19.1 Ohm's law

Electrical Current and Resistance

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Current vs charge over time

Electric current is the rate at which electric charge flows past a point in a circuit. Think of it like water flow: current is how much water passes through a pipe each second, not how much water is in the pipe total.

• Current ($I$) is measured in amperes (A) and calculated with $I = \frac{Q}{t}$, where $Q$ is charge in coulombs and $t$ is time in seconds.
• Charge ($Q$) is measured in coulombs (C). One coulomb equals the charge of roughly $6.24 \times 10^{18}$ electrons. That's a huge number of electrons for just one coulomb, which tells you individual electrons carry a tiny amount of charge.

There are two types of current you should know:

• Direct current (DC) flows in one constant direction. Batteries and solar cells produce DC, and it's what powers things like flashlights and phones.
• Alternating current (AC) periodically reverses direction. This is what comes out of wall outlets. AC is used for power grids because it can be transmitted over long distances much more efficiently than DC.

Resistance and Ohm's law

Resistance ($R$) measures how much a material opposes the flow of current through it. It's measured in ohms ($\Omega$). A higher resistance means less current gets through for a given voltage.

What determines resistance? Three physical factors:

• The resistivity of the material (copper has low resistivity; rubber has high resistivity)
• The length of the conductor (longer = more resistance)
• The cross-sectional area (thinner = more resistance)

Conductivity is the inverse of resistivity. A highly conductive material has low resistance.

Ohm's law ties together the three core quantities in a circuit:

$V = IR$

where $V$ is voltage in volts (V), $I$ is current in amperes (A), and $R$ is resistance in ohms ($\Omega$).

Voltage here is the potential difference between two points in a circuit. It's what "pushes" the current through the resistor. The relationship is linear: double the voltage across a fixed resistance and you double the current. This proportionality only holds for ohmic materials (materials where resistance stays constant regardless of voltage). Non-ohmic devices like diodes and lightbulb filaments don't follow this neat linear relationship.

Applications of Ohm's law

You can rearrange $V = IR$ to solve for any of the three variables. Here are the three forms with worked examples:

Finding current when voltage and resistance are known:

$I = \frac{V}{R}$

Example: A 12 V battery is connected to a 4 $\Omega$ resistor. The current is $I = \frac{12\text{ V}}{4\text{ }\Omega} = 3\text{ A}$.

Finding voltage when current and resistance are known:

$V = IR$

Example: A 2 A current flows through a 6 $\Omega$ resistor. The voltage across it is $V = 2\text{ A} \times 6\text{ }\Omega = 12\text{ V}$.

Finding resistance when voltage and current are known:

$R = \frac{V}{I}$

Example: A 9 V battery drives a 1.5 A current. The resistance is $R = \frac{9\text{ V}}{1.5\text{ A}} = 6\text{ }\Omega$.

In practice, Ohm's law is how you figure out what components a circuit needs. If you know the voltage of your power supply and the current your device requires, you can calculate the right resistor value to use.

Power and Electric Fields in Circuits

Electrical power is the rate at which energy is transferred in a circuit, measured in watts (W). The base formula is:

$P = VI$

By substituting Ohm's law into this equation, you get two additional forms that are useful depending on which quantities you know:

• $P = I^2R$ (useful when you know current and resistance but not voltage)
• $P = \frac{V^2}{R}$ (useful when you know voltage and resistance but not current)

All three formulas give the same answer for the same circuit; they're just rearrangements. For example, that 12 V battery pushing 3 A through a 4 $\Omega$ resistor dissipates $P = 12\text{ V} \times 3\text{ A} = 36\text{ W}$, and you can verify: $P = (3)^2 \times 4 = 36\text{ W}$.

Electric fields inside a circuit are what actually drive charge carriers (usually electrons) through the wire. The electric field points from high potential to low potential, and its strength is proportional to the voltage gradient (how quickly voltage changes over distance). A larger potential difference across a shorter conductor means a stronger electric field and more force on each charge carrier.