Chapter 27

Current and Resistance

Introduction to chapter 25

• In this chapter we will start the study of charges in motion.

• ELECTRIC CURRENT consists of charges in motion from one region to another

• ELECTRIC CIRCUIT is a closed loop conducting path through which charges (i.e. el. current) move.

• In electric circuits, ENERGY is conveyed from one place to another:• Conveying energy of a battery to emit light in flashlight• Conveying energy from power lines to power heater and produce heat • Elevator’s motion, computers, bulbs, home appliances, cell phones,

nerve cells

Average Electric CurrentAssume charges are moving perpendicular to a surface of area A.

If ΔQ is the amount of charge that passes through A in time Δt, then the average current is

avg

Q

t

I

Electric Current

Electric current is the rate of flow of charge through some region of space.

The SI unit of current is the ampere (A).

1 A = 1 C / s (The symbol for electric current is I.)

Instantaneous Electric CurrentIf the rate at which the charge flows varies with time, the instantaneous current, I, is defined as the differential limit of average current as Δt→0. dQ

dtI

Current

E = 0 in a conductor in electrostatic equilibrium

Free electric charge (electrons) move randomly in all directions (sharp changes due to collisions) no net flow of charge in any directions NO CURRENT

If static electric field E is established in the conductor, all free charges (electrons) move under the force

There are collisions with massive ions that slow them down.

As a result, there is net motion or DRIFT of charges in the direction of force NET CURRENT IN CONDUCTOR

vd is an average speed called the drift speed.

¡!F = q¢

¡!E

Current is the net charge moving through the area per unit time.

I = dQdt = [1A = 1C

s ]Ampere

v= 106ms

vd =10¡ 4ms

Direction of Current

The charged particles passing through the surface could be positive, negative or both.

Conventionally, direction of current is direction of flow of positive charge.

This is regardless of the sign of the actual charged particles in motion.

It is common to refer to any moving charge as a charge carrier.

EXAMPLE: In a common conductor (like copper), the current is due to the motion of negatively charged electrons. In this case direction of current is opposite to the direction of flow of electrons.

Direction of current is always from the higher potential to lower, the same as the direction of electric field E.

Consider positive and negative charges moving horizontally through the four regions shown in Figure 27.4. Rank the current in these four regions from highest to lowest.

Figure 27.4

1. (b) > (c) >(d) > (a)

2. (c) > (b) > (a) > (d)

3. (d) > (b) = (c) > (a)

4. (a) > (b) = (c) > (d)

Current and Motion of Charged Particles

n – density of charge carriers

|q| - amount of charge carried by charge in motion

vd – drift velocity of moving charges

A – cross-sectional area of conductor

j – density of current

I = n ¢jqj ¢vd ¢A

j = IA

Problem 27.4.

In the Bohr model of the hydrogen atom, an electron in the lowest energy state moves at a speed of 2.19e6 m/s in a circular path of radius 5.29e-11 m. What is the effective current associated with this orbiting electron?

Conductivity

A current density j and an electric field are established in a conductor whenever a potential difference is maintained across the conductor.

For some materials, the current density is directly proportional to the field.

The constant of proportionality, σ, is called the conductivity of the conductor.

The inverse of the conductivity is the resistivity:

ρ = 1 / σ

Resistivity has SI units of ohm-meters (Ω . m)

Resistance

When charges (electrons) move in a conductor under the influence of electric field E (i.e. current), they collide with positive (nearly fixed) ions, that slow them down. This property is called RESITISTIVITY .

Resistivity depends on the properties of material, and it is A CONSTANT for a given material.

In a given conductor of cross-sectional area A and length L, the resistance to current is defined as:

R = ½¢LA = [1 ] (ohm)

r - electric resistivity of conducting materialL – length of conductorA – cross-section of conductor

IMPORTANT: the direction of current is ALWAYS from higher potential end to lower potential end E field points in the direction of decreasing electric potential!

A cylindrical wire has a radius r and length ℓ. If both r and ℓ are doubled, the resistance of the wire:

1. increases. 2. decreases. 3. remains the same.

Resistors

Most electric circuits use circuit elements called resistors to control the current in the various parts of the circuit.

Stand-alone resistors are widely used.

Resistors can be built into integrated circuit chips.

Values of resistors are normally indicated by colored bands.

The first two bands give the first two digits in the resistance value.

The third band represents the power of ten for the multiplier band.

The last band is the tolerance.Section 27.2

Resistor Color Code Example

Red (=2) and blue (=6) give the first two digits: 26

Green (=5) gives the power of ten in the multiplier: 105

The value of the resistor then is 26 x 105 Ω (or 2.6 MΩ)

The tolerance is 10% (silver = 10%) or 2.6 x 105 Ω

Section 27.2

Resistivity Values

Section 27.2

Georg Simon Ohm

1789 -1854

German physicist

Formulated idea of resistance

Discovered the proportionalities now known as forms of Ohm’s Law

Section 27.2

Ohm’s Law

For large class of materials, voltage V is directly proportional to the current I:

Ohm’s law is only valid for certain materials whose resistance DOES NOT depend on voltage applied.

A circuit device with specific value of resistance R between ends is called RESISTOR.

V = I ¢R

Resistance and Resistivity, Summary

Every ohmic material has a characteristic resistivity that depends on the properties of the material and on temperature.

Resistivity is a property of substances.

The resistance of a material depends on its geometry and its resistivity.

Resistance is a property of an object.

An ideal conductor would have zero resistivity.

An ideal insulator would have infinite resistivity.

Section 27.2

Electrical Power

Assume a circuit as shown

The entire circuit is the system.

As a charge moves from a to b, the electric potential energy of the system increases by QDV.

The chemical energy in the battery must decrease by this same amount.

This electric potential energy is transformed into internal energy in the resistor.

Corresponds to increased vibrational motion of the atoms in the resistor

Section 27.6

Electric Power, 2

The resistor is normally in contact with the air, so its increased temperature will result in a transfer of energy by heat into the air.

The resistor also emits thermal radiation.

After some time interval, the resistor reaches a constant temperature.

The input of energy from the battery is balanced by the output of energy by heat and radiation.

The rate at which the system’s potential energy decreases as the charge passes through the resistor is equal to the rate at which the system gains internal energy in the resistor.

The power is the rate at which the energy is delivered to the resistor.

Energy and power in Electric Circuits

If charge q passes through a circuit element from higher potential Vb to lower potential Va, its electric potential energy decreases.

The time rate at which energy is either delivered or extracted from circuit element is called POWER.

If the circuit element has fixed resistance R (like a resistor):

Units: I is in A, R is in Ω, ΔV is in V, and P is in W

P = Vba ¢I

P = Vba ¢I = I 2 ¢R = V 2baR

Vba =Vcd

For the two light bulbs shown in Figure 27.13, rank the current values at points a through f from greatest to least.

Figure 27.13

1. Ia = Ib > Ic = Id > Ie = If

2. Ic > Id > Ie > If > Ia > Ib

3. Ic = Id > Ia = Ib > Ie > If

4. Ia = Ib > Ic > Id > Ie > If

Electric Power Transmission

Real power lines have resistance.

Power companies transmit electricity at high voltages and low currents to minimize power losses.

Problem 27.37.

An 11 W energy efficient fluorescent light bulb is designed to produce the same illumination as a conventional 40 W incandescent light bulb. Assuming a cost of $0.11 /kWh for energy from the electric company, how much money does the user of the energy-efficient bulb save during 100 h of use?

Example 27.1.

The 12-gauge copper wire in a typical residential building has a cross-sectional area of 3.31e-6 m2. It carries a constant current of 10 A. What is the drift speed of the electrons in the wire? Assume each copper atom contributes one free electron to the current. The density of copper is 8.92 g/cm3.