12. Ohm and resistance

Biologist: Engineer: Physicist:

𝑅 ≡𝑉

𝐼

Who will win the superbowl?

A. Patriots

B. Rams

vs.

𝐼 =∆𝑄

∆𝑡= ሶ𝑄

𝐼 =𝑉

𝑅=𝐴𝑉

𝜌𝐿=𝜎𝐴𝑉

𝐿= 𝐺𝑉

𝐼 = 𝑗𝐴

Ԧ𝐼 = 𝐴Ԧ𝑗 = 𝐴𝜎𝐸 = 𝐴𝑛𝑒𝑣𝑑 ⇒ 𝜎 = 𝑛𝑒𝑣𝑑𝐸

Ohm summary

IV demo

𝐼~𝑉

Ohm’s results:

𝐼~𝐴

𝐼~1/𝑳

Current proportional to voltage

Current proportional to area of wire

Current inversely proportional to length of wire (keep this

explanation for later -> now)

When is the current not inversely proportional to the length?

A. When the conductor is a very pure metal

B. When the conductor is an insulator

C. When the conductor is ballistic

D. When the conductor is very hot

E. When the conductor is vey cold

No scattering: 𝐼~1/𝑳

With a lot of scattering: 𝐼~1/𝑳

(no bumps)

(many bumps)

A typical metal has a lot of scattering => 𝐼~1/𝑳

What is the main source of scattering in a pure metal at room

temperature

A. Electrons scatter off the vibrations of the

atoms

B. Electrons scatter off each other

C. Electrons scatter off the periodic

structure of the atoms

D. Electrons scatter off the surface of the

conductor

E. Electrons scatter off defects in the

conductor

Electrical Conduction – Drude Model (a lot of scattering)

Treat a conductor as an assembly of atoms plus a collection of free electrons.

▪ The free electrons are often called conduction electrons. They are only weakly attached to their host atom.

▪ These electrons become free when the atoms are bound in the solid.

In the absence of an electric field, the motion of the conduction electrons is random.

▪ Their speed is on the order of 106 m/s.

Section 27.3

Drude Model, 2

When an electric field is applied, the conduction electrons are given a drift velocity.

Assumptions:

▪ The electron’s motion after a collision is independent of its motion before the collision.

▪ The excess energy acquired by the electrons in the electric field is transferred to the atoms of the conductor when the electrons and atoms collide, which induces vibrations in the atoms.

▪ This causes the temperature of the conductor to increase.

Section 27.3

𝐼 =∆𝑄

∆𝑡= ሶ𝑄

𝐼 =𝑉

𝑅=𝐴𝑉

𝜌𝐿=𝜎𝐴𝑉

𝐿= 𝐺𝑉

𝐼 = 𝑗𝐴

Ԧ𝐼 = 𝐴Ԧ𝑗 = 𝐴𝜎𝐸 = 𝐴𝑛𝑒𝑣𝑑 ⇒ 𝜎 = 𝑛𝑒𝑣𝑑𝐸

Ohm summary

We want to find

the value of 𝒗𝒅

Drude Model – Calculating the Drift Velocity

The force experienced by an electron is

From Newton’s Second Law, the acceleration is

Applying a motion equation

▪ Since the initial velocities are random, their average value is zero.

q=F E

𝐚 =Ԧ𝐅

𝑚=𝑞𝐄

𝑚𝑒

f i f i

e

q= + t or = + t

m

Ev v a v v

Section 27.3

Drude Model, 4

Let t be the average time interval between successive collisions.

The average value of the final velocity is the drift velocity.

This is also related to the current density: J = nqvd = (nq2E / me)t

▪ n is the number of charge carriers per unit volume.

,f avg d

e

q

mt= =

Ev v

Section 27.3

Drude Model

Using Ohm’s Law, expressions for the conductivity and resistivity of a conductor can be found:

Note, according to this classical model, the conductivity and the resistivity do not depend on the strength of the field.

▪ This feature is characteristic of a conductor obeying Ohm’s Law.

2

2

1 e

e

mnq

m nq

t

t= = =

Section 27.3

IV demo

𝐼~𝑉

Ohm’s results:

𝐼~𝐴

𝐼~1/𝑳

Current proportional to voltage (let’s examine this!)

Current proportional to area of wire

Current inversely proportional to length of wire (only with

lots of scattering – typical in metal wires)

Demo summary:

For a resistor: 𝐼~𝑉

(Or any Ohmic device,

including a human)

For a diode: 𝐼~𝑉

(Or any non-Ohmic device)

Resistance and Temperature

Over a limited temperature range, the resistivity of a conductor varies approximately linearly with the temperature.

▪ ρo is the resistivity at some reference temperature To

▪ To is usually taken to be 20° C

▪ α is the temperature coefficient of resistivity

▪ SI units of α are oC-1

The temperature coefficient of resistivity can be expressed as

[1 ( )]o oρ ρ α T T= + −

1

o T

=

Section 27.4

Temperature Variation of Resistance (like heating the filament of a light

bulb)

Since the resistance of a conductor with uniform cross sectional area is proportional to the resistivity, you can find the effect of temperature on resistance.

R = Ro[1 + α(T - To)]

Use of this property enables precise temperature measurements through careful monitoring of the resistance of a probe made from a particular material.

Section 27.4

=> A light bulb is non-ohmic because the current heats the

filament, which increases the resistance of the filament the

larger the current.

Resistivity and Temperature, Graphical View

For some metals, the resistivity is nearly

proportional to the temperature.

A nonlinear region always exists at very

low temperatures.

The resistivity usually reaches some finite

value as the temperature approaches

absolute zero.

Section 27.4

Residual Resistivity

The residual resistivity near absolute zero is caused primarily by the collisions of

electrons with impurities and imperfections in the metal and collisions between

electrons.

High temperature resistivity is predominantly characterized by collisions between

the electrons and the vibrations of the metal atoms.

▪ This is the linear range on the graph.

Section 27.4