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
Currents and Charge
Electric Current i
Current Density J
Drift Speed, Charge Carrier Collisions
Resistance, Resistivity, Conductivity
Ohm’s Law
Power in Electric Circuits
Examples
Circuit Element Definitions
Kirchhoff’s Rules
EMF’s - “Pumping” Charges, Ideal and real EMFs
Work, Energy, and EMF
Simple Single Loop and Multi-Loop Circuits using Kirchhoff Rules
Equivalent Series and Parallel Resistance formulas using Kirchhoff Rules.

2. Definition of Current : Net rate of charge flow through some area
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
Current is the same across each cross-section of a wire + - i
Current density J may vary
[J] = current/area
At any Node
(Junction) i1 i2 i3
i1+i2=i3
Kirchhoff’s
Rules:
(preview)
Current (Node) Rule: S currents in = S currents out at any node
Voltage (Loop) Rule: S DV’s = 0 for any closed path
Voltage sources (EMFs, electro-motive forces) or current sources
can supply energy to a circuit (or absorb it)
Resistances dissipate energy as heat
Capacitances and Inductances store energy in E or B fields
Energy or
Power
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
3 A
1 A
= 7 A
8 A

4. For non-uniform density: What makes current flow
Current density J: Current / Unit Area (Vector)
Conductor width might vary but…
…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
in a conductor?
conductivity
resistivity
Yes, for isolated charges in vacuum.
No, in a conducting solid, liquid, gas
Recall: terminal velocity for falling object (collisions, drift speed)
Do electrons in a current keep accelerating?

5. E field in solid wire drives current. Charges in random motion
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.
Copper drift speed is 10-8 – m/s.
E field in solid wire drives current.
APPLIED FIELD = ZERO
Charges in random motion
left flow = right flow + -
APPLIED FIELD NOT ZERO
Accelerating charges collide with fixed ions and flow with drift velocity vd
Drift Speed Formulas:
q = charge on each carrier
Note: Electrons flow leftward in sketch but vd and J are still to the right. E qn v D s =

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
Definition of Resistance : Ratio of current flowing through a specific conductor to applied 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 wire made of a good conductor.
- very large current flows so R is small.
Apply voltage to a poor conductor like carbon
- tiny current flows so R is very large.
Ohm’s Law means: R is independent of applied voltage
Resistivity “r” : Property of a material itself
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. Example: calculating resistance or 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. 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 linear model of resistivity: a = temperature coefficient
Conductivity is the reciprocal of resistivity i DV E L A
Definition:

11. Resistivity Tables

12. Current Through a Resistor
7-3: What is the current through the resistor in the following circuit, if DV = 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 in the circuit is doubled, which of
the following might be responsible?
The voltage across the resistor might have doubled.
The resistance of the resistor might have doubled.
The resistance and voltage might have both doubled.
The voltage across the resistor may have dropped by a factor of 2.
The resistance of the resistor may have dropped 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
Definitions of resistance:
(R could depend on applied V)
(r could depend on E)
Definition: For OHMIC conductors and devices:
Ratio of voltage drop to current is constant – NO dependance on applied voltage.
Current through circuit element equals CONSTANT resistance times 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. A Resistor Dissipating PowerV + -
LOAD 1 2
Apply voltage drop V across load resistor
Current flows through load which dissipates energy - irreversible
The EMF (battery) does work by expending potential energy,
It holds potential V and current i constant,
As charge dq flows from 1 to 2 it loses P.E. = dU
- Potential V is PE / unit charge
- Charge = current x time
Circuit element absorbs power for V opposed to i
Circuit element supplies power for V parallel to i 1 2 i V + -
Voltage rise:
Pabsorbed = dot product of Vrise with i (active sign convention)
Note: Some EE circuits texts use passive convention: treat Vdrop as positive.
P = - Vi
P = + Vi

16. EXAMPLE:
EXAMPLE:
EXAMPLE:

17. Basic Circuit Elements and Symbols
Basic circuit elements have two terminals 1 2
Generic symbol:
Ideal basic circuit elements: L C or R
Passive: resistance, capacitance, inductance
Active: voltage sources (EMF’s), current sources is
Ideal independent voltage source:
zero internal resistance
constant vs for any load
Ideal independent current source:
constant is regardless of load vs + - E DC
battery AC
Dependent voltage or current sources: depend on a control voltage or current generated elsewhere within a circuit

18. Circuit Analysis / Circuit Model
Circuits consist of:
ESSENTIAL NODES (junctions)...and …
ESSENTIAL BRANCHES (elements connected in series, one current/branch)
ANALYSIS METHOD: Kirchhoff’S LAWS or RULES
Current (Junction) Rule:
Charge conservation
Loop (Voltage) Rule:
Energy conservation i DV
slope
= 1/R
RESISTANCE:
POWER:
OHM’s LAW:
R is independent of DV or i

19. EMFs raise charges to higher energy
EMFs (independent voltage sources):
move + charges from low to high potential (raise potential energy).
maintain constant potential at terminals for any value of current
do work dW = Edq on charges (source of the energy in batteries is chemical)
Unit: volts (V). Variable name: script E.
Types of EMFs: batteries, electric generators, solar cells, fuel cells, etc.
DC versus AC
Power dissipated by resistor:
Power supplied by EMF: + - R E i
CONVENTION: Positive charges flow
- from + to – outside of EMF
- from – to + inside EMF

20. Ideal EMF device Real EMF device Multiple EMFs
Zero internal battery resistance
Open switch: EMF = E
- zero current, zero power
Closed switch: EMF E is
applied across load circuit
- current & power not zero
Real EMF
device
Open switch: EMF still = E
- r = internal EMF resistance in
series, usually small
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 (CCW)
EB does + work, loses energy
EA is charged up, does negative work
R converts PE to heat
Load (motor or other) produces
motion and/or heat

21. Rules for generating circuit equations from the Kirchhoff Loop Rule
The sum of voltage changes = zero around every closed loop in a multi-loop circuit.
Define: A branch (essential branch) is a series combination of circuit elements
between essential nodes.
Each branch has exactly one current flowing in it through all components in series.
Traverse one closed loop to generate each equation. May involve several branches.
Assume either current direction in each branch. Minus signs may appear in the result.
While traversing a closed loop, a test charge may traverse many branches.
Within each branch a test charge may flow with or against assumed current direction.
Across resistances, write an equation term for a voltage drop DV = - iR if following
assumed current direction. Otherwise, write voltage change DV = +iR.
When crossing EMFs from – to +, write DV = +E, since the potential rises.
If traversing EMF from + to minus write DV= -E.
The ‘dot’ product i.E determines whether power is actually supplied or dissipated.

22. D Potential around the circuit = 0
Kirchhoff Loop Rule example:
Single loop circuit with battery & internal resistance r
Follow circuit from a to b to a, same direction as i
D Potential around the circuit = 0 E
P = iE – i2r
Power in external ckt
circuit
dissipation
battery
drain
P = iVba = i(E – ir)

23. Example: Equivalent resistance for resistors in series
Junction Rule: The current through all of the resistances in series (a single essential branch) is identical. No information from Junction/Current/Node Rule
Loop Rule: The sum of the potential differences around a closed loop path equals zero. Only one loop path exists:
The equivalent circuit replaces the series resistors with a single equivalent resistance Req: 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

24. Example: Equivalent resistance for resistors in parallel
Loop Rule: The potential differences across each of the 4 parallel branches are the same. There are 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

25. 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

26. 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

27. + - + - 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

28. 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

29. 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