How Much Does a Smartphone Battery’s Mass Increase When Charged?

A smartphone battery is slightly heavier when charged than when discharged, because stored energy contributes to the mass of the battery. The change is far too small to notice with an ordinary scale. You can estimate it by converting the energy retained by the battery into its mass equivalent.

The mass–energy relation

Einstein’s relation between energy and mass is

$$E=mc^2,$$

where $c$ is the speed of light, approximately $3.00\times10^8\ \text{m/s}$. For a change in stored energy, the corresponding mass change is

$$\Delta m=\frac{\Delta E}{c^2}.$$

This applies to a battery as a physical system: energy held in its chemical state contributes to its total mass. Charging does not create extra material or add a noticeable number of electrons to the battery. It changes the battery’s energy and chemical state.

Worked example: a 4,000 mAh phone battery

Suppose a battery is rated at 4,000 milliamp-hours and has a nominal voltage of 3.85 volts. First convert the capacity to amp-hours:

$$4000\ \text{mAh}=4.0\ \text{Ah}.$$

The battery’s approximate stored energy is its capacity multiplied by its nominal voltage:

$$\begin{aligned} E&=(4.0\ \text{Ah})(3.85\ \text{V})\\ &=15.4\ \text{Wh}\\ &=15.4(3600)\ \text{J}\\ &=55{,}440\ \text{J}. \end{aligned}$$

Since $c^2\approx9.00\times10^{16}\ \text{m}^2/\text{s}^2$,

$$\begin{aligned} \Delta m&=\frac{55{,}440\ \text{J}}{9.00\times10^{16}\ \text{m}^2/\text{s}^2}\\ &\approx6.2\times10^{-13}\ \text{kg}\\ &\approx0.62\ \text{ng}. \end{aligned}$$

So the rough estimate is about six-tenths of a nanogram. A nanogram is one billionth of a gram; this change is tiny compared with a phone’s mass, which is typically measured in hundreds of grams.

What the estimate means

The calculation uses the battery’s rated energy as an estimate of the energy difference between charged and discharged states. A battery rating is not a precise measurement of the change in total internal energy: actual usable energy depends on temperature, discharge conditions, battery age, and the voltage range used. The exact mass change also depends on the battery’s energy and heat exchange with its surroundings.

Energy supplied by the charger is not all retained in the battery. Some is lost as heat in the battery, charger, and phone. If some heat escapes, it does not remain as energy in the battery-phone system, so it does not contribute to that system’s final mass. If the battery is warmer after charging, its extra thermal energy also contributes to its mass while that heat remains in it. For a fair comparison, compare charged and discharged states at the same temperature and use the energy difference actually retained.

In principle, a sufficiently precise weighing experiment could detect a mass difference associated with energy. In practice, changes in temperature, mechanical stress, air currents, and the scale’s resolution overwhelm a sub-nanogram effect.

Common mistakes

• Using milliamp-hours as if they were energy. Capacity alone is not enough; multiply amp-hours by voltage to obtain watt-hours.
• Forgetting the conversion to joules. In the mass–energy equation, use energy in joules and $c$ in metres per second. One watt-hour equals 3,600 joules.
• Using $c$ instead of $c^2$. The mass change is energy divided by the square of the speed of light, which makes the result extremely small.
• Assuming all charging electricity stays in the battery. Charging losses become heat or other energy transfers; only energy retained by the system increases its mass in the final state.

The exact value varies, but for a typical smartphone battery the energy-related mass increase is on the order of $10^{-13}$ kilograms—not grams or milligrams.