Current and resistance

Electric current

Flow of electrons



arc

Air is typically a poor conductor, but with a high enough potential difference, electricity can flow through air.



light bulb

Electric current is what makes the filament of a light bulb glow.



water current

Water current is simply how much water passes per second. Electrical current is similar; in this case, it is the flow of electric charges, measured in coloumbs per second.



current defined

One Ampere is defined as 1 Coulomb of charge per second.

One Coulomb is equal to the charge of about 6.25 billion billion electrons. (No, that wasn't a typo...)



Sample question



How fast are those electrons moving? In a typical household current:



   A. at the speed of light

   B. as fast as a supersonic jet

   C. as fast as a race car

   D. as fast as a person can run

   E. as fast as a snail crawls




Drift speed



The electromagnetic signal moves at the speed of light, but the electrons move much more slowly. We define the drift speed as the speed that the electrons move through the wire.



drift speed diagram

This diagram depicts electrons moving through a wire of cross-section A, from a time span from t to t + Δt.

The electrons undergo collisions and have some random motion, but move as a group at drift speed vd.



drift speed equations

We define the electron current as the number of electrons per time passing a cross-section.

We can also write the electron current in terms of the number density (electrons per volume) and the drift speed.



lattice

Recall our model of a conductor as a lattice of atoms, nuclei with electron shells. These atoms are ions, since the valence electrons are free to move within the lattice. The potential difference from a battery or charged capacitor sets up a nonuniform charge distribution. When the circuit is completed, the electrons have a pathway to flow, and move through the lattice.



model collision

While moving through the lattice, electrons undergo collisions with the ions in the metal. Valence electrons are free to move within the lattice. We can model a collision using this PHeT simulation.

drift speed equations

The electric field provides the force on the electrons to speed them up between collisions. Recall that the electric force can be defined as F = qE. Here we define the charge of an electron as e, to write F = eE.

We can use kinematics to write the equation of motion for electrons under the acceleration provided by the electric field.

drift speed equations

The average velocity is equal to the drift speed. The time between collisions is defined as τ.

We can use substitution to write the electron current in terms of the drift speed and τ.



drift speed equations

We can relate our electron current to the current I. The total charge of Ne electrons is just the charge of an electron times the number of electrons.

The electron current ie is the rate of electrons flowing through the wire.

The current I is the rate of charge flowing through the wire.

The direction of the current is (incorrectly) defined to be the direction of flow of positive charges.

The direction was defined by Benjamin Franklin, who assumed that the flow was caused by an excess (+) of one kind of charge toward a deficit (-) of charge. This idea was basically correct, but when electrons were later discovered and assigned to have negative charge, the sign convention was determined to be wrong, but kept for historical reasons.



Sample problem



Consider this simple circuit, with a charged capacitor and two light bulbs. When the switch is closed, electrons will flow through the wire. Which light bulb is brighter?

simple circuit

A. bulb A

B. bulb B

C. they will be equally bright



Conservation in electric current



Electrons flowing through a wire obey conservation laws. Electrons cannot be created or destroyed as they flow through a wire or a component of an electrical circuit like a light bulb or resistor. Each electron carries its fundamental charge, so the amount of charge flowing is also conserved.

Energy carried by the electrons is used up as they pass through the filament of a light bulb. Some of the energy given to the electrons by the battery or other power source is converted to thermal energy. The current is not used up. The rate of flow of electrons into the filament of a light bulb is the same as the rate of flow of electrons out of the bulb. Current is constant everywhere in a circuit, as long as there are no junctions involved.

junction rule

Current into a junction equals current out.