Most electrical devices are not electrostatic devices.  Most electrical devices require the flow of a current.
What is a current?
A current requires moving charges.  Let r₊ be the density of the positive charges in some region, i.e. the amount of positive charge per unit volume, and let r_- be the density of the negative charges.

r₊ = n₊q₊,      r_- = n_-q_-

Here n₊ and n_- are the number of positively and negatively charged particles per unit volume and q₊ and q_- are the charge of each positively and negatively charged particle, respectively.  In neutral ordinary matter r₊ + r_- = 0, i.e. the net charge per unit volume is zero.

The current density j is defined as

j = r₊<v₊> + r_-<v_->,

where <v₊> and <v_-> are the average velocities of the positive and negative charges, respectively.

The unit of current density is (C/m³)(m/s) = C/(m²s).  The current density is a vector.  It represents the amount of net charge that crosses a unit area perpendicular to the flow per second.  If just as many negative as positive charges move across a unit area in the same direction per second, then the current density is zero.  If <v_-> = 0 and only positive charges are moving, then the direction of the current density is the direction of the velocity of the positive charges.  In ordinary conductors the positive charges have zero average velocity and only the negative charges are moving.  Then the direction of the current density is opposite to the direction of the velocity of the negative charges, as shown in the diagram below.

The electric current I is the net amount of charge, which flows through a surface.

I = òj·dA = dQ/dt

The unit of current is Ampere = Coulomb/second (A = C/s).  The current I is not a vector.  It is a scalar, but it can be a negative or positive number.  When obtaining the current from the current density using I = òj·dA, we have to choose a direction for the unit vector n normal to the surface.  If the net flow of positive charge is in the direction of n, (or the net flow of negative charge is in a direction opposite to the direction of n), then the current I is positive.  If the reverse is true, then the current I is negative.

Assume that a steady current flows in a wire with cross-sectional area A.  The electrons move with constant average velocity to the left as shown in the diagram above.  The current density is a vector pointing towards the right.  Let the normal to the cross-sectional area point towards the right.  Then

I = òj·dA = jA = dQ/dt,

and the current is a positive number.
(If the electrons were moving towards the right, then j would point towards the left and I would be a negative number.)

The magnitude of the current I moving through a wire is given by

I = |r_-|<v_->A = n_-|q_-|<v_->A = n_ee<v_->A,

where n_e is the number of free electrons per unit volume.  The current I equals the number of electrons that pass any point along the wire per second times the unit of charge e.  <v-> is the drift speed of the electrons.  It is their average speed, as they move along the wire.

What causes the electrons to move in a wire?
Particles accelerate when they acted on by forces.  When a particle with charge q is placed in an electric field E, a force F = qE is acting on the particle and it accelerates.  To produce an electric field, something has to do work and separate charges.  Such a device is called an emf.  Examples of an emf are a battery or a power supply.  The emf separates charges by moving electrons internally from its positive to its negative terminal.  This produces an electric field, which points from the positive to the negative terminal outside and inside the emf.  Electrons in a wire placed in this electric field close to the battery will accelerate toward the end of the wire closest to the positive terminal.  But if the wire is not connected to the terminals of the emf, they will pile up at one end of the wire, leaving net positive charge at the other end.  The separated charges in the wire then produce their own electric field opposing the field produced by the emf.  The interior of the wire will be field free and charges will no longer accelerate.  If, however, the wire is connected to the emf, then the emf can pump the electrons through its interior back from the positive to the negative terminal.  Then field inside the wire will not be zero, and electrons will continue to accelerate towards the positive terminal.

Steady currents can only flow in continuous loops.  At any point, just as much charge has to flow out of a small volume surrounding the point as flows into the volume.  If this were not so, charge would accumulate at the point, setting up its own electric field.  This field would exert an additional force on the moving charges, disrupting the steady current.  The electric field in a homogeneous wire with constant cross-sectional area carrying a steady current is the same everywhere.  If it were not, electrons would move with different velocities in different sections, and charges would accumulate in certain regions.  The field produced by these charges would disrupt the steady current.  The diagram below shows the field in a wire carrying a steady current.

Even though an electric force is continuously acting on each electrons in a wire when a steady current is flowing, the electrons do not continue to accelerate indefinitely, but reach an average terminal velocity, called the drift velocity <v_->.  When the electrons move with the drift velocity, then the resistive forces (frictional forces) acting on them are equal in magnitude and opposite in direction to the electric forces.  Typical drift speeds are on the order of 1mm/s.

The resistive forces depend on the type of material the wire is made of.  The terminal velocity depends on the electric force and the resistive force.  The higher the terminal velocity, the larger is the current.  The current will increase if the voltage across the wire increases, (this increases the strength of the electric field), and decrease if the resistive forces increase in strength.  We write

I = DV/R,  or  R = DV/I.

This expression defines the resistance R of the wire.  The resistance is a measure of the resistive forces.  The unit of resistance is Ohm = Volt/Ampere (W = V/A).

The resistivity r of a material is the resistance of a piece of material with unit cross sectional area and unit length.  The resistance of a wire is proportional to its length l and inverse proportional to its cross-sectional area A.  We have

R = rl/A.

The unit of resistivity is Wm.  The conductivity s of a material is s = 1/r.

The resistivity is a property of the material a conductor is made of.  For many conductors it only weakly depends on the voltage DV across the conductor or the current I flowing through the conductor over a wide range of voltages and currents as long as the temperature of the material is held constant.  This is called Ohm's law.  R = DV/I is approximately constant for many conductors at constant temperature.  Ohm's law breaks down under extreme conditions for all conductors.  Many modern materials do not obey Ohms law under any circumstances.  (Such materials are called non-ohmic materials.)

The resistivity and therefore the resistance of most materials depends on the temperature.  Tables often list the resistivity at some reference temperature T₀, (for example at 20^oC), and then list a temperature coefficient a = (1/r₀)Dr/DT.  To find the resistance at some temperature T use r = r₀[1+a(T-T₀)].

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Problems:

Electrical energy and power

The amount of work done by an emf in moving an amount of charge dQ through a potential difference DV is dW = dQDV.  The potential energy gained by dQ is dU = dW.  The rate at which the emf does work is dW/dt = (dQ/dt)DV = IDV.  This is the power supplied by the emf.

P = IDV.

It provides the potential energy of the separated charges.  The electric field then does work on the separated charges, converting the potential energy into some other form.  For a current flowing in a wire with resistance R, the potential energy is converted into thermal energy through frictional forces.  The power dissipated by the wire is

P = IDV = I²R = (DV)²/R,

using our definition of resistance.

Problems:

The flashlight

A basic flashlight has four components, a battery, a bulb, a switch and a metal strip, that connects the other components together.  Electric current flows from the negative to the positive terminal of the battery through the battery, through the filament of the bulb, and through the switch and metal strip back to the negative terminal in one continuous circuit.  (The electrons which carry the current actually flow in a continuous circuit in the opposite direction.)

A typical alkaline battery pumps electrons until the potential difference between its positive and negative terminals is 1.5V.  Electrochemical reactions provide the energy that an alkaline battery converts into electrical energy. Powdered zinc attached to the negative terminal forms a zinc-oxide complex with a manganese dioxide paste attached the positive terminal.  The reaction process resembles controlled combustion.  In effect, the battery "burns" zinc to obtain the energy needed to pump electrons from the negative to the positive terminal.

Often two batteries are connected together in series, so that the positive terminal of one battery touches the negative terminal of the other.  Each battery pumps charges until its positive terminal is 1.5V above its negative terminal.  Therefore the chain’s positive terminal is 3.0V above its negative terminal.  The voltages of the two batteries add because each battery adds energy to the charges as it pumps them through its interior.  The more batteries are in series, the greater is the potential difference between the chain’s positive terminal and its negative terminal.  If one of the batteries in a chain is reversed, this battery will extract energy from the current passing through it.  If a chain has 3 batteries, 2 of them will add energy to the electrons while the one that is reversed will extracts energy from the electrons.  The potential difference across the chain then is only 1.5 V.

Rechargeable batteries can turn some of the energy they extract from a current flowing backwards through their interior back into chemical potential energy.  Normal alkaline batteries are not rechargeable.  They turn most of the energy they extract from the recharging current into thermal energy instead of chemical potential energy.  Non-rechargeable batteries may overheat and explode during recharging.

The bulb's filament is a coil of fine tungsten wire, which has a large resistance R.  The metal strips connecting the batteries and the bulb have negligible resistance.  As a current I = V/R flows through the circuit electrical energy is converted into thermal energy and light in the bulb.  The power output is P = IV.

A typical current for a flashlight with two batteries in series is 1.0A.  The resistance of the bulb determines the current.  If a 1.0A current flows through a flashlight with two batteries in series, the flashlight consumes 3W of power. The bulb has been designed to become white hot when it dissipates 3J of energy per second.  If the wrong bulb is put into a flashlight, it will dissipate too much or too little energy per second.  If it consumes too much power, will burn out quickly.

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