1

TOPIC 6Electric currentand resistance

2

Conductors have free electrons, which

• Are in continuous rapid motion – thermal and quantum effects

• Undergo frequent scattering from the crystal lattice (positive ions)

• Random motion does not constitute a current

• An applied electric field results in a small drift velocity superimposed on the random motion

• This drift gives a net movement of charge – an electric current, I

Electrons in Conductors

d

d

QI

t

3

Unit of current is Amp (Ampère), 1 A = 1 C s–1

Continuous current through conductor potential difference between ends – eg due to battery

Battery raises positive charges from low potential (negative terminal) to high potential (positive terminal)

As much charge enters one end of the conductor (eg wire) as leaves at the other end – it does not charge up!

Current only flows in a closed loop or circuit

Current density J = current flow per unit area (perpendicular to current) J = I/A

Electric Current

4

The current I flowing through a component depends on the potential difference between its ends, V.

We can define the resistance R of the component from

Unit of resistance : ohm (1 = 1 volt per amp)

Conductance G = 1 / R (units –1, mho or siemens)

Electrical Resistance

VR

I

5

Assume n electrons (of charge q = –e) per unit volume

Applied electric field E gives acceleration

Mean time between collisions with lattice =

Drift velocity (superimposed on random motion)

Charge Q in dotted cylinder = n A q, passes end plane in time t = /vd, so I = n A q vd

Hence

Drude Model

vd

A q

q

mE

a

dq

m E

v

2

dnq

J nqv Em

6

Drude model

We can write this

is the resistivity, with units m

= 1 / = conductivity

Note J = I / A

V = E

Resistivity & Conductivity 2nq

J Em

1J E E

vd

A q

V ER

I JA A

7

Example 1 – Resistance, drift velocity

Copper has a resistivity of 1.710–8 m, and 8.5 1028 m–3 free electrons. What is the mean time between collisions between a conduction electron and the lattice? What will the drift velocity be when 2.0 V is applied across a 5.0 m sample of copper?

What resistance will a copper coil have if it is formed of 1000 turns of wire, of diameter 1.0mm, wrapped around a tube of radius 3.0cm?

8

Increased temperature increased lattice vibrations

increased electron scattering

increased resistivity in most materials

Approximation for modest temperature changes:

Here is the temperature coefficient of resistivity.

Example 2: A sample of platinum has a resistance of 30.00 at 20C, and 39.41 at 100C. What is the coefficient of resistivity for platinum? What would the resistance be at 0C?

Temperature Coefficient of Resistivity

0 01T T T

9

The energy loss of a charge Q falling through potential difference V is Q V. The power dissipated (rate of energy loss) is therefore P = dQ/dt V = I V.

Using V = I R, this can be expressed in a variety of useful ways:

Example 3

A resistance of 3.0 is connected across a potential difference of 2.0 V. How large is the current that flows, and how much power is dissipated?

Electrical Power

22 V

P IV I RR