Current & Resistance Current & Resistance

- Current and current density- Ohm’s Law- Resistivity- Resistance

Electrical CurrentElectrical CurrentCURRENT I is the amount of positive charge flowing past a fixed point in the wire per unit time :

if charge dQ flows in time dt

Units: 1 ampere (A) = 1 C/s

Direction: by convention, current is the direction of movement of positive charge

+ -+

+ +

+

+ ---

--

I I

dQI

dt

Electron Electron VelocitiesVelocities

• Random velocities of electrons are large (several km/s)

• Drift velocity is a slow, average motion parallel to E

start

end

start

end

net displacement

+( )F e E

Eno field

Charge ΔQ in length L of wire passes through the shaded disk of area A in time Δt :

ΔQ = (number of charge carriers/volume) x (charge on each one) x volume

+ ++ +

+ ++ +

+ ++ +

L = vd t

E

Determining the current

Current: I = Q/t = nqAvd t /t

So,

Charge: Q = n q V = n q (AL) = n q A vd t (since L=vt)

I = nqAvd

vd = average (“drift”) velocity of each chargeq = charge on each particlen = number of charge carriers per unit volumeA = cross section areaL = lengthAL = volume

ExampleExample

3melectrons/ 2910n3C/m . 101061 nenq

Take Area = 1 mm2, assume I = 1 A

0.06 mm/sdrift

v

The mobile charges in most metals are electrons, with about one or two electrons per atom being free to move. So there are about 1023 charges per cm3 (or 10 29 m-3).

Which way are the electrons moving?

E = 0 E = 0 inside ?inside ?

Note: in electrostatics, we had E=0 inside a conductor, if not, charges would move, the conductor would not be in equilibrium and there would be a current.

For a wire to carry a current, we must have an electric field inside the conductor, which is caused by the potential difference between the ends of the wire. This is no longer electrostatics!

ExamplExamplee

A copper wire of cross sectional area 3x10-6 m2 carries a current of 10A. Find the drift velocity of the electrons in this wire. Copper has a density of 8.95g/cm3, and atomic weight of 63.5g/mole. Avagadro’s number is 6.02x1023 atoms.Assume each atom contributes one free electron.

QuestioQuestionsns

•When you turn on a flashlight, how long does it take for the electrons from the battery to reach the bulb?

• What happens to the wire as the electrons go through it?

•If you double the electric field in the wire, does the acceleration of the electrons double?

•If the electric field remains constant, how do the electron kinetic energies change with time?

Current Density JCurrent Density J

IJ

A

EEEEEEEEEEEEEE

is proportional to the electric fielddnow v

dJ nqvEEEEEEEEEEEEEEEEEEEEEEEEEEEE

so J EEEEEEEEEEEEEEEEEEEEEEEEEEEEE

Where is a constant called the conductivity of the material.

The current density J and the electric field E are bothestablished in a conductor as a result of a potential difference across the conductor.

The current density is defined as the current per unit area in a conductor, where A is the cross section of conductor.The current density is a vector quantity, units: Amps/m2.

And since I=nqAvd;

L

Uniform E V EL

I V

J EA L

L

V IA

“Resistance”, R

Because J is proportional to the field, current in a wire is proportional to the potential difference between the ends of the wire.

A

V

E

RESISTIVITY: the inverse of conductivity.

1

whereAL

R (Uniform wire, Length L, cross-section area A)

L

V I IRA

(this depends on the type of material)

ampvolt

)( ohm 11 Unit of resistance R is:

Ohm’s Law

Ohm’s Law

•Current density field: J = E•Current potential difference: V = IR

mmV

mA

12

units, = “conductivity”

= “resistivity”, has units of m

ρ (20°C)

(Ω·m)

Cu 1.7 x 10-8

Al 2.8 x 10-8

Graphite 3500 x 10-8

Si 640

Quartz ~ 1018

Resistivities of a few materials

ExampleExample

A copper wire, 2 mm in diameter and 30 m in length, has a current of 5A. Find:

a) resistance

b) potential difference between the ends

c) electric field

d) current density

Cu for m 8107.1

Quiz

The wire in the previous example is replaced with a wire of the same length and half the diameter, carrying the same current. By what factor will the resistance change?

A) increase by 4B) increase by 2C) stay the sameD) decrease by 2E) decrease by 4

Question

The wire in the previous example is replaced with a wire of the same length and half the diameter, carrying the same current. By what factor will each of the following change?

A) resistance

B) potential difference between the ends

C) electric field

D) electron number density

E) electron drift speed

F) current density

SummarySummary

Current Density:

Conductivity:

Resistivity:

Resistance:

Ohm’s Law:

dnqvAIJ

:) (defines EJ

1

AL

R

V IR