DC Circuits

Currents. Resistors. Batteries.

Kirchhoff’s Loop Rules. Power

Examples of Circuits

t

QI

av

Current: flow of charge

Average current: Iav= Charge Q

flowing across area A in time t

t

QI

d

d

Instantaneous current: differential limit of Iav

Units of current: Coulomb/second=Ampere (A)

Direction of the current

Direction of current is in direction of flow of positive charge

or, opposite direction of flow of negative charge

Current density J (1)J: current/unit area

A

IJ

points in the direction of the current

Units A/m2

J

Current density J (2)If the ara is not perpendicular to the current

An area or surface perpendicular to the direction of the current

)cos(n JAJAI

: angle between normal to A and current

nA

IJ

)cos(n AA

Why does current flow?If an electric field is set up in a conductor, charge will move (making a current in the direction of E)

Note that when current is flowing, the conductor is not an equipotential surface (and Einside0)!

Microscopic picture (1)

Drift speed is velocity forced by applied electric field in the presence of collisions, it is typically 4x10-5m/s, or 0.04 mm/s!

To go one meter at this speed takes about 10 hours

Thermal velocity is around 103 km/s !

How can this be?

Microscopic picture (2)

vd : drift velocity

n : number of carriers per unit volume

q : charge of each carrier (normally e)

A : suface perpendicular to vd

AqnvI d

dqnvJ

Conductivity and resistiviy

Ability of current to flow depends on density of charges & rate of scattering.

Two quantities summarize this:

: conductivity

: resistivity 1

Microscopic Ohm’s Law

And depend only on the microscopic properties of the material,

not on its shape

1

or JEEJ

The voltage drops in a resistorThe electric field brings about a voltage drop in a resistor

ElVVV ab

What is the relationship between V and current?

ElVVV ab

Ohm’s Law (1)

A

lIl

A

IJlElV

R has units of ohms ()=Volt/Amp

Ohm’s Law (2)

A

lR IRV

Then, units of : m and : -1m-1

Examples of Circuits

Symbols for circuit elements

Electromotive force -Battery Moving from the negative to positive terminal of a battery increases your potential

Think:

Ski Lift

ab VVV

Internal resistanceReal batteries have an internal resistance, r, which is small but non-zero

IrVVV ab Terminal voltage:

Real battery=ideal battery in series with a resistance

Sign conventions - ResistorMoving across a resistor in the direction of current decreases your potential

ab VV Think:

Ski Slope

Voltage drop:

Voltage decreases in the direction of the current

Sign conventions - CapacitorMoving across a capacitor from the negatively to positively charged plate increases your potential

ab VVV

Resistors in seriesThe same current I must flow through both resistors

eq2121 )( IRRRIIRIRVVV ca

21eq RRR

N

iiRR

1eq

or

Resistors in parallelVoltage drop across the resistors must be the same

eq221121 IRRIRIVVV

N

i iRR 1eq

11

eq212121

11

R

V

RRV

R

V

R

VIII

21eq

111

RRR or

Measuring V, I, R

Measuring Potential Difference A voltmeter must be hooked in parallel across the element you want to measure the potential difference across

Voltmeters have a very large resistance, so that they don’t affect the circuit too much

Measuring CurrentAn ammeter must be hooked in series with the element you want to measure the current through

Ammeters have a very low resistance, so that they don’t affect the circuit too much

Measuring ResistanceAn ohmmeter must be hooked in parallel across the element you want to measure the resistance of

Here we are measuring R1

Ohmmeters apply a voltage and measure the current that flows. They typically won’t work if the resistor is powered (connected to a battery)