Electricity and Magnetism

Electric Circuits

CH 25: Current and Resistance

We have been discussing electrostatics. Stationary charges and the resulting forces, electric fields, electric potential and capacitance for these charged objects. We will now begin a discussion about charges that are moving.Charges can be forced to move from point A to point B through a certain region of space. The amount of charge that passes through that specified region within a certain period of time will define a flow rate for charge (similar to how we would describe the flow of a fluid). The flow rate of charge is more commonly called Current.

Current can be mathematically defined as the amount of charge per unit time.

Av

QI

t

DQ – amount of chargeDt – time intervalIAv – Average current

If we look at the instantaneous value of the current (a infinitesimally small time interval).

dQI

dt

I – instantaneous value of the currentdQ – infinitesimally small quantity of chargedt – infinitesimally small time interval

The current is measured in Amperes (or Amps). CI A

s

We know that electrons typically have much greater freedom of movement, but when we discuss current it is important to remember that current is defined as the flow of positive charge.

Moving charges will be traveling at some velocity, therefore there must be a way to relate the current to the speed of the charges.

x

We can examine the rate at which a certain amount of charge will move through a specified volume.

Volume charges are moving through

The total amount of charge that is within that volume can be determined from the number density of charge carriers, the volume containing the charge carriers and the amount of charge on each charge carrier.

Q Nq nVq

Number density of charge carriers

nqAx Q nqA x

Amount of charge in the specified volume.

The amount of charge contained in the specified volume can be used to determine the rate at which charge flows through that region.

Av

QI

t

nqA x

t

xnqA

t

dnqA v

Drift velocity – Average speed charges move through material

The average current is proportional to the average speed at which the charges move.It is sometimes more convenient to discuss the current that passes through a specified area, called the Current Density.

Avd

IJ nqv

A

I

JA

J – Current Density [A/m2]I – Current [A]A – Cross-Sectional Area [m2]n – Carrier Density [1/m3]q – Charge [C]vd – Drift Velocity [m/s]

This microscopic view of current is used in certain situations, but most of the time is not necessary.

• Used by solid state, quantum and nuclear physicists to look at the motion of small groups of charges.

• Used by chemists to look at charges moving through a variety of mediums in a variety of shapes.

Charges do not move through materials in straight lines. They follow a random path that is defined through the interactions (e.g. collisions) of that charge with all other charges in the material, as well as any external influences. The net effect is for the charge to drift in the direction of current flow. The drift velocity for charges tends to be relatively slow as compared to the observable effects (vd for copper wire is approximately 0.2 mm/s).

If you turn on a light by flipping a switch how long does it take for the light to turn on?

The light turns on nearly instantaneously.

This means that there must be some other effects that are responsible for the light turning on. What causes the charges to move through the wire in the first place?

• When the switch is flipped the charges begin moving in the wire. • The charges must have undergone an acceleration.• A force must be applied to the charges to cause an acceleration.

Where does the force come from?• The presence of an external electric field will exert a force on the charges.• The external electric field travels through the wires from the positive plate to the negative plate of the power supply (battery), pushing positive charges towards the negative plate.

We should therefore be able to relate the electric field from the power supply to the current in the wires.

F qE ma q

a Em

0v v at q

E tm

d

qv v E

m

dJ nqv q

nq Em

2nqJ E

m

Acceleration of a charge due to an electric field.

t is the average time between collisions.

2nq

m

s - Conductivity

Conductivity is how easily charges move through a material. It is a property of the material.

J E

Ohm’s Law

It is sometimes more convenient to look at the potential difference instead of the electric field. Let us look at how this expression changes.

The “collisions” between charges is the primary reason that the drift speed is so slow. These collisions hinder the motion of the charges through the wire. This impedance to the flow of charge through a material is called Resistance.

Resistance is essentially a decrease in the amount of current that can pass through a material.

V Ed E V

E

J E I V

A

1V I

A

1/s = r - Resistivity

Resistivity is how hard it is to move charges through a material.Resistance is an impedance to the flow of charge.

R - Resistance

RA

R – Resistance [W]

r – Resistivity [Wm] l – Length of material [m]A – Cross-sectional area of material [m2]

V IR Ohm’s Law

VR

A

Ohm’s law is only valid for materials where there exists a linear relationship between voltage and current. The depends primarily on the properties of the material.If you apply a voltage across an electrolyte (conducting liquid) the current is exponentially related to the voltage. Ohm’s law is not valid for this case.

VI eExample: A section of copper wire that is 2 m long with a radius of 0.5 mm has a potential

difference of 12 V applied across the two ends. (rcu = 1.7x10-8 Wm)a) How much current is flowing through this section of wire?b) What is the current density contained in the wire?c) What is the magnitude of the electric field traveling through the wire?

a) V IR

RA

} V VA

IR

277A

This is an extremely large current and could not be supplied by most power supplies.

b) IJ

A 8

23.53 10

Ax

m

c)J E

1E

6

V

m

A very small electric field is necessary to set up this large current within a very good conducting material.

1 A is typically considered a moderately high current10 A is considered a very high current

JE