The letter “e” shows up across physics with at least three distinct meanings: the elementary charge (the tiny amount of electric charge carried by a single proton or electron), the mathematical constant 2.718… used in exponential equations, and the variable E (capitalized) representing energy in formulas like E = mc². Which one you’re dealing with depends entirely on the equation in front of you. Here’s how each one works and where you’ll encounter it.
Lowercase e: The Elementary Charge
In electricity and particle physics, a lowercase italic e represents the elementary charge, the smallest unit of electric charge found in nature. Its exact value is 1.602 176 634 × 10⁻¹⁹ coulombs. A proton carries a charge of +e, and an electron carries a charge of −e. Every charged particle ever observed has a charge that is a whole-number multiple of this value: q = ne, where n is an integer. This is what physicists mean when they say charge is “quantized.”
This value wasn’t always known. J. J. Thomson discovered the electron in 1897 but couldn’t pin down its charge precisely. It took Robert Millikan’s famous oil drop experiment, published in its definitive form in 1913, to nail it down. Millikan suspended tiny oil droplets between electrically charged plates, measured how they moved, and showed that every droplet’s charge was a whole-number multiple of one fundamental unit. His measurement landed within half a percent of today’s accepted value.
The elementary charge also gives us a convenient energy unit: the electron volt (eV). One electron volt is the energy a single electron gains when it moves through a voltage difference of 1 volt, equal to about 1.6 × 10⁻¹⁹ joules. That’s vanishingly small by everyday standards, but it’s the natural scale for describing energy at the atomic level. Particle physicists use it constantly.
You’ll see e appear directly in Coulomb’s law, which describes the force between two charged objects. If a nucleus has 46 protons, its total charge is 46 × 1.602 × 10⁻¹⁹ C. Plug that into Coulomb’s law alongside another charge and a distance, and you get the electrostatic force between them.
Lowercase e: Euler’s Number (2.718…)
The other lowercase e in physics isn’t a physical quantity at all. It’s the mathematical constant 2.71828…, sometimes called Euler’s number. It serves as the base of the natural logarithm and appears in any formula describing exponential growth or decay. You can usually tell this e apart from the elementary charge because it sits as the base of an exponent, like e⁻ᵏᵗ.
Radioactive decay is the classic example. The number of undecayed atoms in a sample follows the formula N = N₀e⁻ᵏᵗ, where N₀ is the starting count, k is a decay constant specific to the isotope, and t is time. The constant e makes this equation work because it naturally describes processes where the rate of change is proportional to the current amount.
The same math governs how capacitors charge and discharge in electrical circuits. When a capacitor discharges through a resistor, the charge drops as Q(t) = Q₀e⁻ᵗ/τ, where τ (the “time constant”) equals the resistance times the capacitance. After one time constant, the current has fallen to about 36.8% of its initial value. After three time constants, it’s down to roughly 5%. The shape of that curve, a rapid initial drop that gradually flattens out, is the signature of Euler’s number at work.
You’ll find this same exponential pattern in cooling objects, damped vibrations, atmospheric pressure changes with altitude, and absorption of light passing through a material. Anywhere something fades or grows at a rate proportional to its current value, e is embedded in the math.
Capital E: Energy
When you see a capital E, it almost always stands for energy. This is by far the most common use of the letter in physics, and it appears in several foundational equations.
E = mc² (Mass-Energy Equivalence)
Einstein’s famous equation states that mass and energy are two forms of the same thing. E is the energy contained in a given amount of mass, m is the mass, and c is the speed of light (about 300 million meters per second). Because c² is an enormous number, even a tiny amount of mass corresponds to a staggering amount of energy.
This isn’t just theoretical. In positron-electron annihilation, a particle and its antiparticle collide and convert their entire mass into radiation, pure energy in the form of photons. The reverse also happens: a high-energy photon passing near an atomic nucleus can spontaneously produce an electron and a positron, creating matter from energy. These processes convert mass to energy and back again in exact accordance with E = mc².
E = hf (Photon Energy)
In quantum physics, E = hf describes the energy of a single photon. Here, h is Planck’s constant and f is the frequency of the light. Higher frequency means higher energy, which is why ultraviolet light causes sunburns while radio waves pass harmlessly through your body. The equation is simple, but it was revolutionary: it established that light delivers energy in discrete packets (photons) rather than as a continuous wave.
E in Thermodynamics
In thermodynamics, E often represents internal energy, the total energy stored within a system due to the motion and interactions of its molecules. The first law of thermodynamics ties it together neatly: the change in internal energy equals the heat added to a system minus the work the system does on its surroundings (E₂ − E₁ = Q − W). Some textbooks use U instead of E for internal energy, but the meaning is identical.
How to Tell Them Apart
Context almost always makes the meaning clear. If you see e raised to a power, it’s Euler’s number. If it appears multiplied by a voltage or alongside charges in an electromagnetic equation, it’s the elementary charge. And if it’s a capital E on one side of an equals sign, with mass, frequency, or heat on the other, it’s energy. In practice, textbooks and equations rarely mix these in a way that causes genuine ambiguity, because they operate in different domains of physics.
One place they do overlap neatly: the electron volt ties the elementary charge directly to energy. The eV is literally defined by combining the charge e with a voltage to produce a unit of E. That connection between the two meanings of the letter isn’t a coincidence. It reflects how deeply charge and energy are linked at the subatomic scale.