Visual walkthrough — Efficiency

We will follow one machine the whole way: a box of energy going in at the top, splitting into a useful part and a wasted part. Watch the water level, not the algebra.

Step 1 — Draw the energy going in

WHAT. Picture a single tall bucket. We pour a fixed amount of energy into it. Let us give that amount a name: $E_{\text{in}}$.

WHY. Before we talk about ratios, fractions, or percentages, we need something to be a fraction of. Every efficiency question starts with "how much did you put in?" — so we name it first. The little word "in" written under the $E$ is doing real work: it reminds us this is the total we started with, the whole £1 of pocket money, nothing hidden.

PICTURE. The full lavender column below is $E_{\text{in}}$. Read the symbol as: $E$ = an amount of energy (measured in joules, J), and the subscript $\text{in}$ = "the stuff we supplied."

Step 2 — The energy must go somewhere

WHAT. The energy we poured in does not stay in the bucket. It flows out, and it splits into exactly two streams: the part that did the job we wanted ($E_{\text{useful}}$), and the part that leaked away ($E_{\text{wasted}}$).

WHY THIS TOOL — conservation of energy. We could have invented any rule for how energy splits. We don't — we use one law: Conservation of Energy. It says energy is never created or destroyed, only moved or changed in form. That is the ONLY reason we can be sure the two output streams add back up to exactly what went in. No leaks to a magic void, no bonus energy from nowhere.

$\underset{\text{poured in}}{\underbrace{E_{\text{in}}}}=\underset{\text{did the job}}{\underbrace{E_{\text{useful}}}}+\underset{\text{leaked as heat, sound}}{\underbrace{E_{\text{wasted}}}}$

Term by term, right where it sits:

• $E_{\text{in}}$ — the whole column from Step 1.
• $E_{\text{useful}}$ — the mint slice that came out the useful spout (e.g. a box actually lifted).
• $E_{\text{wasted}}$ — the coral slice lost as heat in wires, sound, Friction.

PICTURE. The single column of Step 1 splits into two coloured slices. Their heights add up to the original — that "add up" is conservation made visible.

Step 3 — Ask the real question: what fraction was useful?

WHAT. We don't actually care about raw joules for comparing machines. A giant crane and a toy motor both "waste 200 J" — but that's tiny for the crane and huge for the toy. We want a fair comparison. So we ask: of everything I put in, what fraction came out useful?

WHY THIS TOOL — dividing by $E_{\text{in}}$. To turn absolute joules into a fair fraction, we divide every slice by the same total, $E_{\text{in}}$. Dividing by the total is the universal move that turns "amounts" into "shares of the whole" — like turning "300 marks" into "60 percent" by dividing by the 500 total. Take the conservation line and divide every term by $E_{\text{in}}$:

$\frac{E_{\text{in}}}{E_{\text{in}}}=\frac{E_{\text{useful}}}{E_{\text{in}}}+\frac{E_{\text{wasted}}}{E_{\text{in}}}$

The left side is a number divided by itself, which is just $1$:

$1=\underset{\text{useful share}}{\underbrace{\frac{E_{\text{useful}}}{E_{\text{in}}}}}+\underset{\text{wasted share}}{\underbrace{\frac{E_{\text{wasted}}}{E_{\text{in}}}}}$

Reading it: the whole thing (1, or 100%) is split into a useful share plus a wasted share.

PICTURE. The same two slices, but now the total column is re-labelled as "1 whole." The mint slice is a fraction, the coral slice is a fraction, and they fill the bar completely.

Step 4 — Name the useful share: this is efficiency

WHAT. The useful share $\frac{E_{\text{useful}}}{E_{\text{in}}}$ is exactly the number we've been hunting. We give it its own symbol, the Greek letter $\eta$ ("eta").

WHY. A quantity we use over and over deserves a short name. From now on, whenever you see $\eta$, read it out loud as "the useful fraction."

Every symbol, one last time:

• $\eta$ — the answer, "how good is this machine."
• $E_{\text{useful}}$ — top of the fraction: what you got.
• $E_{\text{in}}$ — bottom of the fraction: what you paid.

PICTURE. We zoom on just the mint slice from Step 3 and stamp the label $\eta$ on it. Efficiency is literally the height of the useful slice when the whole bar is 1.

Step 5 — The waste form: $\eta =1-$ (wasted share)

WHAT. Rearrange the Step 3 equation. Since useful share + wasted share $=1$:

$\boxed{\eta =1-\frac{E_{\text{wasted}}}{E_{\text{in}}}}$

WHY. This second form is not a new idea — it's the same picture read from the top down instead of the bottom up. Sometimes a question tells you the waste directly (e.g. "40 J is lost as heat"), and this form lets you get $\eta$ without computing the useful part.

Term by term:

• Start at $1$ (the full bar).
• $\frac{E_{\text{wasted}}}{E_{\text{in}}}$ — the coral fraction we subtract off.
• What's left is $\eta$.

PICTURE. Same bar, but now an arrow starts at the top ($1$) and drops down by the coral wasted amount; where it lands is $\eta$.

Step 6 — The ceiling case: why $\eta$ can never beat 1

WHAT (edge case). What is the best a machine can do? Push the coral slice to zero — no waste at all.

WHY this matters. This is the degenerate, limiting case, and it sets the hard ceiling. From Step 5, if $E_{\text{wasted}}=0$, then: $\eta =1-\frac{0}{E_{\text{in}}}=1-0=1(=100\%)$ And $E_{\text{wasted}}$ can never be negative (you can't "un-waste" energy — that would mean the machine creates energy, breaking Conservation of Energy). Since the smallest possible waste is $0$, the largest possible $\eta$ is $1$:

$\boxed{0\le \eta \le 1}$

Real machines always have some waste (friction, resistance, heat — the Heat Engines & Carnot Efficiency limit even forbids $100\%$ for engines), so in practice $\eta <1$ always.

PICTURE. Three bars side by side: a real machine (big coral waste), an ideal machine (coral gone, $\eta =1$), and an impossible machine (mint spills over the top — forbidden, crossed out).

Step 7 — The power version: time cancels

WHAT. Everything above used energy $E$. But questions often give power $P$ — energy per second. From Work and Power, $P=\frac{E}{t}$, where $t$ is time in seconds.

WHY THIS TOOL — same $t$ top and bottom. The input and the output happen over the same time interval $t$ (the machine runs once, both streams flow during that same run). So when we form the ratio, the $t$ on top and the $t$ on bottom cancel:

$\eta =\frac{E_{\text{useful}}}{E_{\text{in}}}=\frac{E_{\text{useful}}/t}{E_{\text{in}}/t}=\frac{P_{\text{useful}}}{P_{\text{in}}}$

• $E_{\text{useful}}/t=P_{\text{useful}}$ — useful power out (watts, W).
• $E_{\text{in}}/t=P_{\text{in}}$ — power supplied.
• The $t$'s cancel because it's the same run, so both formulas give the same $\eta$.

PICTURE. The same two slices, but the axis is now "per second." The bar looks identical because dividing both slices by the same $t$ keeps their ratio unchanged.

Step 8 — Walk one machine through, start to finish

This is Steps 1–5 with real numbers: pour in 500, 300 comes out mint-useful, 200 leaks coral-wasted, and the mint slice is $60\%$ of the bar.

The one-picture summary

Everything on this page is one bar splitting into two slices. The whole bar is what you put in. The mint slice is what you wanted — its fraction of the bar is $\eta$. The coral slice is the leak. Divide by the total to get fractions; the ceiling is $\eta =1$ because the coral slice can shrink to zero but never go negative.

Recall

Connections

• Efficiency (parent topic) — this page is its visual derivation.
• Conservation of Energy — the one law that lets the slices add back to the whole.
• Work and Power — source of $P=E/t$ for Step 7.
• Gravitational Potential Energy — the useful output $mgh$ in Step 8.
• Heat and Internal Energy — what the coral wasted slice becomes.
• Friction — a major reason the coral slice is never zero.
• Heat Engines & Carnot Efficiency — why even ideal engines fall short of $\eta =1$.