Since h is an energy divided by a frequency, the dimensional formula of Planck's constant works out to [ML²T⁻¹], with every power shown [M¹ L² T⁻¹]. This is the bracket of angular momentum too, and of the quantity called action.
Planck's constant links the energy of a photon to its frequency, E = hf, and its value is 6.626 × 10⁻³⁴ J s. It sits at the heart of Class 12 dual nature of radiation and matter and of atomic structure. Below are the derivation, the unit J s and its eV form, the matching quantities, checks on de Broglie and photon energy, and the slips. The same constant drives the photoelectric effect on the work function formula page.
Quick facts
Dimensional formula
[M¹ L² T⁻¹], short form [ML²T⁻¹]Defining relation
Value
6.626 × 10⁻³⁴ J sSI unit
J s = kg m² s⁻¹Same dimensions as
Angular momentum, ħChapter
Dual nature of radiation and matter, Class 12
From E = hf to the dimensional formula of Planck's constant
- Photon energy: E = hf, so h = E/f.
- Energy has the bracket [ML²T⁻²], as for any energy.
- Frequency counts cycles per second, which makes it [T⁻¹].
- Divide: [ML²T⁻²] ÷ [T⁻¹]. The time power rises from −2 to −1, so h is [ML²T⁻¹].
The step that catches students is dividing by T⁻¹: dividing by a negative power is the same as multiplying by a positive one, so one power of time comes back. The frequency and energy pages hold the two inputs.
J s, eV s and erg s
In base units, [ML²T⁻¹] becomes kg m² s⁻¹. Grouped as energy times time, it reads J s (joule second), the form every data table uses.
| Unit | Value |
|---|---|
| J s | 6.626 × 10⁻³⁴ |
| eV s | 4.136 × 10⁻¹⁵ |
| erg s (CGS) | 6.626 × 10⁻²⁷ |
The eV s value comes from dividing by 1.602 × 10⁻¹⁹ J/eV. The CGS value uses 1 J = 10⁷ erg. The reduced constant ħ = h/2π = 1.055 × 10⁻³⁴ J s has the same bracket, since 2π is a pure number.
Quantities with the bracket of h
| Quantity | Relation | Why it matches |
|---|---|---|
| Angular momentum | L = mvr | Mass × speed × radius |
| Reduced constant ħ | h/2π | Number times h |
| Action | energy × time | A name used in advanced mechanics |
| Angular impulse | τΔt | Torque × time |
The angular momentum match is used directly in the Bohr model, where the allowed orbits are those for which mvr equals n multiplied by h/2π. Without matching brackets that rule would make no sense. See the angular momentum dimensional formula.
Two checks from modern physics
Check: de Broglie wavelength λ = h/p
h is [ML²T⁻¹] and momentum is [MLT⁻¹].
Ratio: M and T⁻¹ cancel, one L is left, so λ is [L], a length. The relation passes.
Find the energy of a 500 nm photon in joules and electron volts
Use E = hc/λ. Dimensions first: [ML²T⁻¹][LT⁻¹] ÷ [L] = [ML²T⁻²], an energy.
Numbers: E = (6.626 × 10⁻³⁴)(3.00 × 10⁸) ÷ (5.00 × 10⁻⁷) = 3.98 × 10⁻¹⁹ J.
In electron volts: 3.98 × 10⁻¹⁹ ÷ 1.602 × 10⁻¹⁹ = 2.48 eV.
Common slips with h
- Writing [ML²T⁻²]. That is energy. Dividing by frequency brings one power of time back.
- Writing [ML²T⁻³]. That comes from multiplying by frequency instead of dividing.
- Using J/s for the unit. That is a watt. Planck's constant is J s, a joule times a second.
- Thinking ħ has different dimensions. Dividing by 2π never changes a bracket.
Other constants, such as the Boltzmann constant, sit beside it in the dimensional formulas list.
Frequently asked questions
In what unit is h quoted?
The joule second, J s. Broken into base units, kg m² s⁻¹.
Why do Planck's constant and angular momentum share dimensions?
Each can be written as energy multiplied by time. Angular momentum mvr and h = E/f each come to [ML²T⁻¹].
Is Planck's constant a scalar?
Yes. It is a fixed scalar constant of nature with no direction.
What is the value of Planck's constant in eV s?
About 4.136 × 10⁻¹⁵ eV s, found by dividing 6.626 × 10⁻³⁴ J s by 1.602 × 10⁻¹⁹ J per eV.
Does the reduced constant ħ have the same dimensions as h?
Yes. ħ = h/2π, and a pure number cannot change dimensions, so ħ is also [ML²T⁻¹].