Here’s a weird question: does your phone weigh more when it’s “full” than when it’s “empty”? It sounds almost ridiculously naïve – of course, a phone isn’t like a jug of water, with each photo or contact adding an extra few milliliters to the total. How could information have a weight?

But like many apparently simple questions, this one has a surprisingly complex answer – and it all comes down to one of the most fundamental laws of the universe. So, with a bit of special relativity, a lot of math, and a little dip into computer science, let’s actually take a serious look at the weight of your phone – before, and after, you’ve filled it with lolcats and Hide the Pain Harolds.

In theory: yes

It might sound counterintuitive, but the data on your phone does in fact have a weight. And the even more surprising thing is, you already know the scientific reason why.

“Information is stored [on electrons],” explained NPR science correspondent Robert Krulwich back in 2011. “And electrons are very, small. But they do have mass. Einstein taught us that. So it's possible to take all the energy (E)… and, using Einstein's equation, (E = mc²) turn that energy into something we can weigh.”

It’s one of the most famous equations in science – but unless you’re a working physicist or cosmologist, you probably didn’t think it had much use in day-to-day life. Here, though, it’s exactly what we need: if we can work out the change in energy levels between a “full” phone and an “empty” one, Einstein’s most famous soundbite will allow us to figure out the difference in mass between the two.

First, though, we need to understand how and why the energy levels change at all. When we add information to non-electronic systems – like, say, a book, or a photo album – the difference is simple: a blank page versus a full one. But with our phones, e-readers, tablets and so on, it’s more complicated: in these cases, the data is stored as binary information, encoded in series of zeroes and ones.

When you add or remove data from your device’s flash memory, you’re not adding or removing these digits, but switching them – changing ones to zeroes and zeroes to ones. “Another way of thinking about it is that the atoms in the memory have magnetic-like properties,” explains Gareth Mitchell for BBC Science Focus. “Groups of atoms align in one direction or the other according to whether they are storing a 1 or a 0 and will possess differing amounts of energy according to how they are aligned.”

Technically, flash memory works by either holding electrons in place or not. While they’re being held in place – that is, when they’re encoding information – they become more energetic. And as Einstein’s equation tells us: more energy equals more mass.

Technically, then, filling your phone with photos, music, and messages does indeed make it heavier. But don’t go swapping out your dumbbells for an old iPhone just yet – because…

In practice: not really

It’s one thing to know that special relativity means your phone gets heavier with more data. But what would actually happen if you weighed your device before and after loading it up with memes?

Luckily, it’s pretty easy to work this out using Einstein’s equation: we have energy, E, and we want to work out mass, m. There’s just one letter in the equation to go: the constant, c – which refers to the speed of light.

Now, as you may recall, the speed of light is an extremely large number – around 3 × 10⁸ meters/second. When you plug this into the equation for mass-energy equivalence, it results in something interesting: it means that a little tiny piece of mass will be the equivalent of just vast amounts of energy.

We’re working the other way around, but it’s still true: it would take a mind-boggling amount of energy to result in a noticeable difference in mass. And while modern mobile phones can hold quite a lot of information – there’s easily six orders of magnitude more memory in your smartphone than there were in the computers that put humans on the moon, for a bit of perspective – when we run the numbers, we find that even the latest devices can’t hold enough data to actually feel “heavier” when full.

So, what kind of sizes are we talking? Well, according to John D. Kubiatowicz, a professor of computer science at the University of California, Berkeley, who answered a similar question for The New York Times back in 2011, a conservative estimate for the difference in energy between a trapped electron and a free one is around 10^–15 joules per bit. Plugging that into Einstein’s equation, he calculated that a full 4-gigabyte Kindle e-reader will weigh more than an empty one, but “the amount is very small, on the order of an attogram,” or 10^–18 grams. “This amount is effectively unmeasurable.”

Of course, that was more than a decade ago, and even cheap phones usually have quite a lot more than 4 gigabytes of memory today. But even allowing for 512 gigabytes of data – the amount available on the latest iteration of the iPhone – will only result in a difference of around 10^-16 grams, or 0.1 femtograms.

To put that in perspective, it’s slightly heavier than a single tobacco mosaic virus, the first virus ever discovered. Alternatively, it’s about one-tenth the weight of a single HIV-1 virus. In other words, it’s not the kind of mass you’d notice even if it was dropped on your head from the top of the Empire State Building.

Weighty information

So just how much data would we need to be carrying on our phone before we’d actually feel a difference? Well, to answer that, we need to know something else: exactly how much heavier does something need to be before your average human would notice?

This is where the extremely coolly-named science of psychophysics comes in – the branch of experimental psychology focused on sense, sensation, and perception. To psychophysicists, the value we’re looking for is called the “just-noticeable difference”, or JND: the amount a value must be changed in order for a difference to be detectable at least half the time.

While it isn’t great at super-low or -high frequencies, for most physical quantities there’s a pretty good rule of thumb for figuring out the JND: it’s called the Weber–Fechner law, and it states that “the ratio of the increment threshold to the background intensity is a constant.”

It makes sense: it’s saying that if you have two small quantities, you need less of a difference for it to be noticeable than if you have two larger quantities. Think about talking in a quiet library versus a loud birthday party – it’s way easier to tell a louder sound from a quieter one in the first place than the second.

When applied to weight, the law reveals something very convenient: to be “noticeable”, the difference between two objects must be at least five percent of one of their weights. 

The new iPhone 14, according to official specs, weighs 172 grams – meaning that it would take a weight increase of 8.6 grams before you’d notice that it had been made heavier. If 512 gigabytes weigh about 10^-16 grams, that would mean you’d need – deep breath here folks – about 44,000,000,000,000,000,000 gigabytes, or 44 million extra zettabytes of information stored on your phone.

How long would that take to download? Well, quite apart from the fact that your phone would never be able to contain such a wodge of information – in fact, you’d need over 30 billion human brains just to contain it all – it would be quite the long-term commitment. Even with optimum 5G speeds of around 200 megabits per second, it would take around 56,000 years to download that amount of data onto your phone – and you’d probably need that extra time, since the entire internet is currently estimated to contain less than 100 zettabytes.

So, does your phone weigh more when it’s full of data? Yes, technically – but not so much that you’d ever notice it. You can carry on snapping those photos and videos without having to hit the gym afterwards – you’ll probably run out of interesting things to document before your phone becomes heavier by even a microgram.