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Current and Resistance

Electric Current

Current is moving charge. Electrons which are free to move in a conductor move randomly at speeds of the order of one-thousandth that of light ($10^5$ m $s^{-1}$). But this random motion does not constitute a current. Only when a potential is placed across a length of conductor do the electrons flow in the same direction to form a current.

The electric current (unit amp A) is defined as the amount of charge (unit coulomb C) which moves through a cross-sectional area of a wire per second:

$$I = {ΔQ}/{Δt}$$

where I is the electric current, ΔQ the amount of charge flowing in time Δt.

Circuits

An electrical circuit is created when the two poles of a voltage differential are electrically connected.

A circuit can be composed of any number of electrical devices connected by wires, which makes a complete sequence, with no gaps, and returns to the same point. Circuits need a source of electrical energy, such as a battery. When electrons enter the circuit from the negative electrode of a battery, they travel around the circuit and return to the positive electrode of the battery.

An electrical circuit may be very short, such as when a battery is 'shorted' by a wire connecting the two electrodes directly, or very long, such as when power is generated in a power station and transported hundreds or thousands of kilometers to a household.

Circuit
Circuits have a power source, a continuous conductive path. The electrons travel in the opposite direction to the current.

An example is a battery with wires passing to a lamp. Electrons pass from the negative electrode of the battery along a wire to one pole of the lamp. The electrons then enter the lamp, pass through the filament, exciting it to emit light and heat, then pass out of the lamp via the other pole. The electrons then continue back to the battery, entering it at the positive electrode.

Electric Resistance

When a voltage is applied across a length of conductor, the free electrons which were moving randomly are accelerated in a single direction, parallel to and in the opposite direction to the potential difference. As the electrons gain speed, the frequency of their inelastic collisions with the atoms of the conductor increases.

There is therefore a transfer of kinetic energy of the electrons to the atoms, which gain energy in the form of vibration, which manifests macroscopically as heat. A current through a wire warms the wire. A wire of high resistance will glow. That is why an incandescent bulb creates both heat and light.

The resistance is proportional the voltage and inversely proportional to the current:

$$R = V/I$$

where V is the voltage difference across a resistance R, and I the resulting current. The unit is volt per amp, which is defined as the ohm (Ω).

Ohm's Law
$$I ∝ V$$

The current I is proportional to the voltage V in a metallic conductor, at constant temperature.

If you think of current as being like water passing through a pipe, then the resistance can be seen to decrease the wider the pipe, and increase the longer the pipe.

$$R ∝ L/A$$

The resistance R of a length of conductor L, with cross-section area A, and resistivity ρ, is:

$$R = {L⋅ ρ}/A$$

Electric Power

As a current flows through a circuit with resistance, power is dissipated as $P = {work}/{time} = {VΔQ}/{Δt} = VI$.

$$P = VI$$

Since $R = V/I$, $P = RI^2 = {V^2}/R$

$$P = RI^2$$

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