Chapter 27

Current and Resistance

Intro

• Up until now, our study of electricity has been focused Electrostatics (charges at equilibrium conditions).

• We will now look at non-equilibrium conditions, and we will define electric current as the rate of flow of charge.

• We will look at current at the microscopic levels and investigate factors oppose current as well.

27.1 Electric Current

• Current- any net flow of charge through some region. – A similar analogy would be water current, or the

volume of water flowing past a given point per unit time (shower heads, rivers etc.)

• The rate of charge passingperpendicularly through a given area.

27.1

• The average current

• The instantaneous current

• The SI unit of current is the Ampere (A)

t

QIavg

dt

dQI

s 1

C 1A 1

27.1

• Current Direction-– Traditional- in the direction the flow of positive

charge carriers.– Conducting Circuits- Electrons are the flowing

charge, current is in the opposite direction of the flow of negative charge carriers (electrons).

– Particle Accelerator- with the beam of positive charges

– Gases and Electrolytes- the result of both positive and negative flowing charge carriers.

27.1

• At the microscopic level we can relate the current, to the motion of the charge carriers.– The charge that passes through a given region of

area A and length Δx is

– Where n is the number of charge carriers per unit volume and q is the charge carried by each.

qxnAQ

27.1

27.1

• If the carriers move with a speed of vd, (drift velocity) such that

and

• So the passing charge is also given as

t

xvd

tvx d

qtnAvQ d

27.1

• If we divide both sides by time we get another expression for average current

Anqvt

QI davg

27.1

• Drift Velocity- – Charge carrier: electron– The net velocity will be in the opposite direction of

the E-field created by the battery

27.1

• We can think of the collisions as a sort of internal friction, opposing the motion of the electrons.

• The energy transferred via collision increases the Avg Kinetic Energy, and therefore temperature.

• Quick Quiz p 834• Example 27.1

27.2 Resistance

• E-Field in a conductor = 0 when at equilibrium≠ 0 under a potential difference

• Consider a conductor of cross-sectional area A, carrying a current I.

• We can define a new term called current density

• Units A/m2 dnqA

IvJ

27.2

• Because this current density arises from a potential difference across, and therefore an E-field within the conductor we often see

• Many conductors exhibit a Current density directly proportional to the E-field.

• The constant of proportionality σ, is called the “conductivity”

EJ

27.2

• This relationship is known as Ohm’s Law.• Not all materials follow Ohm’s Law– Ohmic- most conductors/metals– Nonohmic- material/device does not have a linear

relationship between E and J.

27.2

• From this expression we can create the more practical version of Ohm’s Law

• Consider a conductor of length l

EV

V

EJ

27.2

• So the voltage equals

• The term l/σA will be defined as the resistance R, measured in ohms (1 Ω = 1 Volt/Amp)

IRA

IJ

V

I

VR

IRV

27.2

• We will define the inverse of the conductivity (σ) as the resistivity (ρ) and is unique for each ohmic material.

• The resistance for a given ohmic conductor can be calculated

AR

27.2

• Resistors are very common circuit elements used to control current levels.

• Color Code

27.2

• Quick Quizzes, p. 838-839• Examples 27.2-27.4

27.4 Resistance and Temperature

• Over a limited temperature range, resistivity, and therefore resistance vary linearly with temperature.

• Where ρ is the resistivity at temperature T (in oC), ρo is the resistivity at temperature To, and α is the temperature coefficient of resistivity.

• See table 27.1 pg 837

oo TT 1

27.4

• Since Resistance is proportional to resistivity we can also use

oo TTRR 1

27.4

• For most conducting metals the resistivity varies linearly over a wide range of temperatures.• There is a nonlinear region as T approaches absolute zero where the resitivity will reach a finite value.

27.4

• There are a few materials who have negative temperature coefficients

• Semiconductors will decrease in resistivity with increasing temps. • The charge carrier density increases with temp.

27.4

• Quick Quiz p 843• Example 27.6

27.5 Superconductors

• Superconductors- a class of metals and compounds whose resistance drops to zero below a certain temperature, Tc.

• The material often acts like a normal conductor above Tc, but falls of to zero, below Tc.

27.5

27.5

• There are basically two recognized types of superconductors– Metals very low Tc.– Ceramics much higher Tc.

27.5

• Electric Current is known to continue in a superconducting loop for YEARS after the applied potential difference is removed, with no sign of decay.

• Applications: Superconducting Magnets (used in MRI)

27.6 Electrical Power

• When a battery is used to establish a current through a circuit, there is a constant transformation of energy.– Chemical Kinetic Internal (inc. temp)

• In a typical circuit, energy is transferred from a source (battery) and a device or load (resistor, light bulb, etc.)

27.6

• Follow a quanity of charge Q through the circuit below.

• As the charge moves from a to b, the electric potential energy increase by U = QΔV, while the chemical potential energy decrease by the same amount.

27.6

• As the charge moves through the resistor, thesystem loses this potential energy due to the collisions occuring within the resistor. (Internal/Temp)• We neglect the resistance in the wires and assume that any energy lost between bc and da is zero.

27.6

• This energy is then lost to the surroundings.• The rate at which the system energy is

delivered is given by

• Power the rate at which the battery delivers energy to the resistor.

VIVdt

dQVQ

dt

d

dt

dU

VIP

27.6

• Applying the practical version of Ohm’s Law (ΔV = IR) we can also describe the rate at which energy is dissipated by the resistor.

• When I is in Amps, V is in Volts, and R is in Ohms, power will be measured in Watts.

R

VRIP

22

27.6

• Quick Quizzes p. 847• Examples 27.7-27.9