Angular motion
Equations of angular motion are relevant wherever you have rotational motions around an axis. When the object has rotated through an angle of θ with an angular velocity of ω and an angular acceleration of α, then you can use these equations to tie these values together.
Carnot engines
A heat engine takes heat, Qh, from a high temperature source at temperature Th and moves it to a low temperature sink (temperature Tc) at a rate Qc and, in the process, does mechanical work, W. (This process can be reversed such that work can be performed to move the heat in the opposite direction — a heat pump.) The amount of work performed in proportion to the amount of heat extracted from the heat source is the efficiency of the engine. A Carnot engine is reversible and has the maximum possible efficiency, given by the following equations. The equivalent of efficiency for a heat pump is the coefficient of performance.
Fluids
A volume, V, of fluid with mass, m, has density, ρ. A force, F, over an area, A, gives rise to a pressure,P. The pressure of a fluid at a depth of h depends on the density and the gravitational constant, g. Objects immersed in a fluid causing a mass of weight, Wwater displaced, give rise to an upward directed buoyancy force, Fbuoyancy. Because of the conservation of mass, the volume flow rate of a fluid moving with velocity, v, through a cross-sectional area, A, is constant. Bernoulli’s equation relates the pressure and speed of a fluid.
Forces
A mass, m, accelerates at a rate, a, due to a force, F, acting. Frictional forces, FF, are in proportion to the normal force between the materials, FN, with a coefficient of friction, μ. Two masses, m1 and m2, separated by a distance, r, attract each other with a gravitational force, given by the following equations, in proportion to the gravitational constant G:
Electron = -1.602 19
× 10-19 C = 9.11 × 10-31 kg
Proton = 1.602 19 ×
10-19 C = 1.67 × 10-27 kg
Neutron = 0 C = 1.67 × 10-27 kg
6.022 × 1023 atoms in one atomic mass unit
e is the elementary charge: 1.602 19 × 10-19 C
Potential Energy, velocity of electron: PE = eV
= ½mv2
1V = 1J/C 1N/C = 1V/m 1J = 1 N·m = 1 C·V
1 amp = 6.21 × 1018 electrons/second = 1 Coulomb/second 1 hp = 0.756 kW
1 N = 1 T·A·m
1 Pa = 1 N/m2
Power = Joules/second =
I2R = IV [watts W]
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Quadratic 2 Kinetic Energy [J]
Equation: x = 2a KE = 2 mv2
[Natural Log: when eb = x, ln x = b ]
m: 10-3 m: 10-6 n: 10-9 p: 10-12 f: 10-15 a: 10-18
Addition of Multiple Vectors:
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R = A+ B +C Resultant = Sum of the vectors
Rx = Ax + Bx + Cx x-component Ax = A cosq
Ry = Ay + By + Cy y-component Ay = A sin q
R = Rx2 + Ry2 Magnitude (length) of R
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qR = tan-1 Ry or tanqR = Ry Angle of the resultant
Multiplication of Vectors:
Cross Product or Vector Product: Positive direction:
i ´ j = k j ´ i = -k
i ´ i = 0 j k
Derivative of Vectors:
Velocity is the derivative of position with respect to time:
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v = dt (xi+ yj+ zk) = dt i+ dt j+ dt k
Acceleration is the derivative of velocity with respect to time:
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a = dt (vxi+ vy j+ vzk) = dt i+ dty j+ dtz k
Rectangular Notation: Z = R± jX where +j represents
inductive reactance and -j
represents capacitive reactance. For example, Z =8+ j6W
means that a resistor of 8W is
in series with an inductive reactance of 6W.
Polar Notation: Z = M Ðq, where M is the magnitude of the reactance and
q is the direction with respect to
the horizontal (pure resistance) axis. For example, a resistor of
4W in series with a capacitor with a reactance of 3W
would be expressed as 5 Ð-36.9° W.
In the descriptions above, impedance is used as an example. Rectangular and Polar
Notation can also be
used to
express
amperage, voltage, and
power.
To convert from rectangular to polar notation: Given: X - jY (careful with
the sign before the ”j”)
Magnitude: X2 +Y2 = M
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Angle: (negative sign carried over from rectangular notation in this example)
Note: Due to the way the calculator works, if X is negative, you must add 180° after taking the inverse tangent.
If the result is greater
than 180°, you may optionally subtract 360° to obtain the value closest
to the reference angle.
To convert from polar to rectangular (j) notation:
Given: M Ðq
X Value: M cosq
Y (j) Value: M sinq Y
In conversions, the j value will have the same sign as the q value for angles having a magnitude < 180°.
Use rectangular notation when adding and subtracting.
Use polar notation for multiplication and division. Multiply in polar notation
by multiplying the magnitudes and adding the angles. Divide in polar notation by dividing the
magnitudes
and subtracting the denominator angle
from
the numerator
angle.
ELECTRIC CHARGES AND FIELDS Coulomb's Law: [Newtons N]
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where: F = force on one charge by 2
the other[N]
k = 8.99 × 109 [N·m2/C2] q1 = charge [C]
q2 = charge [C] r = distance [m]
Electric Field: [Newtons/Coulomb or Volts/Meter]
where: E = electric field [N/C or V/m]
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E = k = k = 8.99 × 109 [N·m2/C2] q = charge [C]
r = distance [m] F = force
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Electric field
lines radiate outward from
positive charges. The electric field is zero inside a conductorSpeed = distance / time.
Average speed = total distance / total time.
(m/s or cm/s)
v=d/t
Velocity= displacement / time .
(m/s + direction)
a=(vf-vi)/t or Δv/t
Acceleration is equal to (final v - initial v) / time.
(m/s²)
g=10 m/s²
Gravity; acceleration of a free falling object is 10 m/s².
(m/s² or N/kg)
F=ma
Newton's 2nd Law; the force on an object is equal to the product of its mass and acceleration.
(Newtons (N))
Fg=mg=W
Force of gravity on an object is the product of the mass and gravity, which is its weight.
(N)
F=kx
Hooke's law: The extension of a spring (x) is proportional to the load placed on the spring (F), until it reaches the limit of proportionality. K is the spring constant; it is in N/m or N/cm. It is a number that describes how hard it is to stretch the spring.
(F in N, K in N/cm or N/m, x in cm or m)
Ek=1/2mv²
Kinetic Energy; energy of motion = 1/2 x mass x velocity².
(Joules (J))
Ep=Ek(mgh=1/2mv²)
Conservation of energy for a falling body, energy at top (mgh) is equal to the energy at the bottom (1/2 mv²) and vice versa.
W=Fd
Work is force times distance, measured in Joules.
(J)
W=ΔE
When you do work on an object you change its energy, either its kinetic energy or potential energy.
(J)
W=Fd=ΔEk or ΔEp
Conservation of energy for work done on an object.
P=W/t or ΔE/t
Power is the rate of work or energy transfer.
(Watts (W) or J/s)
Efficiency
Efficiency = (useful work or power output) / (total work or power input) x 100%.
(%)
Pressure=F/A
P=pgh density x gravity x height (depth in the water).
(Pa or N/m²)
Q=mc Δ T
The heat energy is joules required to change temperature of matter with no phase change.
(J)
Q= mL
Lf or Lv; The heat energy in joules required for a phase change (either melting, Lf, or boiling Lv).
| Name of the quantity | Formula | Quantities | Unit |
| Voltage (V) | V = IR | I = Current R = Resistance | Volts (V) |
| Power (P) | P = I V | V = Voltage | Watts (W) |
| Power (P) | P = $\frac{V^{2}}{R}$ | R = Resistance | Watts (W) |
| Power (P) | P = I2 R | I = Current | Watts(W) |
Mechanics
Mechanics is the oldest branch of physics. Mechanics deals with all kinds and complexities of motion. It includes various techniques, which can simplify the solution of a mechanical problem.
Motion in One Dimension
The formulas for motion in one dimension (Also called Kinematical equations of motion) are as follows. (Here 'u' is initial velocity, 'v' is final velocity, 'a' is acceleration and t is time):
s = ut + ½ at2
v = u + at
v2 = u2 + 2as
vav (Average Velocity) = (v+u)/2
Momentum, Force and Impulse
Formulas for momentum, impulse and force concerning a particle moving in 3 dimensions are as follows (Here force, momentum and velocity are vectors ):
Momentum is the product of mass and velocity of a body. Momentum is calculate using the formula: P = m (mass) x v (velocity)
Force can defined as something which causes a change in momentum of a body. Force is given by the celebrated newton's law of motion: F = m (mass) x a (acceleration)
Impulse is a large force applied in a very short time period. The strike of a hammer is an impulse. Impulse is given by I = m(v-u)
Pressure
Pressure is defined as force per unit area:
Pressure (P) = Force (F)
Area (A)
Density
Density is the mass contained in a body per unit volume.
The formula for density is:
Density (D) = Mass(M)
Volume (V)
Angular Momentum
Angular momentum is an analogous quantity to linear momentum in which the body is undergoing rotational motion. The formula for angular momentum (J) is given by:
J = r x p
where J denotes angular momentum, r is radius vector and p is linear momentum.
Torque
Torque can be defined as moment of force. Torque causes rotational motion. The formula for torque is: τ = r x F, where τ is torque, r is the radius vector and F is linear force.
Circular Motion
The formulas for circular motion of an object of mass 'm' moving in a circle of radius 'r' at a tangential velocity 'v' are as follows:
Centripetal force (F) = mv2
r
Centripetal Acceleration (a) = v2
r
Center of Mass
General Formula for Center of mass of a rigid body is :
R = ΣNi = 1 miri
ΣNi = 1mi
where R is the position vector for center of mass, r is the generic position vector for all the particles of the object and N is the total number of particles.
Reduced Mass for two Interacting Bodies
The physics formula for reduced mass (μ) is :
μ = m1m2
m1 + m2
where m1 is mass of the first body, m2 is the mass of the second body.
Work and Energy
Formulas for work and energy in case of one dimensional motion are as follows:
W (Work Done) = F (Force) x D (Displacement)
Energy can be broadly classified into two types, Potential Energy and Kinetic Energy. In case of gravitational force, the potential energy is given by
P.E.(Gravitational) = m (Mass) x g (Acceleration due to Gravity) x h (Height)
The transitional kinetic energy is given by ½ m (mass) x v2(velocity squared)
Power
Power is, work done per unit time. The formula for power is given as
Power (P) = V2
R =I2R
where P=power, W = Work, t = time.
Friction can be classified to be of two kinds : Static friction and dynamic friction.
Static Friction: Static friction is characterized by a coefficient of static friction μ . Coefficient of static friction is defined as the ratio of applied tangential force (F) which can induce sliding, to the normal force between surfaces in contact with each other. The formula to calculate this static coefficient is as follows:
μ = Applied Tangential Force (F)
Normal Force(N)
The amount of force required to slide a solid resting on flat surface depends on the co efficient of static friction and is given by the formula:
FHorizontal = μ x M(Mass of solid) x g (acceleration)
Dynamic Friction:
Dynamic friction is also characterized by the same coefficient of friction as static friction and therefore formula for calculating coefficient of dynamic friction is also the same as above. Only the dynamic friction coefficient is generally lower than the static one as the applied force required to overcome normal force is lesser.
Formulas for Moments of Inertia of different objects. (M stands for mass, R for radius and L for length):
| Object | Axis | Moment of Inertia |
| Disk | Axis parallel to disc, passing through the center | MR2/2 |
| Disk | Axis passing through the center and perpendicular to disc | MR2/2 |
| Thin Rod | Axis perpendicular to the Rod and passing through center | ML2/12 |
| Solid Sphere | Axis passing through the center | 2MR2/5 |
| Solid Shell | Axis passing through the center | 2MR2/3 |
Newton's Law of universal Gravitation:
Fg = Gm1m2
r2
where
m1, m2 are the masses of two bodies
G is the universal gravitational constant which has a value of 6.67300 × 10-11 m3 kg-1 s-2
r is distance between the two bodies
Formula for escape velocity (vesc) = (2GM / R)1/2where,
M is mass of central gravitating body
R is radius of the central body
Projectile motion:
(v = velocity of particle, v0 = initial velocity, g is acceleration due to gravity, θ is angle of projection, h is maximum height and l is the range of the projectile.)
Maximum height of projectile (h) = v0 2sin2θ
2g
Horizontal range of projectile (l) = v0 2sin 2θ / g
The physics formula for the period of a simple pendulum (T) = 2π √(l/g)where
l is the length of the pendulum
g is acceleration due to gravity
Conical Pendulum
The Period of a conical pendulum (T) = 2π √(lcosθ/g)
where
l is the length of the pendulum
g is acceleration due to gravity
Half angle of the conical pendulum
Electricity.
Ohm's Law
Ohm's law gives a relation between the voltage applied a current flowing across a solid conductor:
V (Voltage) = I (Current) x R (Resistance)
Power
In case of a closed electrical circuit with applied voltage V and resistance R, through which current I is flowing,
Power (P) = V2
R
= I2R. . . (because V = IR, Ohm's Law)
Kirchoff's Voltage Law
For every loop in an electrical circuit:
ΣiVi = 0
where Vi are all the voltages applied across the circuit.
Kirchoff's Current Law
At every node of an electrical circuit:
ΣiIi = 0
where Ii are all the currents flowing towards or away from the node in the circuit.
Resistance
The physics formulas for equivalent resistance in case of parallel and series combination are as follows:
Resistances R1, R2, R3 in series:
Req = R1 + R2 + R3
Resistances R1 and R2 in parallel:
Req = R1R2
R1 + R2
For n number of resistors, R1, R2...Rn, the formula will be:
1/Req = 1/R1 + 1/R2 + 1/R3...+ 1/Rn
Capacitors
A capacitor stores electrical energy, when placed in an electric field. A typical capacitor consists of two conductors separated by a dielectric or insulating material. Here are the most important formulas related to capacitors. Unit of capacitance is Farad (F) and its values are generally specified in mF (micro Farad = 10 -6 F).
Capacitance (C) = Q / V
Energy Stored in a Capacitor (Ecap) = 1/2 CV2 = 1/2 (Q2 / C) = 1/2 (QV)
Current Flowing Through a Capacitor I = C (dV / dt)
Equivalent capacitance for 'n' capacitors connected in parallel:
Ceq (Parallel) = C1 + C2 + C3...+ Cn = Σi=1 to n Ci
Equivalent capacitance for 'n' capacitors in series:
1 / Ceq (Series) = 1 / C1 + 1 / C2...+ 1 / Cn = Σi=1 to n (1 / Ci)
Here
C is the capacitance
Q is the charge stored on each conductor in the capacitor
V is the potential difference across the capacitor
Parallel Plate Capacitor Formula:
C = kε0 (A/d)
Where
k = dielectric constant (k = 1 in vacuum)
ε0 = Permittivity of Free Space (= 8.85 × 10-12 C2 / Nm2)
A = Plate Area (in square meters)
d = Plate Separation (in meters)
Cylinrical Capacitor Formula:
C = 2π kε0 [L / ln(b / a)]
Where
k = dielectric constant (k = 1 in vacuum)
ε0 = Permittivity of Free Space (= 8.85 × 10-12 C2 / Nm2)
L = Capacitor Length
a = Inner conductor radius
b = Outer conductor radius
Spherical Capacitor Formula:
C = 4π kε0 [(ab)/(b-a)]
Where
k = dielectric constant (k = 1 in vacuum)
ε0 = Permittivity of Free Space (= 8.85 × 10-12 C2 / Nm2)
a = Inner conductor radius
b = Outer conductor radius
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 inductance measured in Henries
N is the number of turns on the coil
A is cross-sectional area of the coil
m0 is the permeability of free space (= 4π × 10-7 H/m)
K is the Nagaoka coefficient
l is the length of coil
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
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 |
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| Internal Energy (U) | dU = TdS − pdV + µdN | |
| Enthalpy (H) | H = U + pV dH = TdS + Vdp + µdN |
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| 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 |
|
| Boyle's Law | PV = Constant or P1V1 = P2V2 (At Constant Temperature) |
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| Charles's Law | V / T = Constant or V1 / T1 = V2 / T2 (At Constant Pressure) |
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| Amontons' Law of Pressure-Temperature | P / T = Constant or P1 / T1 = P2 / T2 (At Constant Volume) |
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| Equation of State For An Ideal Gas | PV = nRT = NkT |
Kinetic Theory of Gases
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
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
Formulas related to the very basics of quantum physics.
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
2π
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
2π
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
2π
where ΔE is the uncertainty in energy and Δt is the uncertainty in time.
