Electrical Current and Charge Movement

Electrical current flows from high to low potential due to a field pushing charge.
Potential difference can be supplied by generators, capacitors, or batteries.

Think of electrical current like a hydrologist thinks about water flow.
Mississippi River (large flow) vs. a garden hose (smaller flow).
The river has a larger flow of water but not necessarily a higher velocity.
The garden hose might have a higher velocity but lower overall flow.
Electrical current is not the speed of electrical charges, just like water flow isn't the speed of water molecules.
Current is about the total amount of charge passing a location per unit of time.

Electrical current is defined as the total amount of charge ( $\Delta Q$ ) that passes a location divided by the total time ( $\Delta t$ ) it takes:
$I = \frac{\Delta Q}{\Delta t}$
The Mississippi River has a larger cross-sectional area, allowing more charge to pass in the same amount of time.
A garden hose has a smaller cross-sectional area, restricting the amount of water (charge) that can pass through.

The concept of current was developed before the understanding of atomic structure.
Consider a conducting object exposed to an electrical field, becoming positive on one end and negative on the other.
Upon removing the field, the object becomes neutral.
Reversing the field reverses the polarity of the object.
In reality, electrons move to create these charges.

Historically, scientists didn't know about electrons and protons.
They arbitrarily decided that positive charges were moving.
This led to the concept of conventional current, where current is defined as the direction that positive charge seems to flow.
Positive charges (protons) do not actually move in solid conductors.
Conventional current is an artifact of history.
The direction of conventional current (I) is opposite to the actual flow of electrons in a wire.

It's useful to be able to calculate the actual speed of charges.
Drift current refers to the actual speed of charges.
Consider a cylindrical wire of cross-sectional area $A$ , where charges move with a drift velocity $v_d$ .

Charge density (n) is the total number of charges divided by the volume (V): $n = \frac{N}{V}$ .
The total number of charges can also be calculated as the total charge (Q) divided by the charge of a single carrier (q): $N = \frac{Q}{q}$ .
Volume of a small cylinder section of wire: $V = A \Delta l$
Therefore, $n = \frac{Q}{q} / (A \Delta l)$ .
Rearranging, $nq = \frac{Q}{A \Delta l}$ . This relates charge density and single charge to total charge.

$I = \frac{\Delta Q}{\Delta t}$
Multiply by $A \Delta l$ divided by $A \Delta l$ : $I = \frac{\Delta Q}{A \Delta l} \cdot \frac{A \Delta l}{\Delta t}$
$\frac{\Delta Q}{A \Delta l}$ is the charge per volume, so $\frac{\Delta Q}{A \Delta l} = nq$
$\frac{\Delta l}{\Delta t}$ is the drift velocity, or $v_d$
Therefore $I = nqAv_d$

The equation $I = nqAv_d$ describes the current as density multiplied by single charge, area, and drift velocity.
Using typical values for copper wires (1-2 mm diameter), drift velocities are very slow (less than 1 mm/s).

If electron drift velocity is so slow, how do lights turn on instantly when a switch is flipped?

Water flows from a pipe, through a hose, and out a nozzle.
Initially, turning on a valve results in a delay before water comes out, due to the time it takes for water to travel the length of the hose.
However, if the hose is already full of water, turning the valve on results in an immediate flow.

Electrons are like the water in the hose.
When a light switch is flipped, the electrons are already present throughout the circuit.
The flow of electrons starts immediately because the