## The physics behind lava lamps

##### Apr 04, 18 - 21:30pm, 2 months and 2 weeks ago

The last time someone asked me to explain how those warm and tubular, alienesque floaty boyes worked I ended up being pinned down on their bed. Anyway, lava lamps, physics. I’m assuming many a time you’ve asked yourself: “how does it work”? - or not.

### tl;dr

Change in temperature of wax causes a change in density which causes a change in resultant force of upthrust extered upon floaty balls such that upthrust > weight and therefore positive acceleration.

#### Electricity -> floatyness

A lava lamp is doing work

At the core of a lava lamp we’re doing work done against gravity, albeit incredibly inefficiently. Work simply means the amount of energy transformed from one form to another. In this case electrical energy into kinetic energy.

Our total work done is the amount of energy required to move a wax ball of mass $m$ to a height of $h$, $mg\Delta h$.

In the simplest scenario, we’ve got the following going on.

In order for there to be positive vertical acceleration the vertical component of force being exerted upon the wax ball must be greater than weight of the ball.

In any instance where there is a moving object inside a fluid, viscous drag occurs (assuming laminar flow). Modeling the ball as a spherical object we can apply Stokes Law to provide a rough estimate for the size of this force. Resistive forces always act opposite to the direction of motion, such that:

The only positive vertical component of force acting upon the wax ball is upthrust, the force felt by an object partially/submerged in a fluid due to the difference in densities of the fluid. Therefore;

Okay that’s all nice and all - but it doesn’t explain why the wax ball rises and then falls.
Introduce, Pressure Law and Charles’s Law which state that:

The heating element at the bottom of a lava lamp provides any nearby wax molecules with energy, kinetic energy. This rise in KE means particle vibrate more, thus higher temperature. Since we know that:

Lets consider what happens when the wax is at the bottom of the lamp. The wax sphere is closer to the lamp and so heats up, and since density is inversely proportional to temperature, the density of the wax ball decreases.

Archimedes principle states that:

In order to delve into a bit more of how lava lamps work, we need to gather some data about their attributes, what they’re made of and whatnot.

#### Floaty balls

Wax balls. balls made of wax, that’s what they are, or more verbosely:

A formula from 1968 US patent consisted of water and a transparent, translucent, or opaque mix of mineral oil, paraffin wax, and carbon tetrachloride.

I’ll just say it’s paraffin wax. Paraffin wax has a density of 900 $kgm^{-3}$.

#### Container liquid

Probably water, which has a density of 1000 $kgm^{-3}$.

#### Device wattage

Your run-of-the-mill uses about 25-40 watts, for this I’ll use 40, because muh power.

First off: some assumptions, I’m going to model the floaty things as completely spherical, uh, spheres which removes any non-laminar-y, fluid mechanics crap (miss me with that Rayleigh-Taylor instability) from the equation - we’re going to be dealing with ideal liquids/states and ideal ambient environments.

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