Plan:
1) Review of resistance & preflights
2) Work out a circuit problem in detail
Electricity & Magnetism Lecture 9, Slide 4
I
A
s
L
Conductivity – high for good conductors.
V
Ohm’s Law: J  s E
Observables:
V  EL
I  JA
I/A  sV/L
R  Resistance
R  1/s
I  V/R
I  V/(L/sA)
R
L
sA
Electricity & Magnetism Lecture 9, Slide 5
Civil Engineering Analogy
I is like flow rate of water
V is like pressure
R is how hard it is for water to flow in a pipe
L
R
sA
To make R big, make L long or A small
To make R small, make L short or A big
Battery is like a
pump
Electricity & Magnetism Lecture 9, Slide 6
1
CheckPoint: Two Resistors
2
Same current through
both resistors
Compare voltages across
resistors
L
R
A
L
V  IR 
A
A2  4 A1  V2  14 V1
L2  2L1  V2  2V1
Electricity & Magnetism Lecture 9, Slide 7
CheckPoint: Current Density
The SAME amount of current I passes through three different resistors. R2 has twice
the cross-sectional area and the same length as R1, and R3 is three times as long as R1
but has the same cross-sectional area as R1. In which case is the CURRENT DENSITY
through the resistor the smallest?
Electricity & Magnetism Lecture 9, Slide 8
Circuit
c
b
VB
R1
E is conservative force
 go completely around circuit
Vf = Vo
R2
d
a
+VB
V
-IR1
-IR2
a
b
c
d
a
Unit 9, Slide 9
Resistor Summary
Series
Parallel
R1
R1
R2
R2
Wiring
Each resistor on the
same wire.
Each resistor on a
different wire.
Voltage
Different for each resistor.
Vtotal  V1 + V2
Same for each resistor.
Vtotal  V1  V2
Current
Same for each resistor
Itotal  I1  I2
Different for each resistor
Itotal  I1 + I2
Increases
Req  R1 + R2
Decreases
1/Req  1/R1 + 1/R2
Resistance
Electricity & Magnetism Lecture 9, Slide 10
CheckPoint: Resistor Network 1
Three resistors are connected to a battery with emf V as shown. The resistances of the
resistors are all the same, i.e. R1= R2 = R3 = R.
Compare the current through R2 with
the current through R3:
A. I2 > I3
B. I2 = I3
C. I2 < I3
Electricity & Magnetism Lecture 9, Slide 11
CheckPoint: Resistor Network 2
Three resistors are connected to a battery with emf V as shown. The resistances of the
resistors are all the same, i.e. R1= R2 = R3 = R.
Compare the current through R1 with
the current through R2:
A. I1/I2=1/2
B. I1/I2=1/3
C. I1 = I2
D. I1/I2=2
E. I1/I2=3
Electricity & Magnetism Lecture 9, Slide 12
CheckPoint: Resistor Network 3
Three resistors are connected to a battery with emf V as shown. The resistances of the
resistors are all the same, i.e. R1= R2 = R3 = R.
Compare the voltage across R2 with the
voltage across R3:
A. V2 > V3
B. V2 = V3 = V
C. V2 = V3 < V
D. V2 < V3
Electricity & Magnetism Lecture 9, Slide 13
R1  R2  R3  R
CheckPoint 2
Compare the current through R1
with the current through R2
I1
I2
CheckPoint 3
CheckPoint 4
Compare the voltage across R2
with the voltage across R3
Compare the voltage across R1
with the voltage across R2
V2
V3
V1
V2
Electricity & Magnetism Lecture 9, Slide 14
CheckPoint: Resistor Network 4
Three resistors are connected to a battery with emf V as shown. The resistances of the
resistors are all the same, i.e. R1= R2 = R3 = R.
Compare the voltage across R1 with the
voltage across R2.
A. V1 = V2 = V
B. V1 = 1/2 V2 = V
C. V1 = 2V2 = V
D. V1 =1/2 V2 =1/5 V
E. V1 =1/2 V2 = 1/2 V
Electricity & Magnetism Lecture 9, Slide 15
Example 9.1
R1
R2
In the circuit shown: V  18V,
R1  1W, R2  2W, R3  3W, and R4  4W.
R4
What is V2, the voltage across R2?
V
R3
Conceptual Analysis:
Ohm’s Law: when current I flows through resistance R, the potential drop V is given by:
V  IR.
Resistances are combined in series and parallel combinations
Rseries  Ra + Rb
(1/Rparallel)  (1/Ra) + (1/Rb)
Strategic Analysis:
 Combine resistances to form equivalent resistances
 Evaluate voltages or currents from Ohm’s Law
 Expand circuit back using knowledge of voltages and currents
Electricity & Magnetism Lecture 9, Slide 16
Example 9.1
R2
R1
V
R2
R1
V
R3
R24
R3
R4
R1
V
V
R234
R1234
Electricity & Magnetism Lecture 9, Slide 17
Quick Follow-Ups
I1
R1
I2
R2
R1
I3
V

R3
a
V
R234
R4
b
What is I3 ?
A) I3  2 A
B) I3  3 A
V3 = V234 = 12V
C) I3  4 A
I3  V3/R3  12V/3W  4A
V  18V
R1  1W
R2  2W
R3  3W
R4  4W
R24  6W
R234  2W
V234= 12V
V2 = 4V
I1234 = 6 Amps
What is I1 ?
We know I1  I1234  6 A
I1  I2 + I3
Make Sense?
Electricity & Magnetism Lecture 9, Slide 18
Power In Resistors
Electro-Motive Force (EMF),

Energy provided to make charges move, units of V
dW

dq
dW
dq
Power 

dt
dt
=VI (for a battery)
= I2R (for resistor)
Unit 9, Slide 19
Example 9.2
R1
R2
In the circuit shown: V  18V,
R1  1W, R2  2W, R3  3W, and R4  4W.
R4
How much electrical energy does the
battery put into the circuit every second
in the previous problem?
V
R3
How much electrical energy does each resistor turn into
thermal energy every second?
Parallel and Series (with color)
R1
R2
V
R3
R4
If every electron that goes
through one element must go
through another, those two are
in series.
If two sides of two elements can
be connected by different
colored lines, those two are in
parallel. .
If two points are connected by a line not
containing any circuit elements those point are at
the same potential.
Unit 9, Slide 21