Energy Wavelength Frequency Equation

This calculator computes the frequency of a photon from its energy, or the energy from its frequency, using the Planck-Einstein relation. You enter either the energy in joules or the frequency in hertz, and the calculator returns the corresponding frequency or energy, along with the wavelength in a vacuum. The core equation is E = hf, where h is Planck’s constant (6.626 × 10⁻³⁴ J·s).

Formula

The relationship between energy (E), frequency (f), and wavelength (λ) is given by the Planck-Einstein relation:

E = h × f

where:

• E = photon energy, in joules (J)
• h = Planck’s constant, 6.626 × 10⁻³⁴ J·s
• f = frequency, in hertz (Hz, or s⁻¹)

To connect wavelength, use the wave equation:

c = λ × f

where c = speed of light in vacuum (2.99792458 × 10⁸ m/s), λ = wavelength in meters, and f = frequency in Hz. Combining both gives:

E = h × c / λ

So if you know any one of energy, frequency, or wavelength, you can find the other two.

Worked Example

Using the calculator’s default energy value of 3.0 × 10⁻¹⁹ J:

1. Start with E = 3.0 × 10⁻¹⁹ J.
2. Apply E = hf, so f = E / h.
3. f = (3.0 × 10⁻¹⁹) / (6.626 × 10⁻³⁴) = 4.528 × 10¹⁴ Hz.
4. Find wavelength: λ = c / f = (2.99792458 × 10⁸) / (4.528 × 10¹⁴) = 6.62 × 10⁻⁷ m (662 nm, red light).

The calculator gives the same frequency (rounded to 4.528 × 10¹⁴ Hz) and wavelength (6.62 × 10⁻⁷ m) for that energy.

When to Use This Equation

This equation is fundamental in quantum physics, chemistry, and electronics. Use it whenever you need to:

• Convert photon energy to frequency (e.g., for spectroscopy).
• Find the energy of a photon from its wavelength (e.g., for solar panels or LEDs).
• Determine the frequency of electromagnetic radiation from an energy transition (e.g., in atomic physics).

It applies to all electromagnetic radiation—radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays—because each photon’s energy is directly proportional to its frequency. It also applies to other quantum particles like electrons, but then you need the de Broglie relation for matter waves.

How to Read the Results

The calculator returns frequency in hertz and wavelength in meters. For practical use:

• Visible light ranges from about 400 nm (violet) to 700 nm (red), corresponding to 7.5 × 10¹⁴ Hz to 4.3 × 10¹⁴ Hz.
• Energies of visible photons are around 2–3 eV (1 eV = 1.602 × 10⁻¹⁹ J).
• For chemistry, you often convert joules to electronvolts by dividing by 1.602 × 10⁻¹⁹.

If you get a wavelength in meters, convert to nanometers (multiply by 10⁹) for visible light, or to micrometers (multiply by 10⁶) for infrared.

Common Mistakes

Unit mix-ups cause most errors:

• Using frequency in MHz or GHz without converting to Hz (multiply by 10⁶ or 10⁹).
• Using wavelength in nm without converting to meters (divide by 10⁹).
• Forgetting that Planck’s constant is in J·s, so energy must be in joules, not electronvolts.
• Confusing frequency with angular frequency (ω = 2πf). The equation E = hf uses ordinary frequency, not angular frequency.

Also, remember that the speed of light in a medium is slower than in vacuum, so the wavelength changes, but the frequency (and thus energy) stays the same. This calculator assumes a vacuum.

Limits of the Formula

The Planck-Einstein relation is exact for single photons, but it has practical limits:

• It does not account for relativistic effects—it assumes the photon is massless, which is true.
• For very high energies (gamma rays), the photon can convert into matter, but the relation still holds for the initial photon.
• For matter waves (electrons, neutrons), you need the de Broglie wavelength: λ = h / p, where p is momentum.

In everyday life, the energy of a single photon is tiny; a typical light bulb emits about 10²⁰ photons per second. So this equation is most useful at the atomic and subatomic scale, or for very precise measurements like in GPS satellite corrections.

Frequently Asked Questions

Why is Planck’s constant so small?

Planck’s constant (6.626 × 10⁻³⁴ J·s) is small because quantum effects only become noticeable at the atomic scale. It sets the scale for the graininess of energy. In macroscopic systems, the energy steps are so tiny that they appear continuous.

What is the difference between frequency and wavelength?

Frequency is how many wave cycles pass a point per second (Hz). Wavelength is the physical distance between two consecutive crests (m). They are inversely proportional: as frequency increases, wavelength decreases, because the speed of light is constant.

Can I use this for sound waves?

No. The equation E = hf applies only to photons (electromagnetic waves) and other quantum particles. Sound is a mechanical wave, not a particle. Its energy depends on amplitude and medium, not on a fixed constant like Planck’s.

How do I convert electronvolts to joules?

1 electronvolt (eV) = 1.602 × 10⁻¹⁹ J. So multiply the energy in eV by 1.602 × 10⁻¹⁹ to get joules. For example, a 3 eV photon has 4.806 × 10⁻¹⁹ J.

What is the wavelength of a 100 MHz radio wave?

Use λ = c / f. 100 MHz = 100 × 10⁶ Hz. λ = (3.00 × 10⁸) / (100 × 10⁶) = 3.0 meters. That’s why FM radio antennas are about a meter long.

Does the equation work for any photon, regardless of its origin?

Yes, any photon—from a radio transmitter, a star, or a laser—obeys E = hf. The origin does not matter; only the frequency (or wavelength) determines its energy.

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in e | Energy | J | 3.0e-19 | 0
in f | Frequency | Hz | 0 | 0
out freq | Frequency | Hz | e / 6.626e-34 | 4
out energy | Energy | J | f * 6.626e-34 | 4
out wl | Wavelength (vacuum) | m | 2.99792458e8 / (e / 6.626e-34) | 6