Joule

The joule ($\mathrm{J}$) is the derived SI unit for Energy and Work.

One joule is the energy transferred when a force of one Newton moves an object one Metre in the direction of the Force:

$$1\mathrm{J}=1\mathrm{N}\mathrm{m}$$

Since $1\mathrm{N}=1\mathrm{kg}\mathrm{m}{\mathrm{s}}^{-2}$,

$$1\mathrm{J}=1\mathrm{kg}{\mathrm{m}}^2{\mathrm{s}}^{-2}$$

Read that as “mass times speed squared”. That is why kinetic energy has the form $E_k=\frac{1}{2}m{v}^2$.

Useful equivalent definitions:

• Electrical: moving one Coulomb of Charge through one Volt transfers one joule.

$$1\mathrm{J}=1\mathrm{C}\mathrm{V}$$

• Power over time: one Watt sustained for one Second transfers one joule.

$$1\mathrm{J}=1\mathrm{W}\mathrm{s}$$

• Heating in a resistor follows from power:

$$E=Pt=I^{2}Rt$$

Andrew-friendly check: joules are about “how much changed”. Watt is “how fast the change is happening”. A $100\mathrm{W}$ device uses $100\mathrm{J}$ every second.

Common mistake: newton-metres can also appear as torque units, but torque is not energy unless there is angular displacement.

Quick check

A joule is also a newton metre:

$$1\mathrm{J}=1\mathrm{N}\mathrm{m}.$$

That does not mean torque and energy are the same concept, even though they share dimensions. Context matters: work/energy comes from force through distance, while torque comes from force at a lever arm.