Joules, watts or calories? Energy and power without fear

♣ Newbie

Imagine energy is water. To heat something, move it or switch on a light, you have to use up a bit of that water. And, like water, it can be measured with containers of different sizes:

• The joule is the teaspoon: roughly what it takes to lift an apple one metre.
• The kilocalorie, the one on a yoghurt label, is a glass: 4,184 teaspoons.
• The kilowatt hour, the one on your electricity bill, is a bucket: 3.6 million teaspoons.

All three measure the same thing. When a biscuit says “kcal” and the bill says “kWh”, they’re talking about the same stuff with a different glass.

One important detail: water doesn’t vanish when you use it, it spills. Energy isn’t destroyed either. It changes form (from food to your muscles, from electricity to hot water) and almost always ends up as heat that drifts away.

♦ Intermediate

Energy is what you have to supply to change something: lift it, speed it up, heat it or break it. It has two very useful properties. It is conserved: it neither appears nor disappears, it only passes from one form to another. And all its forms are measured with the same unit, the joule (J).

A joule is the energy of pushing with a force of one newton over one metre. In practice, lifting a 100 g apple by one metre costs almost exactly one joule: 0.1 kg × 9.8 m/s² × 1 m ≈ 0.98 J.

The other units are the joule in a different size:

A kilocalorie is also almost exactly what it takes to warm a litre of water by one degree: 4,184 J against about 4,180 J.

On the ruler above, each tick multiplies by a thousand. That’s how an apple and a tonne of TNT fit in the same drawing.

♠ Technical

The work done by a force along a path is W = ∫ F · dx, and the work-energy theorem equates it to the change in kinetic energy, W = ΔE_k. The joule is the SI derived unit for energy, work and heat: 1 J = 1 N·m = 1 kg·m²·s⁻². That energy is conserved in an isolated system follows from the laws of physics not changing over time (Noether’s theorem).

Several conversions are exact by definition. Since the 2019 SI redefinition, the elementary charge is fixed at e = 1.602 176 634×10⁻¹⁹ C, and the electronvolt with it. The thermochemical calorie is defined as 4.184 J. Careful: the IT calorie is 4.1868 J and the “mean” one about 4.190 J (NIST SP 811), so in a fine calculation you have to say which one you’re using.

Dimensional analysis: any energy expression must come out in kg·m²·s⁻². For example, m·g·h = kg · m·s⁻² · m. If a formula doesn’t balance in units, it’s wrong before you plug in any numbers.

♣ Newbie

Now the second question: not how much water, but how fast it comes out. That’s power, and it’s measured in watts.

Picture three identical bathtubs. The first gets a dripper and takes days to fill. The second, a normal tap: a few minutes. The third gets a burst dam: it fills in one second and bursts. All three receive the same water; the only thing that changes is the rate.

That’s why a kettle with 2,000 W on its label doesn’t hold 2,000 of anything. That number says how fast it turns electricity into heat. Leave it on longer and it uses more energy; shorter, less. Watts are the tap; joules, the water.

Some electricity bills, like Spain’s, charge for both separately. The contracted power, in kW, is the biggest tap you can open at once before the power cuts out. The energy, in kWh, is the water you actually used.

♦ Intermediate

Power is energy divided by time: P = E / t. A watt (W) is one joule per second. The other way round, energy is power times time: E = P · t.

That’s where the kilowatt hour comes from: 1,000 W for 3,600 s is 3.6 million joules. A 2,000 W kettle on for 3 minutes uses 2,000 × 180 = 360,000 J, which is 0.1 kWh.

If you draw power going up and time going across, energy is the area of the rectangle. The same energy fits in a long, low rectangle or in a narrow, very tall one: 1 kWh is 1 kW for an hour, 60 kW for a minute or 3.6 megawatts (MW, millions of watts) for a second.

Some reference powers:

Look at the cyclist: the shorter the effort, the more power they can hold. Power isn’t only how much you can do, but for how long.

Lightning is the extreme case. A typical bolt carries about 950 million joules, about 260 kWh (other estimates go as low as 100 million): the same as a 2,000 W kettle running for about 130 hours. It isn’t a huge amount of energy. What makes it terrible is that it releases it in a fraction of a second.

♠ Technical

Instantaneous power is the time derivative of energy, P = dE/dt, and the energy transferred over an interval is its integral, E = ∫ P(t) dt. The mean power P̄ = ΔE/Δt is the height of the rectangle with the same area as the real profile. For a force acting on a moving body, P = F · v.

The power on an appliance label is the rated power: a maximum or an average under test conditions. For actual consumption you have to integrate the profile P(t), which is what a meter does.

Physics and engineering always separate two limits: energy (how much fuel or battery) and power (how much the motor, the cable or the heat sink can take). A problem that’s feasible in energy can be impossible in power. Lightning shows it: about 9.5×10⁸ J for a typical four-stroke flash (Cooray, 1997; the EPA lists estimates from 10⁸ J), a household-sized energy delivered in a fraction of a second.

The cyclist’s power-duration curve is the same phenomenon in a body: 836–895 W as the best 15 s average against 342–394 W over 30 min (Vogt et al., 2007). The maximum sustainable power falls as the effort gets longer.

♣ Newbie

Almost all the energy we use ends up as heat. And here’s a trap: heat and temperature are not the same thing.

Back to the bathtub, now with a drain. The tap is the energy coming in. The drain is the heat the object gives off to the air. And the water level is the temperature: how hot it is. If as much comes in as goes out, the level stays put. If you open the tap more than the drain can swallow, the level rises… until it overflows.

Your body works like this: it produces about 80 watts of heat doing nothing and gives it off through the skin. If you run, the tap opens wider and the body widens the drain: you sweat.

That’s why doing something very fast tends to burn: you put the energy in faster than it can escape.

Another thing that confuses people: a spark from a bonfire is very hot, and yet you barely feel it if it lands on your hand. A warm bath, on the other hand, heats you all over. The spark is so small that it carries very little energy. Temperature tells you how hot something is, not how much energy it carries.

♦ Intermediate

Temperature measures how agitated the particles of a body are. Heat is energy that passes from one body to another because they’re at different temperatures, always from hot to cold. Heat is measured in joules; temperature, in degrees Celsius or in kelvin (K). A degree Celsius and a kelvin are the same step, and 0 °C = 273.15 K.

How much the temperature rises depends on the energy that stays (Q), the mass (m) and the specific heat (c), which is what it takes to raise one gram of that substance by one degree: ΔT = Q / (m · c).

Water has a high specific heat, about 4.18 J per gram per degree. Boiling a litre from 20 °C costs 1,000 g × 4.18 × 80 ≈ 334,000 J. With a 2,000 W kettle and no losses that’s 167 seconds, about 2 minutes 47. In real life it takes a bit longer, because some heat escapes in the meantime.

Heat leaves in three ways: by conduction (contact), by convection (a fluid, like air or blood, carrying it away) and by radiation (infrared light, no contact). At rest, your body gives off its 68–87 W mostly by radiation, around 60%.

The rule that explains why fast means hot is a balance: what comes in minus what goes out is what stays, and that raises the temperature. A kilo of water receiving 1,000 W without losing any rises about 0.24 degrees per second: 14 degrees per minute.

♠ Technical

Energy balance for a body of mass m and specific heat c receiving power P: m·c·dT/dt = P − P_out(T). If the loss is proportional to the difference with the surroundings (Newton’s law of cooling, P_out = hA(T − T₀)), the solution is T(t) = T₀ + (P/hA)(1 − e^(−t/τ)), with τ = m·c/(hA). The equilibrium T* = T₀ + P/(hA) grows linearly with P.

Radiation follows the Stefan-Boltzmann law, εσA(T⁴ − T₀⁴). For small differences it linearises to 4εσT₀³(T − T₀) and adds to h; at body temperature it’s the main route of heat loss, around 60% at rest.

Why speed is a heat problem and not an energy problem: with a fixed E delivered over a time t, P = E/t. If t ≪ τ, the system is almost adiabatic and ΔT ≈ E/(m·c): all the energy stays. If t ≫ τ, the temperature follows the power, ΔT ≈ P/(hA) = E/(hA·t). Shortening t raises the equilibrium ΔT up to the adiabatic ceiling.

Since 2019 the kelvin is defined by fixing k_B = 1.380 649×10⁻²³ J/K, which ties temperature to energy: at 300 K, k_B·T ≈ 4.1×10⁻²¹ J ≈ 0.026 eV.

♣ Newbie

Real-life powers range from the tiny to the enormous. A ladder to get your bearings:

• You, sitting still: about 80 watts.
• A professional cyclist on a half-hour climb: nearly 400 watts.
• A kettle: 2,000 watts.
• A nuclear reactor: a billion watts.
• The Sun: so many zeros they don’t fit on the ladder.

Each rung isn’t “a bit more”. The reactor equals more than eleven million people at rest. To compare things this different you have to count zeros, which is exactly what the big-numbers lesson teaches.

A fool’s trick that works: when someone says something “needs a huge amount of energy”, ask in how long. Divide, and compare with the ladder.

♦ Intermediate

To fit a person and the Sun in the same drawing you use a log scale: each equal step along the axis multiplies by the same amount, here by a thousand (watt, kilowatt, megawatt, gigawatt). The prefixes help: kilo is a thousand, mega a million and giga a billion.

On that scale you can see the reactor is seven steps of ten above you, and the Sun nearly eighteen steps beyond the reactor.

The method for judging any claim has three steps:

1. Work out the energy needed.
2. Divide it by the time.
3. Compare with the table of references.

If it comes out around a kettle, it’s a household thing. If it comes out around a reactor, it’s a power-station thing.

♠ Technical

On a log scale the position is x = log₁₀(P/1 W), so a constant ratio is a constant distance. Person at rest: log₁₀ 87 ≈ 1.9. Reactor: 9. Sun, with the IAU nominal luminosity (3.828×10²⁶ W): ≈ 26.6. Between person and reactor there are 7.1 orders of magnitude, which is the log of the ratio.

A Fermi estimate pins down the order of magnitude with known quantities. P ≈ E/t with E and t within a factor of 2 or 3 of their real values is enough to classify a process, because the references in the table are several orders of magnitude apart.

An example to sharpen it: total solar irradiance at 1 au is 1361 W/m² (IAU 2015). Matching 1 GW takes collecting 10⁹/1361 ≈ 7.3×10⁵ m², about 0.73 km² outside the atmosphere, with no losses and no night. Every real correction (atmosphere, angle, night, efficiency) multiplies that area.