Charge, Current, and Voltage

Voltage as Energy per Charge

the battery's budget

Voltage is the energy a battery gives to each coulomb of charge. Divide the energy delivered by the charge moved and you get volts.

Example

Voltage is the energy a battery gives to each coulomb of charge. Divide the energy delivered by the charge moved and you get volts. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

A battery is an energy source

A battery does not make charge. It takes the charge already in the wire and gives each bit of it a fixed amount of energy, pushing it around the loop. That energy-per-charge is what we call voltage.

battery: energy in  →  charge pushed round\text{battery: energy in} \;\rightarrow\; \text{charge pushed round}
A battery gives each bit of charge its energyA loop with a battery and a resistor completing the path; plus and minus signs mark where the voltage is measured across the battery.R+-

Voltage is energy divided by charge

Voltage is the energy delivered for each coulomb of charge that passes. Joules per coulomb is exactly a volt.

V=EQ(volt=joule per coulomb)V = \frac{E}{Q} \quad (\text{volt} = \text{joule per coulomb})

Fixed charge turns energy into voltage

Hold the charge at 12 coulombs. More energy spread across the same charge means more energy per coulomb, so the voltage rises.

EQV12 J12 C1 V24 J12 C2 V36 J12 C3 V\begin{array}{c|c|c}E & Q & V \\12\ \text{J} & 12\ \text{C} & 1\ \text{V} \\24\ \text{J} & 12\ \text{C} & 2\ \text{V} \\36\ \text{J} & 12\ \text{C} & 3\ \text{V}\end{array}
A battery gives each bit of charge its energyA loop with a battery and a resistor completing the path; plus and minus signs mark where the voltage is measured across the battery.R+-

Fixed energy spread over more charge lowers voltage

Now keep the energy at 24 joules and share it among more coulombs. Each coulomb gets a smaller share, so the volts go down as the charge column grows.

EQV24 J6 C4 V24 J12 C2 V24 J24 C1 V\begin{array}{c|c|c}E & Q & V \\24\ \text{J} & 6\ \text{C} & 4\ \text{V} \\24\ \text{J} & 12\ \text{C} & 2\ \text{V} \\24\ \text{J} & 24\ \text{C} & 1\ \text{V}\end{array}
A battery gives each bit of charge its energyA loop with a battery and a resistor completing the path; plus and minus signs mark where the voltage is measured across the battery.2 VR+-

Substitute the energy and the charge

A battery delivers 24 joules of energy to 12 coulombs of charge as it passes through.

V=24 J12 CV = \frac{24\ \text{J}}{12\ \text{C}}

Compute the voltage

Dividing gives 2 joules for every coulomb, which is 2 volts. So each coulomb this battery pushes out carries that much energy: it is a 2 volt battery.

V=2 V=JCV = \hl{2}\ \text{V} = \frac{\text{J}}{\text{C}}
A battery gives each bit of charge its energyA loop with a battery and a resistor completing the path; plus and minus signs mark where the voltage is measured across the battery.2 VR+-
electricity A battery delivering 24 J of energy to 12 C of charge gives 2 J per coulomb, which is exactly 2 V: a volt is a joule per coulomb.