Current and Resistance

Alastair McLean

March 10, 2010

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1 Current and ResistanceLearning ObjectivesNew SymbolsCurrent densityDrift velocityElectrical ResistancePowerSummary

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Copyright

This work is licensed under a Creative CommonsAttribution-Noncommercial-Share Alike 3.0 Unported License.

http://creativecommons.org/licenses/by-nc-sa/3.0/

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Reading Assignment

Knight Chapter 31

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Current and Resistance Learning Objectives

To understand electric current and current density.

To understand Ohm’s law that relates the current flowing in amaterial to the potential difference across it.

To understand the concept of resistance and also how it relates toboth material and geometrical parameters.

To be able to calculate the power dissipated in a material whencurrent flows through it.

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Current and Resistance New Symbols

Table: New Symbols

I current A

J current density Am−2

σ conductivity 1/Ωm

ρ resistivity Ωm

R resistance Ω

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Current and Resistance New Symbols

In a conducting wire, mobile charge carriers will move if there is differencein potential between the two ends of the wire Vab. The flow of charge iscalled a current.

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Current and Resistance New Symbols

I

conducting wire

V

V

a

ba b

F e-

E

A

I =dQ

dtUnits : A (Ampères)

The current is the amount of mobile charge that flows through a fixedsurface (e.g. A on the Fig.) per unit time.

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Current and Resistance New Symbols

In a metal the mobile charge carriers are electrons and they will moveto the left because Fx = −eEx .An electron moving to the left with charge −e has the same effect asa positive charge +e moving to the right:

(−e)(−vx) = (+e)(+vx) = evx

and we defined current in terms of positive charge.

You can think in terms of positive charge; +e flowing to the right.This is called conventional current.

Discuss faucets and water with negative mass.

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Current and Resistance Current density

If we normalize the current to the cross sectional area of the wire, we havethe current density:

J ≡ IA

Units : Am−2

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Current and Resistance Drift velocity

There are a lot of electrons in conductors and they do not all travel at thesame speed. They have a range of speeds and they also don’t all move inthe same direction. They move in both directions (left and right).

The electrons that are moving most rapidly can be moving at ≈ c/100.

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Current and Resistance Drift velocity

The effect of the electric field is to change the average electron speed, ordrift velocity

vd =1

N

∑vi ,

from zero to some finite value.

Analogy: Traffic on the 401. Calculate the average speed by countingvehicles that pass a fixed point for an entire day. Then we will raise the401 near Toronto by 5 km (e.g. add a gravitational potential) andcalculate the average speed.

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Current and Resistance Drift velocity

I

Q

t=0 A

t=t

v tdE

I =Q

t=

(Avd t)ne

t= Anevd

Where n is the electron concentration. Therefore, the current density is:

J = nevd .

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Current and Resistance Drift velocity

Example 1. Calculate the drift velocity for electrons in a copper wire witha diameter of 1.0 mm if it is carrying a current of I = 1A. Theconcentration of electrons on copper - basically one per atom - is n = 1.1× 1029 m−3.

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Current and Resistance Electrical Resistance

The ratio of J to E is a constant for many materials and it is called theconductivity (σ). Materials with a high conductivity conduct well. Theinverse of the conductivity is called the resistivity (ρ).

J

E= σ =

1

ρ. Ohm′s Law

Units: σ 1/Ωm; ρ Ωm.

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Current and Resistance Electrical Resistance

Question: What can we say about the ratio V /I ?Starting from Ohm’s law:

J

E=

I

A

L

V=

1

ρ

we have

V

I= ρ

L

A≡ R, Second form of Ohm′s Law

where R is the resistance of the material. Notice that the resistancecontains both geometric (L and A) and materials (ρ) parameters:

R = ρL

A. Units : Ω (Ohms)

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Current and Resistance Electrical Resistance

Separating material parameters from geometrical parameters is a verysmart thing to do.

If we double the length of the wire, the resistance doubles.

If we double the area of the wire, the resistance halves.

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Current and Resistance Electrical Resistance

This is how our resistor would be represented in a circuit diagram:

I A

V

R

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Current and Resistance Power

Considering the mobile charge carriers in the wire.

P =I

eU = I

U

e= I V .

Using the second form of Ohm’s Law V = IR, we have two alternate forms

P = I 2R =V 2

R.

Units: W (Watts) = Js−1.

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Current and Resistance Power

Example 2. A 12 V car starter motor draws I = 200 A. (a) Calculate theamount of power dissipated and (b) calculate the energy dissipated in a 5s burst.

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Current and Resistance Summary

Current is the amount of mobile charge that flows through a fixedsurface per unit time. (I = dQ/dt)

Current density J = I/A

Ohm’s Law: σ = J/E = 1/ρ where σ is the conductivity and ρ is theresistivity.

Resistance R = V /I = ρL//A

Power P = IV = I 2R = V 2/R

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Current and ResistanceLearning ObjectivesNew SymbolsCurrent densityDrift velocityElectrical ResistancePowerSummary