Dimensional formula of any physical quantity
Check equations, explore quantities, find formulas.
Equation checker
Matching dimensions are necessary, not sufficient. A check cannot see pure numbers such as 1/2 or 2π, so a consistent equation can still be wrong.
s = u t + (1/2) a t2s[L]=ut[L]+1/2 at2[L]Every quantity
- l Length [L]
- m Mass [M]
- t Time [T]
- I Electric current [A]
- T Temperature [K]
- n Amount of substance [mol]
- Iv Luminous intensity [cd]
- A Area [L²]
- V Volume [L³]
- x Displacement [L]
- θ Angle [M⁰ L⁰ T⁰]
- Ω Solid angle [M⁰ L⁰ T⁰]
- v Velocity [L T⁻¹]
- a Acceleration [L T⁻²]
- j Jerk [L T⁻³]
- ω Angular velocity [T⁻¹]
- α Angular acceleration [T⁻²]
- p Momentum [M L T⁻¹]
- F Force [M L T⁻²]
- T Tension [M L T⁻²]
- Ft Thrust [M L T⁻²]
- Fc Centripetal force [M L T⁻²]
- J Impulse [M L T⁻¹]
- W Weight [M L T⁻²]
- W Work [M L² T⁻²]
- E Energy [M L² T⁻²]
- K Kinetic energy [M L² T⁻²]
- U Potential energy [M L² T⁻²]
- P Power [M L² T⁻³]
- τ Torque [M L² T⁻²]
- I Moment of inertia [M L²]
- L Angular momentum [M L² T⁻¹]
- J Angular impulse [M L² T⁻¹]
- k Radius of gyration [L]
- μ Coefficient of friction [M⁰ L⁰ T⁰]
- e Coefficient of restitution [M⁰ L⁰ T⁰]
- k Spring constant [M T⁻²]
- C Torsional constant [M L² T⁻²]
- ρ Density [M L⁻³]
- d Relative density [M⁰ L⁰ T⁰]
- u Energy density [M L⁻¹ T⁻²]
- g Acceleration due to gravity [L T⁻²]
- Eg Gravitational field intensity [L T⁻²]
- V Gravitational potential [L² T⁻²]
- U Gravitational potential energy [M L² T⁻²]
- ve Escape velocity [L T⁻¹]
- P Pressure [M L⁻¹ T⁻²]
- dP/dx Pressure gradient [M L⁻² T⁻²]
- σ Stress [M L⁻¹ T⁻²]
- U Strain energy [M L² T⁻²]
- u Strain energy density [M L⁻¹ T⁻²]
- ε Strain [M⁰ L⁰ T⁰]
- Y Young's modulus [M L⁻¹ T⁻²]
- B Bulk modulus [M L⁻¹ T⁻²]
- G Shear modulus [M L⁻¹ T⁻²]
- σ Poisson's ratio [M⁰ L⁰ T⁰]
- κ Compressibility [M⁻¹ L T²]
- S Surface tension [M T⁻²]
- SE Surface energy [M T⁻²]
- η Coefficient of viscosity [M L⁻¹ T⁻¹]
- dv/dx Velocity gradient [T⁻¹]
- ν Kinematic viscosity [L² T⁻¹]
- MF Mass flow rate [M T⁻¹]
- Q Volume flow rate [L³ T⁻¹]
- Re Reynolds number [M⁰ L⁰ T⁰]
- Q Heat [M L² T⁻²]
- C Heat capacity [M L² T⁻² K⁻¹]
- c Specific heat capacity [L² T⁻² K⁻¹]
- C Molar heat capacity [M L² T⁻² K⁻¹ mol⁻¹]
- L Latent heat [L² T⁻²]
- K Thermal conductivity [M L T⁻³ K⁻¹]
- dT/dx Temperature gradient [L⁻¹ K]
- dQ/dt Heat flow rate [M L² T⁻³]
- α Coefficient of linear expansion [K⁻¹]
- γ Coefficient of volume expansion [K⁻¹]
- Rth Thermal resistance [M⁻¹ L⁻² T³ K]
- S Entropy [M L² T⁻² K⁻¹]
- U Internal energy [M L² T⁻²]
- e Emissivity [M⁰ L⁰ T⁰]
- γ Ratio of specific heats [M⁰ L⁰ T⁰]
- n Number density [L⁻³]
- λ Mean free path [L]
- vrms Root mean square velocity [L T⁻¹]
- M Molar mass [M mol⁻¹]
- q Heat flux [M T⁻³]
- S Solar constant [M T⁻³]
- λ Wavelength [L]
- k Wave number [L⁻¹]
- f Frequency [T⁻¹]
- ω Angular frequency [T⁻¹]
- T Time period [T]
- A Amplitude [L]
- I Intensity of a wave [M T⁻³]
- β Sound level [M⁰ L⁰ T⁰]
- b Damping constant [M T⁻¹]
- μ Linear mass density [M L⁻¹]
- q Electric charge [T A]
- E Electric field [M L T⁻³ A⁻¹]
- V Electric potential [M L² T⁻³ A⁻¹]
- V Potential difference [M L² T⁻³ A⁻¹]
- ε Electromotive force (emf) [M L² T⁻³ A⁻¹]
- UE Electric potential energy [M L² T⁻²]
- ΦE Electric flux [M L³ T⁻³ A⁻¹]
- p Electric dipole moment [L T A]
- C Capacitance [M⁻¹ L⁻² T⁴ A²]
- εr Relative permittivity [M⁰ L⁰ T⁰]
- σ Surface charge density [L⁻² T A]
- λ Linear charge density [L⁻¹ T A]
- ρ Volume charge density [L⁻³ T A]
- J Current density [L⁻² A]
- R Electrical resistance [M L² T⁻³ A⁻²]
- G Conductance [M⁻¹ L⁻² T³ A²]
- ρ Resistivity [M L³ T⁻³ A⁻²]
- σ Conductivity [M⁻¹ L⁻³ T³ A²]
- μ Mobility [M⁻¹ T² A]
- α Temperature coefficient of resistance [K⁻¹]
- Z Electrochemical equivalent [M T⁻¹ A⁻¹]
- τ Time constant [T]
- B Magnetic field [M T⁻² A⁻¹]
- ΦB Magnetic flux [M L² T⁻² A⁻¹]
- m Magnetic moment [L² A]
- H Magnetising field [L⁻¹ A]
- M Magnetisation [L⁻¹ A]
- χ Magnetic susceptibility [M⁰ L⁰ T⁰]
- μr Relative permeability [M⁰ L⁰ T⁰]
- mp Magnetic pole strength [L A]
- L Self inductance [M L² T⁻² A⁻²]
- M Mutual inductance [M L² T⁻² A⁻²]
- X Reactance [M L² T⁻³ A⁻²]
- XL Inductive reactance [M L² T⁻³ A⁻²]
- XC Capacitive reactance [M L² T⁻³ A⁻²]
- Z Impedance [M L² T⁻³ A⁻²]
- cos φ Power factor [M⁰ L⁰ T⁰]
- Q Quality factor [M⁰ L⁰ T⁰]
- Id Displacement current [A]
- S Poynting vector [M T⁻³]
- f Focal length [L]
- P Power of a lens [L⁻¹]
- n Refractive index [M⁰ L⁰ T⁰]
- m Magnification [M⁰ L⁰ T⁰]
- Φv Luminous flux [cd]
- Ev Illuminance [L⁻² cd]
- β Fringe width [L]
- φ0 Work function [M L² T⁻²]
- E Photon energy [M L² T⁻²]
- p Photon momentum [M L T⁻¹]
- λ de Broglie wavelength [L]
- V0 Stopping potential [M L² T⁻³ A⁻¹]
- λ Decay constant [T⁻¹]
- R Activity [T⁻¹]
- Thalf Half-life [T]
- Eb Binding energy [M L² T⁻²]
- G Gravitational constant [M⁻¹ L³ T⁻²]
- c Speed of light [L T⁻¹]
- h Planck's constant [M L² T⁻¹]
- ħ Reduced Planck's constant (h-bar) [M L² T⁻¹]
- k Boltzmann constant [M L² T⁻² K⁻¹]
- R Universal gas constant [M L² T⁻² K⁻¹ mol⁻¹]
- NA Avogadro constant [mol⁻¹]
- e Elementary charge [T A]
- ε0 Permittivity of free space [M⁻¹ L⁻³ T⁴ A²]
- μ0 Permeability of free space [M L T⁻² A⁻²]
- k Coulomb constant [M L³ T⁻⁴ A⁻²]
- σ Stefan-Boltzmann constant [M T⁻³ K⁻⁴]
- b Wien's constant [L K]
- RH Rydberg constant [L⁻¹]
- F Faraday constant [T A mol⁻¹]
- a0 Bohr radius [L]
- me Electron mass [M]
- u Atomic mass unit [M]
- μB Bohr magneton [L² A]
Nothing matches. Try a symbol, a unit or another word.
Find a formula
Assume the target is a number k times each chosen quantity raised to some power. Matching the powers of mass, length and time on both sides gives equations, and solving them gives the powers. The number k is beyond a dimension check.
Pick what to find and at least one quantity it depends on.
Dimensional formulas and checks
What is a dimensional formula?
It writes a physical quantity in terms of the seven base quantities: mass M, length L, time T, current A, temperature K, amount of substance mol and luminous intensity cd. Force, for example, is [M L T⁻²].
How do I check an equation with dimensions?
Work out the dimensions of every term that is added or subtracted. They must all match, and both sides of the equals sign must match. Type the equation into the checker and it shows the dimensions of each term.
Does a matching check prove an equation is correct?
No. Pure numbers such as 1/2 or 2π have no dimensions, so a check cannot see them. Two quantities with the same dimensions, such as work and torque, also cannot be told apart.
Why does a symbol like P or T have several meanings?
Books reuse letters. P can be pressure, power or momentum, and T can be time period, temperature or tension. The checker tries the readings, tells you which one made the equation work, and lets you pick another.
Can dimensions give me a formula?
Often. If you know what a quantity depends on, matching the powers of mass, length and time can fix the formula up to a pure number, as for the time period of a pendulum. The formula finder shows the working.