Unit 3: Direct current and electric resistance

Electric current and movement of charges.

Intensity of current and drift speed.

Density of current in homogeneous currents.

Ohm’s law. Resistance of a homogeneous

conductor of constant cross section.

Changes in resistivity depend on temperature

Resistors association.

Series and parallel association.

Tipler, chapter 25, part 25.1

t

qI

∆

∆=

Electric current and movement of charges

An electric current occurs when a conductor is not inelectrostatic equilibrium. This can be achieved by applyingan electric field ≠ 0 to the conductor.

Intensity is a base quantity. Unit: Ampère

I is the same for any cross section of the conductor

E++

++

Although the moving particles in a conductor are electrons,for historical reasons electric current is studied as positivecharges moving in the direction of the electric field.

Electric current and movement of charges

The charges collide withthe nuclei of the atoms,causing chaotic movement.

These collisions set up aresistance to themovement of the charges(electric resistance).

An average speed (drift speed Vd) can becomputed for the charges, Vd being relatedto E. This rate is called mobility (µ):

Vd

E

vd=µ

Drift speed is the modulus of vector drift velocity

dd Aqn

t

tAqn

t

qI v

v=

∆

∆=

∆

∆=

Electric current and movement of charges

A characteristic of each conductor is the number of freeelectrons per unit of volume (n) (density of charge carriers),which depends on the chemical structure of the material. Ourhipothesys is that all the electrons are moving at once (like alog) inside the conductor. Obviously, I and Vd are relatedthrough n:

Vdt: distance covered by an electron during Δt

A: cross section of conductor

AVdt: volume crossing A during Δt

nqAVdt: charge crossing A during Δt

Current density (J) is the intensity of currentthrough a cross section of the conductor, dividedinto the section. This rate can be computed at anypoint of the cross section:

Density of electric current in homogeneous conductors.

dd nqv

A

nqAv

A

IJ === [J]=IL-2

I

is a vector with the same direction as at anypoint:J

E

EJ

σ= Omh’s Law

J

E

σ conductivity ≅ (Ωm)-1

Materials obeying Ohm’s law are called ohmic materials.

Tipler, chapter 25, part 25.2

Ohm’s Law

ρ=1/σ resistivity ≅ Ωm

Ω-1 = S (Siemens)

a bV V L

R RI A

ρ−= = =

Electric Resistance of a homogeneous conductor of constant cross section.

A

LIdr

A

Idr

A

Idr

JEdrrdEVV

b

a

b

a

b

a

ba

ρ

σσσ

b

a

b

a

======−

Tipler, chapter 25, part 25.2

E is always pointing tolower potentials (Va>Vb)

R is called electrical resistance ofconductor, depending on itsmaterial, shape and size. Ismeasured in Ohms

[R]=ML2T-1Q-2

Changes in resistivity depending on temperature

Tipler, chapter 25, section 25.2

The resistance of conductors increases with temperature dueto the higher number of collisions between free electronsand nuclei.

[ ]CT º( 20120

−+= αρρ

Combinations of resistors. Resistors in series.

=

=n

i

ieq RR1

Tipler, chapter 25, section 25.4

1 2 1 2( )

a cV V IR IR I R R− = + = +

a c eqV V IR− =

1 2eqR R R= +

=

=n

i ieq RR 1

11

Combinations of resistors. Resistors in parallel.

1 2

1 2 1 2

1 1( )( )a b a b

a b

V V V VI I I V V

R R R R

− −= + = + = − +

a b

eq

V VI

R

−=

1 2

1 1 1

eqR R R

= +