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Modern Physics formula solvers

Photon Energy (E = hf)

E=hfE = h f

Modern PhysicsPhysicsThe energy of a single photon: Planck's constant times the light's frequency.

Photon Energy from Wavelength (E = hc/λ)

E=hcλE = \frac{h c}{\lambda}

Modern PhysicsPhysicsPhoton energy written in terms of wavelength: shorter waves, more energetic photons.

Photon Momentum (p = h/λ)

p=hλp = \frac{h}{\lambda}

Modern PhysicsPhysicsMassless but not momentum-less: a photon carries h divided by its wavelength.

De Broglie Wavelength

λ=hmv\lambda = \frac{h}{m v}

Modern PhysicsPhysicsEvery moving particle has a wavelength: Planck's constant over its momentum mv.

Mass-Energy Equivalence (E = mc²)

E=mc2E = m c^{2}

Modern PhysicsPhysicsThe rest energy locked in mass: multiply by the speed of light squared.

Photoelectric Effect

KEmax=Ephoton−ϕKE_{max} = E_{photon} - \phi

Modern PhysicsPhysicsMaximum kinetic energy of an ejected electron: photon energy (hf) minus the work function.

Lorentz Factor

γ=11−β2\gamma = \frac{1}{\sqrt{1 - \beta^{2}}}

Modern PhysicsPhysicsThe relativistic stretch factor, written in terms of β = v/c.

Time Dilation

Δt=Δt01−β2\Delta t = \frac{\Delta t_0}{\sqrt{1 - \beta^{2}}}

Modern PhysicsPhysicsA moving clock's proper time Δt₀ stretches to Δt for a stationary observer; β = v/c.

Length Contraction

L=L01−β2L = L_0 \sqrt{1 - \beta^{2}}

Modern PhysicsPhysicsA moving object's rest length L₀ contracts to L along its direction of motion; β = v/c.

Radioactive Activity (A = λN)

A=λNA = \lambda N

Modern PhysicsChemistryPhysicsDecays per second: the decay constant times the number of undecayed nuclei.

Wien's Displacement Law

λmax=bT\lambda_{max} = \frac{b}{T}

Modern PhysicsThermodynamicsPhysicsThe peak wavelength of thermal radiation, with b = 2.8978 × 10⁻³ m·K.

Bohr Model Energy Levels

En=−13.606 eVn2E_n = -\frac{13.606\ \mathrm{eV}}{n^{2}}

Modern PhysicsChemistryPhysicsEnergy of the hydrogen atom's nth level: −13.606 eV divided by n².