A Comprehensive List of All the Physics Formulas(Part 2)

Learning physics is all about applying concepts to solve problems. This article provides a comprehensive physics formulas list, that will act as a ready reference, when you are solving physics problems. You can even use this list, for a quick revision before an exam.

Inductors

An inductor is an electrical component which resists the flow of electrons or electric current through it. This property of inductance, in these devices, is caused by the electromotive force, created by magnetic field induced in them. The unit of inductance is Henry. Here are some important formulas associated with inductors.

Energy Stored in Inductor (Estored) = 1/2 (LI2)

Where, L is inductance and I is the current flowing through the inductor.

Inductance of a cylindrical air core Coil (L) = (m0KN2A / l)

Where

  • l is the length of coil
  • m0 is the permeability of free space (= 4π × 10-7 H/m)
  • N is the number of turns on the coil
  • L is inductance measured in Henries
  • A is cross-sectional area of the coil
  • K is the Nagaoka coefficient

Inductors in a Series Network
For inductors, L1, L2…Ln connected in series,

Leq = L1 + L2…+ Ln (L is inductance)

Inductors in a Parallel Network
For inductors, L1, L2…Ln connected in parallel,

1 / Leq = 1 / L1 + 1 / L2…+ 1 / Ln

Thermodynamics Formulas

Thermodynamics is a vast field providing an analysis of the behavior of matter in bulk. It’s a field focused on studying matter and energy in all their manifestations. Here are some of the most important formulas associated with classical thermodynamics and statistical physics.

First Law of Thermodynamics

dU = dQ + dW

where, dU is the change in internal energy, dQ is the heat absorbed by the system and dW is the work done on the system.

Thermodynamic Potentials

All of thermodynamical phenomena can be understood in terms of the changes in five thermodynamic potentials under various physical constraints. They are Internal Energy (U), Enthalpy (H), Helmholtz Free Energy (F), Gibbs Free Energy (G), Landau or Grand Potential (Φ). Each of these scalar quantities represents the potentiality of a thermodynamic system to do work of various kinds under different types of constraints on its physical parameters.

Thermodynamic Potential Defining Equation
  • U is Energy
  • T is temperature
  • S is Entropy
  • N is particle number
  • µ is Chemical Potential
  • p is Pressure
  • V is Volume
  • H is Enthalpy
  • G is Gibbs Free Energy
  • Φ is Grand Potential
  • F is Helmholtz Free Energy
Internal Energy (U) dU = TdS − pdV + µdN
Enthalpy (H) H = U + pV
dH = TdS + Vdp + µdN
Gibbs Free Energy (G) G = U – TS + pV = F + pV = H – TS
dG = -SdT + Vdp + µdN
Helmholtz Free Energy (F) F = U – TS
dF = – SdT – pdV + µdN
Landau or Grand Potential Φ = F – µN
dΦ = – SdT – pdV – Ndµ

Ideal Gas Equations

An ideal gas is a physicist’s conception of a perfect gas composed of non-interacting particles which are easier to analyze, compared to real gases, which are much more complex, consisting of interacting particles. The resulting equations and laws of an ideal gas conform with the nature of real gases under certain conditions, though they fail to make exact predictions due the interactivity of molecules, which is not taken into consideration. Here are some of the most important physics formulas and equations, associated with ideal gases. Let’s begin with the prime ideal gas laws and the equation of state of an ideal gas.

Law Equation
  • P is Pressure
  • V is Volume
  • T is Temperature
  • n is the number of moles
  • R is the ideal gas constant [= 8.3144621(75) J / K mol]
  • N is the number of particles
  • k is the Boltzmann Constant (= 1.3806488(13)×10-23)
Boyle’s Law PV = Constant
or
P1V1 = P2V2
(At Constant Temperature)
Charles’s Law V / T = Constant
or
V1 / T1 = V2 / T2
(At Constant Pressure)
Amontons’ Law of Pressure-Temperature P / T = Constant
or
P1 / T1 = P2 / T2
(At Constant Volume)
Equation of State For An Ideal Gas PV = nRT = NkT

Kinetic Theory of Gases
Based on the primary assumptions that the volume of atoms or molecules is negligible, compared to the container volume and the attractive forces between molecules are negligible, the kinetic theory describes the properties of ideal gases. Here are the most important physics formulas related to the kinetic theory of monatomic gases.

Pressure (P) = 1/3 (Nm v2)

Here, P is pressure, N is the number of molecules and v2 is the mean squared particle velocity.

Internal Energy (U) = 3/2 (NkT)

Heat Capacities

Heat Capacity at Constant Pressure (Cp) = 5/2 Nk = Cv + Nk

Heat Capacity at Constant Volume (Cv) = 3/2 Nk

Ratio of Heat Capacities (γ) = Cp / Cv = 5/3

Velocity Formulas

Mean Molecular Velocity (Vmean) = [(8kT)/(πm)]1/2

Root Mean Square Velocity of a Molecule (Vrms) = (3kT/m)1/2

Most Probable Velocity of a Molecule (Vprob) = (2kT/m)1/2

Mean Free Path of a Molecule (λ) = (kT)/√2πd2P (Here P is in Pascals)

Here N is the number of molecules, k is the Boltzmann constant, P is pressure, d is the molecular diameter, m is mass of the molecule and T is the gas temperature.

Electromagnetism
Here are some of the basic formulas from electromagnetism.
The coulombic force between two charges at rest is
(F) = q1q2
4πε0r2
Here,

  • q1, q2 are charges
  • ε0 is the permittivity of free space
  • r is the distance between the two charges

Lorentz Force
The Lorentz force is the force exerted by an electric and/or magnetic field on a charged particle.

(Lorentz Force) F = q (E + v x B)
where

  • q is the charge on the particle
  • E and B are the electric and magnetic field vectors

Relativistic Mechanics
Here are some of the most important relativistic mechanics formulas. The transition from classical to relativistic mechanics is not at all smooth, as it merges space and time into one by taking away the Newtonian idea of absolute time. If you know what is Einstein’s special theory of relativity, then the following formulas will make sense to you.

Lorentz Transformations
Lorentz transformations can be perceived as rotations in four dimensional space. Just as rotations in 3D space mixes the space coordinates, a Lorentz transformation mixes time and space coordinates. Consider two, three dimensional frames of reference S(x,y,z) and S'(x’,y’,z’) coinciding with each other.

Now consider that frame S’ starts moving with a constant velocity v with respect to S frame. In relativistic mechanics, time is relative! So the time coordinate for the S’ frame will be t’ while that for S frame will be t.

Consider
γ = 1
√(1 – v2/c2)

The coordinate transformations between the two frames are known as Lorentz transformations and are given as follows:
Lorentz Transformations of Space and Time

x = γ (x’ + vt’) and x’ = γ (x – vt)

y = y’

z= z’

t = γ(t’ + vx’/c2) and t’ = γ(t – vx/c2)

Relativistic Velocity Transformations

In the same two frames S and S’, the transformations for velocity components will be as follows (Here (Ux, Uy, Uz) and (Ux’, Uy’, Uz’) are the velocity components in S and S’ frames respectively):

Ux = (Ux’ + v) / (1 + Ux’v / c2)

Uy = (Uy’) / γ(1 + Ux’v / c2)

Uz = (Uz’) / γ(1 + Ux’v / c2) and

Ux’ = (Ux – v) / (1 – Uxv / c2)

Uy’ = (Uy) / γ(1 – Uxv / c2)

Uz’ = (Uz) / γ(1 – Uxv / c2)

Momentum and Energy Transformations in Relativistic Mechanics

Consider the same two frames (S, S’) as in case of Lorentz coordinate transformations above. S’ is moving at a velocity ‘v’ along the x-axis. Here again γis the Lorentz factor. In S frame (Px, Py, Pz) and in S’ frame (Px’, Py’, Pz’) are momentum components. Now we consider formulas for momentum and energy transformations for a particle, between these two reference frames in relativistic regime.

Component wise Momentum Transformations and Energy Transformations

Px = γ(Px’ + vE’ / c2)

Py = Py’

Pz = Pz’

E = γ(E’ + vPx)

and

Px’ = γ(Px – vE’ / c2)

Py’ = Py

Pz’ = Pz

E’ = γ(E – vPx)

Physical Formulas for Quantities in Relativistic Dynamics

All the known quantities in classical mechanics get modified, when we switch over to relativistic mechanics which is based on the special theory of relativity. Here are formulas of quantities in relativistic dynamics.

Relativistic momentum p = γm0v
where m0 is the rest mass of the particle.

Rest mass energy E = m0c2

Total Energy (Relativistic) E = √(p2c2 + m02c4))

Optics

Optics is one of the oldest branches of physics. There are many important optics physics formulas, which we need frequently in solving physics problems. Here are some of the important and frequently needed optics formulas.

Snell’s Law

Sin i
Sin r = n2
n1 = v1
v2

  • where i is angle of incidence
  • r is the angle of refraction
  • n1 is refractive index of medium 1
  • n2 is refractive index of medium 2
  • v1, v2 are the velocities of light in medium 1 and medium 2 respectively

Gauss Lens Formula: 1/u + 1/v = 1/f
where

  • u – object distance
  • v – image distance
  • f – Focal length of the lens

Lens Maker’s Equation

The most fundamental property of any optical lens is its ability to converge or diverge rays of light, which is measued by its focal length. Here is the lens maker’s formula, which can help you calculate the focal length of a lens, from its physical parameters.

1 / f = [n-1][(1 / R1) – (1 / R2) + (n-1) d / nR1R2)]

Here,

  • n is refractive index of the lens material
  • R1 is the radius of curvature of the lens surface, facing the light source
  • R2 is the radius of curvature of the lens surface, facing away from the light source
  • d is the lens thickness

If the lens is very thin, compared to the distances – R1 and R2, the above formula can be approximated to:

(Thin Lens Approximation) 1 / f ≈ (n-1) [1 / R1 – 1 / R2]

Compound Lenses

The combined focal length (f) of two thin lenses, with focal length f1 and f2, in contact with each other:

1 / f = 1 / f1 + 1 / f2

If the two thin lenses are separated by distance d, their combined focal length is provided by the formula:

1 / f = 1 / f1 + 1 / f2 – (d / f1 – f2))

Newton’s Rings Formulas

Here are the important formulas for Newton’s rings experiment which illustrates diffraction.

nth Dark ring formula: r2n = nRλ

nth Bright ring formula: r2n = (n + ½) Rλ

where

  • nth ring radius
  • Radius of curvature of the lens
  • Wavelength of incident light wave

Quantum Physics

Quantum physics is one of the most interesting branches of physics, which describes atoms and molecules, as well as atomic sub-structure. Here are some of the formulas related to the very basics of quantum physics, that you may require frequently.

De Broglie Wave
De Broglie Wavelength:
λ = h
p

where, λ- De Broglie Wavelength, h – Planck’s Constant, p is momentum of the particle.

Bragg’s Law of Diffraction: 2a Sin θ = nλ
where

  • a – Distance between atomic planes
  • n – Order of Diffraction
  • θ – Angle of Diffraction
  • λ – Wavelength of incident radiation

Planck Relation
The plank relation gives the connection between energy and frequency of an electromagnetic wave:
E = hv = hω

where h is Planck’s Constant, v the frequency of radiation and ω = 2πv

Uncertainty Principle
Uncertainty principle is the bedrock on which quantum mechanics is based. It exposes the inherent limitation that nature imposes on how precisely a physical quantity can be measured. Uncertainty relation holds between any two non-commuting variables. Two of the special uncertainty relations are given below.

Position-Momentum Uncertainty

What the position-momentum uncertainty relation says is, you cannot predict where a particle is and how fast it is moving, both, with arbitrary accuracy. The more precise you are about the position, more uncertain will you be about the particle’s momentum and vice versa. The mathematical statement of this relation is given as follows:
Δx.Δp ≥ h

where Δx is the uncertainty in position and Δp is the uncertainty in momentum.

Energy-Time Uncertainty
This is an uncertainty relation between energy and time. This relation gives rise to some astounding results like, creation of virtual particles for arbitrarily short periods of time! It is mathematically stated as follows:
ΔE.Δt ≥ h

where ΔE is the uncertainty in energy and Δt is the uncertainty in time.

This concludes my review of some of the important physics formulas. This list, is only representative and is by no means anywhere near complete. Physics is the basis of all sciences and therefore its domain extends over all sciences. Every branch of physics theory abounds with countless formulas. If you resort to just mugging up all these formulas, you may pass exams, but you will not be doing real physics. If you grasp the underlying theory behind these formulas, physics will be simplified. To view physics through the formulas and laws, you must be good at maths. There is no way you can run away from it. Mathematics is the language of nature!

The more things we find out about nature, more words we need to describe them. This has led to increasing jargonization of science with fields and sub-fields getting generated. You could refer to a glossary of science terms and scientific definitions for any jargon that is beyond your comprehension.

If you really want to get a hang of what it means to be a physicist and get an insight into physicist’s view of things, read ‘Feynman Lectures on Physics’, which is highly recommended reading, for anyone who loves physics. It is written by one of the greatest physicists ever, Prof. Richard Feynman. Read and learn from the master. Solve as many problems as you can, on your own, to get a firm grasp of the subject.

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