1. Electricity and Magnetism Lecture 07 - Physics 121 Current, Resistance, DC Circuits: Y&F Chapter 25 Sect. 1-5 Kirchhoff’s Laws: Y&F Chapter 26 Sect. 1
Circuits and Currents
Electric Current i
Current Density J
Drift Speed
Resistance, Resistivity, Conductivity
Ohm’s Law
Power in Electric Circuits
Examples
Kirchhoff’s Rules applied to Circuits
EMF’s - “Pumping” Charges
Work, Energy, and EMF
Simple Single Loop and Multi-Loop Circuits
Summary

2. Current Definition: Net charge crossing a surface per unit time
Convention: flow is from + to – as if free charges are +
Units: 1 Ampere = 1 Coulomb per second
Charge / current is conserved - charge does not pile up or vanish
At any
junction i1 i2 i3
i1+i2=i3
Kirchhoff’s
Rules:
(summary)
Junction Rule: S currents in = S currents out at any junction
Voltage Rule: S DV’s = 0 for any closed path
Current is the same across each cross-section of a wire + - i
Current density J may vary
[J] = current/area
EMFs provide energy (electro-motive force)
Resistances dissipate energy as heat
Capacitances store energy in E field
Inductances store energy in B field
Energy
in a circuit:

3. Junction Rule Example – Current Conservation
7-1: What is the value of the current marked i?
1 A.
2 A.
5 A.
7 A.
Cannot determine
from information given.
5 A
2 A
3 A
1 A
6 A i

4. + - Current density J: Current / Unit Area (Vector)
Same current crosses larger or smaller
Surfaces, current density J varies + - i A A’
J’ = i / A’
(small)
J = i / A
(large)
High current density
in this region
Small current density
units: Amperes/m2
For uniform density:
For non-uniform density:
What makes current flow?
Microscopic level
E field in solid wire drives current.
APPLIED FIELD = 0
Charges in random motion
flow left = flow right + -
APPLIED FIELD NOT ZERO
Moving charges collide with fixed ions
and flow with drift velocity vd
conductivity
resistivity

5. - + Drift speed: Do electrons in a current keep accelerating?
COLLISIONS with ions, impurities, etc. cause resistance
Charges move at constant drift speed vD:
Thermal motions (random motions) have speed
Drift speed is tiny compared with thermal motions.
Drift speed in copper is 10-8 – m/s. + -
For E = 0 in conductor: no current,
vD=0, J = 0, i = 0
For E not = 0
(battery voltage not 0):
Note: Electrons drift rightward but vd and J are still leftward
|q| = e = 1.6 x C.

6. A./m2 Assume: Doubly charged positive ions
EXAMPLE: Calculate the current density Jions for ions in a gas
Assume:
Doubly charged positive ions
Density n = 2 x 108 ions/cm3
Ion drift speed vd = 105 m/s
Find Jions – the current density for the ions only (forget Jelectrons)
coul/ion
ions/cm3
m/s
cm3/m3
A./m2

7. Increasing the Current
7-2: When you increase the current in a wire, what changes and what is constant?
The density of charge carriers stays the same, and the drift speed increases.
The drift speed stays the same, and the number of charge carriers increases.
The charge carried by each charge carrier increases.
The current density decreases.
None of the above

8. Resistivity “r” : Property of a material itself
Resistance Definition: How much current flows through a device in response to a given potential difference. i DV E L A
R depends on the material & geometry Note: C= Q/DV – inverse to R R
Circuit
Diagram V
Apply voltage to a conducting wire.
- Very large current flows so R is small.
Apply voltage to a poorly conducting material like carbon
- Tiny current flows so R is very large.
Resistivity “r” : Property of a material itself
(as is dielectric constant). Does not depend on dimensions
The resistance of a device depends on resistivity r and also depends on shape
For a given shape, different materials produce different currents for same DV
Assume cylindrical resistors
For insulators: r  infinity

9. Calculating resistance, given the resistivity
proportional to length
inversely proportional
to cross section area
resistivity
EXAMPLE:
Find R for a 10 m long iron wire, 1 mm in diameter
Find the potential difference across R if i = 10 A. (Amperes)
EXAMPLE:
Find resistivity of a wire with R = 50 mW,
diameter d = 1 mm, length L = 2 m
Use a table to identify material. Not Cu or Al, possibly an alloy

10. Resistivity Tables

11. Change the temperature from reference T0 to T
Resistivity depends on temperature:
Resistivity depends on temperature: Higher temperature  greater thermal motion  more collisions  higher resistance.
r in 20o C.
1.72 x copper
9.7 x iron
2.30 x pure silicon
Reference Temperature
SOME SAMPLE
RESISTIVITY
VALUES
Change the temperature from reference T0 to T
Coefficient a depends
on the material
Simple model of resistivity: a = temperature coefficient
Conductivity is the reciprocal of resistivity i DV E L A
Definition:

12. Current Through a Resistor
7-3: What is the current through the resistor in the following circuit, if V = 20 V and R = 100 W?
20 mA.
5 mA.
0.2 A.
200 A.
5 A. R V
Circuit
Diagram

13. Current Through a Resistor
7-4: If the current is doubled, which of the following might also have changed?
The voltage across the resistor doubles.
The resistance of the resistor doubles.
The voltage in the wire between the battery and the resistor doubles.
The voltage across the resistor drops by a factor of 2.
The resistance of the resistor drops by a factor of 2. R V
Circuit
Diagram
Note: there might be
more than 1 right answer

14. Definitions of resistance:
Ohm’s Law and Ohmic materials (a special case)
Definitions of resistance:
but R could depend on applied V
but r could depend on E
Definition: OHMIC conductors and devices:
Ratio of voltage drop to current is constant – it does not depend on applied voltage
i.e., current is proportional to applied V
Resistivity does not depend on magnitude or direction of applied voltage
Ohmic Materials
e.g., metals, carbon,…
Non-Ohmic Materials
e.g., semiconductor diodes
constant slope = 1/R
varying slope = 1/R
band gap
OHMIC
CONDITION
is CONSTANT

15. Resistive Loads in Circuits Dissipate Power+ -
LOAD a b
Apply voltage drop V across load
Current flows through load which dissipates energy
An EMF (e.g., a battery) does work, holding
V and current i constant by expending potential energy
As charge dq flows from a to b it loses P.E. = dU
- potential is PE / unit charge
- charge = current x time
EXAMPLE:
Space heater: Find rate of converting electrical energy to heat

16. EXAMPLE:
EXAMPLE:
EXAMPLE:

17. Circuit analysis with resistances and EMFs
GENERAL ANALYSIS METHOD: Kirchhoff’S LAWS or RULES
Junction Rule
Charge conservation
Loop Rule
Energy conservation
CIRCUIT ELEMENTS:
PASSIVE: RESISTANCE, CAPACITANCE, INDUCTANCE
ACTIVE: EMF’s (SOURCES OF POTENTIAL DIFFERENCE AND ENERGY)
JUNCTIONS
and
BRANCHES i i1 i2
….etc…
RESISTANCE:
POWER: i DV
slope
= 1/R
OHM’s LAW:
R is independent of DV or i

18. EMFs “pump” charges to higher energy
EMFs (electromotive force) such as batteries supply energy:
move charges from low to high potential (potential energy).
maintain constant potential at terminals
do work dW = Edq on charges (source of the energy in batteries is chemical)
EMFs are “charge pumps”
Unit: volts (V). Symbol: script E.
Types of EMFs: batteries, electric generators, solar cells, fuel cells, etc.
DC versus AC + - R E i
CONVENTION:
Current flows CW through circuit
from + to – outside of EMF
from – to + inside EMF
Power dissipated by resistor:
Power supplied by EMF:

19. Ideal EMF device Real EMF device Multiple EMFs
Zero internal battery resistance
Open switch: EMF = E
no current, zero power
Closed switch: EMF E is also
applied across load circuit
Current & power not zero
Real EMF
device
Open switch: EMF still = E
r = internal EMF resistance in
series, usually small ~ 1 W
Closed switch:
V = E – ir across load, Pckt= iV
Power dissipated in EMF
Pemf = i(E-V) = i2r R
Multiple
EMFs
Assume EB > EA (ideal EMF’s)
Which way does current i flow?
Apply kirchhoff Laws to find current
Answer: From EB to EA
EB does work, loses energy
EA is charged up
R converts PE to heat
Load (motor, other) produces
motion and/or heat

20. Potential around the circuit
Generating Circuit Equations with the Kirchhoff Loop Rule
Traversing a closed loop generates one equation. Multiple loops may be needed.
The sum of voltage changes = zero around all closed loops in a multi-loop circuit)
A branch is a series combination of devices between branches.
Assume either current direction in each branch. Minus signs in the result may appear.
Traverse each branch of the circuit with or against assumed current direction.
Across resistances, voltage drop DV = - iR if following assumed current direction.
Otherwise, voltage change is +iR.
When crossing EMFs from – to +, DV = +E. Otherwise DV= -E
Dot product i.E determines whether power is actually supplied or dissipated
Follow circuit from a to b to a, same direction as i
Potential around the circuit
EXAMPLE: Single loop circuit with battery (internal resistance r) E
P = iE – i2r
Power in External Ckt
circuit
dissipation
battery
drain
P = iV = i(E – ir)

21. Equivalent resistance for resistors in series
Junction Rule: The current through all of the resistances in series (a single branch) is identical. No information from Junction Rule
Loop Rule: The sum of the potential differences around a closed loop equals zero. Only one loop path exists:
The equivalent circuit replaces the series resistors with a single equivalent resistance: same E, same i as above
CONCLUSION: The equivalent resistance for a series combination is the sum of the individual resistances and is always greater than any one of them.
inverse of series capacitance rule

22. Equivalent resistance for resistors in parallel
Loop Rule: The potential differences across each of the parallel branches are the same. Four unknown currents. Apply loop rule to 3 paths.
i not in these equations
Junction Rule: The sum of the currents flowing in equals the sum of the currents flowing out. Combine equations for all the upper junctions at “a” (same at “b”).
The equivalent circuit replaces the series resistors with a single equivalent resistance: same E, same i as above.
CONCLUSION: The reciprocal of the equivalent resistance for a parallel combination is the sum of the individual reciprocal resistances and is always smaller than any one of them.
inverse of parallel capacitance rule

23. Resistors in series and parallel
7-7: Four identical resistors are connected as shown in the figure. Find the equivalent resistance between points a and c.
4 R.
3 R.
2.5 R.
0.4 R.
Cannot determine
from information given. c R R R R a

24. Capacitors in series and parallel
7-8: Four identical capacitors are connected as shown in figure. Find the equivalent capacitance between points a and c.
4 C.
3 C.
2.5 C.
0.4 C.
Cannot determine
from information given. c C C C C a

25. + - + - EXAMPLE: Find i, V1, V2, V3, P1, P2, P3 EXAMPLE:
R1= 10 W
R2= 7 W
R3= 8 W
E = 7 V i
EXAMPLE:
Find currents and voltage drops i + -
E = 9 V R1 R2 i1 i2

26. EXAMPLE: MULTIPLE BATTERIES
SINGLE LOOP + -
R1= 10 W
R2= 15 W
E1 = 8 V
E2 = 3 V i
A battery (EMF) absorbs power (charges up) when I is opposite to E

27. So why do electrical signals on wires seem
EXAMPLE: Find the average current density J in a copper wire whose diameter is 1 mm carrying current of i = 1 ma.
Suppose diameter is 2 mm instead. Find J’:
Current i is unchanged
Calculate the drift velocity for the 1 mm wire as above?
About 3 m/year !!
So why do electrical signals on wires seem
to travel at the speed of light (300,000 km/s)?
Calculating n for copper: One conduction electron per atom