From: Kavishka Gihan
Coulomb's law
$$
\begin{align*}
F =\frac{1}{4\pi\epsilon}\frac{Q_1Q_2}{r^2}
\end{align*}
$$
Relative Permittivity
$$
\begin{align*}
Relative\ permitivity &= \frac{\epsilon}{\epsilon_0}\\
\epsilon_0 &= permivitity\ of\ vacuum
\end{align*}
$$
Electric Field Intensity
$$
\begin{align*}
\vec{E} =& \frac{1}{4\pi\epsilon}\frac{Q}{r^2}
\end{align*}
$$
Force generated due to the electrical field intensity
$$
F = \vec{E}q
$$
Net electric flux (Gauss theorem)
$$
\begin{align*} \phi = \frac{Q}{\epsilon} \end{align*}
$$
Electric Flux Density (equates to electrical field intensity)
$$
\begin{align*}
\phi &= \vec{E}A
\end{align*}
$$
$$
\begin{align*}
\therefore \vec{E}A = \frac{Q}{\epsilon} \\
\ For\ a\ cylinder,\ \vec{E} &= \frac{1}{2\pi\epsilon}\frac{\lambda}{r}
\ For\ a\ conducting\ plate,\ \vec{E} &= \frac{\sigma}{\epsilon}
\ For\ an\ insulating\ plate,\ \vec{E} &= \frac{\sigma}{2\epsilon}
\ For parallel\ plate\ capacitors,\ \vec{E} &= \frac{Q}{A\epsilon}
\end{align*}
$$
Electric Potential
$$
\begin{align*}
V &= \frac{1}{4\pi \epsilon} \frac{Q}{r} \\
V &= \vec{E}r
\end{align*}
$$
Potential Difference
$$
\begin{align*}
V_{AB} = \delta v =& \frac{Q}{4\pi \epsilon}(\frac{1}{a} - \frac{1}{b}) \end{align*}
$$
Electric Potential Energy (workdone)
$$
\begin{align*}
E_{p} =&\ \delta vq \\
\end{align*}
$$
Potential gradient (equals to negative value of electric field intensity)
$$
\begin{align*}
P_g = (\frac{\delta v}{ \delta d})
\end{align*}
$$
$$
\vec{E} =-(\frac{\delta v}{ \delta d})
$$
Static electric capacitance
$$
Q = CV_s
$$
$$
\begin{align*}
For\ spherical\ conductor, \\
C &= 4\pi\epsilon R
\\
For\ parallel\ plate\ capacitors, \\
C &= \frac{A\epsilon}{d}
\ Capacitors\ in\ series\ connection , \\
\frac{1}{C} &= \frac{1}{C_1} + \frac{1}{C_2}
\ Capacitors\ in\ parallel\ connection, \\
C &= C_1 + C_2
\ Capacitror\ with\ 2\ dielectric\ media\ - in\ series, \\
C &= \frac{A \epsilon_1\epsilon_2}{d_1\epsilon_2 + d_2\epsilon_1}
\ Capacitror\ with\ 2\ dielectric\ media\ - in\ parallel, \\
C &= \frac{A_1\epsilon_1 + A_2\epsilon_2}{d}
\end{align*}
$$
Static electric potential energy
$$
\begin{align*}
E_p = \frac{1}{2}V_sQ = \frac{1}{2}CV_s^2 = \frac{1}{2}\frac{Q^2}{V_s}
\end{align*}
$$
Newtons Law of Gravitation
$$
F = \frac{Gm_1m_2}{r^2}
$$
Gravitational Field Intensity
$$
g = \frac{GM}{r^2}
$$
Gravitational Potential
$$
U = -\frac{GM}{r}
$$
Gravitational Potential Energy
$$
E_p = -mgr = Um
$$
Velocity of an artificial satellite
$$
v = \sqrt{\frac{GM}{r}}
$$
Angular velocity of an artificial satellite
$$
\omega = \sqrt{\frac{GM}{r^3}}
$$
Time period of an artificial satellite
$$
T = 2\pi\sqrt{\frac{r^3}{GM}}
$$
Total energy of an satellite
$$
\begin{align*}
E_T &= E_k + E_p \\
E_T &= \frac{1}{2}m\frac{GM}{r}-\frac{GMm}{r} \\
E_T &= \frac{GMm}{r}
\end{align*}
$$
Minimum energy
$$
E_{min} = GMm(\frac{1}{R} - \frac{1}{r})
$$
Escape Velocity
$$
v_{esc} = \sqrt{\frac{2GM}{R}}
$$
Triple point of water
$$
1 K = \frac{Temperature\ of\ triple\ point\ of\ water}{273.16}
$$
Solid Expansion
$$
\begin{align*}\delta l_2 =& l_1 ( 1 + \theta\alpha) \ \delta A_2 =& A_1 ( 1 + \theta\beta) \ \delta V_2 =& V_1 ( 1 + \theta\gamma) \ \end{align*}
$$
Liquid Expansion
$$
r = r_R + 3\alpha
$$
Density variation with temperature
$$
\begin{align*}
\rho_2 =& \frac{\rho_1}{(1 + 3\alpha\theta)}
\end{align*}
$$
Boyle's Law - Only for idea gasses
$$
P_1V_1 = P_2V_2
$$
Charles's Law- Only for idea gasses
$$
\frac{V_1}{T_1} = \frac{V_2}{T_2}
$$
Volume expansion of gasses (Under constant pressure)
$$
\begin{align*}
V =& V_o(1 + \gamma_p\theta) \to \gamma_p= 0.003^0C
\end{align*}
$$
Pressure Law - Only for idea gasses
$$
\frac{P_1}{T_1} = \frac{P_2}{T_2}
$$
Combined gas equation
$$
\begin{align*}
\frac{P_1V_1}{T_1} =& \frac{P_2V_2}{T_2}
\end{align*}
$$
Ideal gas equation
$$
\begin{align*}
PV &= nRT \ \\
Other\ forms,\\
PV &= \frac{m}{M}RT \\
\frac{P}{\rho} &= \frac{RT}{M}
\end{align*}
$$
Avogadro Law - for any gass
$$
\begin{align*}
\frac{V}{N} &= k \\
(N &= nL)
\end{align*}
$$
Dalton's Law of partial pressure.
$$
P_T = P_A + P_B + P_C
$$
Kinetic theory equations (optional forms)
$$
\begin{align*}
For\ a\ molecule,\\
PV &= \frac{1}{3}Nm_o\bar{c^2}
\\
For\ a\ gass,\\
PV &= \frac{1}{3}m\bar{c^2}\\
P &= \frac{1}{3}\rho\bar{c^2}
\end{align*}
$$
Relationship between root mean square and absolute temperature
$$
\begin{align*}
c &= \sqrt{\frac{3RT}{M}}
\end{align*}
$$
Kinetic energy of gas molecule
$$
\begin{align*}
E_K &= (\frac{3}{2})(\frac{R}{L})T \ \ \frac{R}{L} =& Boltzman\ Constant (K) \ \ \therefore E_K &= \frac{3}{2}KT
\end{align*}
$$
Relative Humidity
$$
\begin{align*}
In\ terms\ of\ vapour\ density, \\
R.H &= \frac{\rho}{\rho_s} * 100 % \\
In\ terms\ of\ vapour\ mass (constant\ volume), \\
R.H &= \frac{m}{m_s} * 100 % \\
In\ terms\ of\ vapour\ pressure, \\
R.H &= \frac{P}{P_s} * 100 % \\
In\ terms\ of\ dew\ point, \\
R.H &= \frac{S.V.P\ @\ \theta_D}{S.V.P\ @\ \theta_R} = \frac{P_{SD}}{P_{SR}} * 100 % \\
In\ terms\ of\ absolute\ humidity, \\
R.H &= \frac{A.H\ @\ \theta_R}{A.H\ @\ \theta_D} = \frac{P_{R}}{P_{D}} * 100 % \\
\end{align*}
$$
Heat capacity
$$
H = C\theta
$$
Specific Heat Capacity
$$
H = mS\theta
$$
Molar Heat Capacity
$$
H = nC_0\theta
$$
Relationship between molar heat capacity and Molar mass
$$
C_o = MS
$$
Specific Latent Heat
$$
H = mL
$$
Relationship between Molar heat capacity under constant pressure and constant volume
$$
\begin{align*}
C_p - C_v &= P\delta V
\end{align*}
$$
Workdone by a gas
$$
\begin{align*}
For\ a\ whole\ gas,\\
W &= P\delta V \\
For\ a\ 1\ mole\ of\ gas,\\
W_o &= C_p - C_v
\end{align*}
$$
Relationship between Molar heat capacity under constant pressure, constant volume and universal gas constant
$$
C_p - C_v = R
$$
Relationship between specific heat capacity under constant pressure, constant volume and universal gas constant
$$
S_p - S_V = \frac{R}{M}
$$
Atomicity
$$
Atomicity (\gamma) = \frac{S_p}{S_v} =\frac{C_p}{C_v}
$$
First law of thermodynamics
$$
\begin{align*}
\delta Q &= \delta U + \delta W
\\\
When\ work\ is\ done\ by\ the\ gas, \\
\delta Q &= \delta U + \delta W
\\
When\ work\ is\ done\ on\ the\ gas, \\
\delta Q &= \delta U - \delta W
\end{align*}
$$
Thermodynamic processes
$$
\begin{align*}
Isothermal\ process (\delta T = 0), \\
\delta Q &= \delta W \ &\to P_1V_1 = P_2V_2\\
Adiabatic\ process (\delta Q = 0), \\
\delta U &= - \delta W \\
\delta U &= - P\delta V \ &\to \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}\\
Isobaric\ process (\delta P= 0), \\
\delta Q &= \delta U + \delta W \ &\to \frac{V_1}{T_1} = \frac{V_2}{T_2}\\
Isochoric\ process (\delta V = 0), \\
\delta Q &= \delta U \ &\to \frac{P_1}{T_1} = \frac{P_2}{T_2}\\
\end{align*}
$$
PV curves
$$
\begin{align*}
Area\ under\ the\ graph = Workdone = P\delta V
\\
\\
Clockwise\ process \to +W\\
Anti-clockwise\ process \to -W\\
\end{align*}
$$
Excess Temperature
$$
\theta_E = \theta_S - \theta_R
$$
Rate of loosing heat
$$
\begin{align*}
\frac{dH}{dt} &= KA(\theta_S - \theta_R)
\end{align*}
$$
Rate of cooling
$$
\begin{align*}
\frac{d\theta}{dt} &= \frac{KA}{mS}(\theta_S - \theta_R)
\ \\
Rate\ of\ cooling &= \frac{Rate\ of\ loosing\ heat}{mS}
\end{align*}
$$
Cooling Curve
$$
Gradient\ of\ the\ graph = Rate\ of\ Cooling = \frac{d\theta}{dt}\\
$$
Magnetic Flux Density
$$
\begin{align*}
B = \frac{\phi}{A}
\end{align*}
$$
Force generated on a linear current carrying conductor placed in an magnetic field
$$
\begin{align*}
F &= BILSin(\theta)\\\
B &\to Magnetic\ Flux\\
L &\to Length\ of\ the\ conductor\\
I &\to Current\\
\theta &\to angle\ with\ the\ horizontal
\end{align*}
$$
Force generated of a square loop which carries current with number of turns
$$
\begin{align*}
F &= BINACos(\theta)\\\
A &\to Area\\
N &\to Number\ of\ turns
\end{align*}
$$
Current sensitivity of Ammeter
$$
\begin{align*}
\frac{BNA}{k} &= \frac{\theta}{I}
\end{align*}
$$
Voltage sensitivity of Ammeter
$$
\begin{align*}
\frac{BNA}{kR} &= \frac{\theta}{V}
\end{align*}
$$
Force on a charge particle moving in a magnetic field
$$
\begin{align*}
F &= Bq\vec{u}Sin(\theta)\\\
q &\to charge\\
\vec{u} &\to drift\ velocity\\
\theta &\to angle\ with\ the\ horizontal\\
\end{align*}
$$
Bio-Savat laws
$$
\begin{align*}
Around\ a\ straight\ coductor\ &with\ finite\ length\\
B &= \frac{\mu_o I}{4\pi r}(Sin(\alpha_1) + Sin(\alpha_2))\\\
Around\ a\ straight\ coductor\ &with\ infinite\ length\\
B &= \frac{\mu_o I}{2\pi r}\\\
At\ the\ center\ of\ &a\ circular\ loop\\
B &= \frac{\mu_o I}{2r}\\\
At\ the\ center\ of\ a\ circular\ &loop\ with\ N\ turns\\
B &= \frac{\mu_o IN}{2r}\\\
Through\ a\ solonoid\ &with\ N\ turns\\
B &= \mu_o IN\\\
\end{align*}
$$
Hall voltage
$$
\begin{align*}
V_{H} = B\vec{u}d = \frac{BI}{ten}\\
\end{align*}
$$
Current
$$
\begin{align*}
I &= \frac{Q}{t} \ \end{align*}
$$
Current Density
$$
J = \frac{I}{A}
$$
Mean Drift Velocity
$$
\vec{u} = \frac{I}{Ane}
$$
Ohm's Law
$$
V = IR
$$
Electric Resistance
$$
R = \frac{\rho l}{A}\\
$$
Conductivity
$$
C = \frac{1}{\rho}
$$
Resistance variation with temperature
$$
R = R_o(1 + \alpha \theta)
$$
Resistor networks
$$
\begin{align*}
Series\ &network,\\
R &= R_1 + R_2 + R_3\\\
Parallel\ &network,\\
\frac{1}{R} &= \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}
\end{align*}
$$
Root-mean-square for current and voltage (Used in AC current)
$$
\begin{align*}
V_s \to peak\ voltage\\
I_s \to peak\ current\\\
V_{r.m.s} = \frac{V_s}{\sqrt{2}}\\
I_{r.m.s} = \frac{I_s}{\sqrt{2}}
\end{align*}
$$
Electrical Energy
$$
\begin{align*}
E &= VQ = VIt = I^2Rt = \frac{V^2t}{R}
\end{align*}
$$
Electric Power
$$
\begin{align*}
P &= \frac{VQ}{t} = VI = I^2R = \frac{V^2}{R}
\end{align*}
$$
Effective potential of a cell with internal resistance.
$$
\begin{align*}
When\ current\ leaving\ (+)\ terminal,\\
V &= E - Ir\\\
When\ current\ is\ leaving\ (-)\ terminal,\\
V &= E + Ir\\\
\end{align*}
$$
Effective electromotive force of a cell network
$$
\begin{align*}
Series\ Network,\\
E &= E_1 + E_2 + E_3\\
r &= r_1 + r_2 + r_3\ (internal\ resistance)
\\\
Parallel\ Network,\\
\frac{E}{r} &= \frac{E_1}{r_1} + \frac{E_2}{r_2} + \frac{E_3}{r_3}
\end{align*}
$$
Kirchhoff's laws
$$
\begin{align*}
KCL,\\
\Sigma I_{in} &= \Sigma I_{out}\\\
KVL,\\
\Sigma E &= \Sigma IR
\end{align*}
$$
Energy concept of a simple cell.
$$
\begin{align*}
Power\ of\ cell &= Total\ power\ of\ resistors
\end{align*}
$$
Efficiency of a circuit. (Maximum power is given when r = R)
$$
\begin{align*}
<<<<<<< HEAD
Efficiency &= \frac{Power\ of\ external\ resistance}{Power\ of\ cell}*100 % \\
\eta &= \frac{R}{R+r}*100 % Efficiency &= \frac{Power\ of\ external\ resistance}{Power\ of\ cell} * 100 % \\
\eta &= \frac{R}{R+r} * 100 % \
d531fd232cdbf4528d1cca8644bf3e5f53c98f0e
\end{align*}
$$
Wheatstone bridge
$$
\begin{align*}
\frac{R_1}{R_2} = \frac{R_3}{R_4}
\end{align*}
$$
Meter Bridge
$$
\begin{align*}
l \to& balance\ length\ \\
\frac{R_1}{R_2} &= \frac{l}{100 - l}
\end{align*}
$$
Potentiometer
$$
\begin{align*}
l \to& balance\ length\\
K \to& Potential\ Gradient\\\
E &= Kl\\
K &= \frac{IR}{l}
\end{align*}
$$
Speed of an electromagnetic wave
$$
\begin{align*}
\epsilon \to Permitivity\\
\mu \to Permiability\ \\
C = \frac{1}{\sqrt{\epsilon \mu}}
\end{align*}
$$
Surface emissivity
$$
\begin{align*}
e &= \frac{Total\ energy\ emitted\ by\ a\ surface}{Energy\ emitted\ by\ a\ black\ body\ with\ same\ surface\ area}
\\\
&For\ black\ body \to e = 1
\end{align*}
$$
Surface absorptivity
$$
\begin{align*}
a &= \frac{Energy\ absorbed\ by\ a\ surface}{Energy\ falls\ on\ that\ surface}\\
\\
&For\ black\ body \to a = 1
\end{align*}
$$
Intensity of sound
$$
\begin{align*}
I &= \frac{E}{At}
\end{align*}
$$
Stefan's Law
$$
\begin{align*}
I &= \sigma T^4\\
E &= eAt\sigma T^4\\
For\ black\ &body \to E = At\sigma T^4
\end{align*}
$$
Wien's Displacement Law
$$
\begin{align*}
C \to Wien's\ Constant\\
\frac{1}{\lambda_m} \propto T\\
\\
C = \lambda_m T
\end{align*}
$$
Planck-Einstein relation
$$
\begin{align*}
h \to &Planck's\ constant \\\
E &= hf
\end{align*}
$$
Photoelectric effect
$$
\begin{align*}
I_{max} \propto& Intensity\\
V_{s} \propto& frequency\\
\\
Intensity &= \frac{ne}{t}
\end{align*}
$$
Einstein's Hypothesis on photoelectric effect
$$
\begin{align*}
\phi \to &work\ function\\
f_o \to &threshhold\ frequency\\
e \to &charge\ of\ an\ electron\\
V_s \to &Stop\ potential\\\
hf &= \phi + K.E_{max} \\
hf &= hf_o + \frac{1}{2}mv^2 \ \\
hf &= hf_o + eV_s
\end{align*}
$$
Work function
$$
\begin{align*}
c \to& Speed\ of\ light \\
\phi &= \frac{hc}{\lambda}
\end{align*}
$$
De Broglie Wave length
$$
\begin{align*}
p \to& momentum \\\
\lambda &= \frac{h}{p}
\end{align*}
$$
X-ray tube (work done to move a charge b/w terminals)
$$
\begin{align*}
eV = hf
\end{align*}
$$
$\alpha$ decay
$$
\begin{align*}
\alpha\ &particle \to ^4_2\alpha \\\
^b_aX \to& ^{(b-4)}_{(a-2)}Y +\ ^4_2\alpha + Energy
\end{align*}
$$
$\beta^{-}$ decay
$$
\begin{align*}
\beta^{-}\ & particle \to ^0_{-1}\beta \
\vec{V_e} \to& Anti - electro\ neutrino\\
^b_aX \to& ^{(b)}{(a+1)}Y +\ ^{0} {-1}\beta + \vec{V_e}
\\
1\ neutron\ &converts\ to\ 1\ proton
\end{align*}
$$
$\beta^{+}$ decay
$$
\begin{align*}
\beta^{+}\ &particle \to ^0_{+1}\beta \
V_e \to& electro\ neutrino\\
^b_aX \to& ^{(b)}{(a-1)}Y +\ ^{0} {1}\beta + V_e
\\
1\ proton\ &converts\ to\ 1\ neutron
\end{align*}
$$
Rate of disintegration
$$
\begin{align*}
\frac{dN}{dt} &= -\lambda N\\
N &= N_oe^{-\lambda t}
\end{align*}
$$
Activity of radioactive sample
$$
\begin{align*}
A &= \lambda N
\end{align*}
$$
Half life of an atom
$$
\begin{align*}
T\frac{1}{2} = \frac{0.7}{\lambda}
\end{align*}
$$
Stress
$$
\begin{align*}
\sigma = \frac{F}{A}
\end{align*}
$$
Strain
$$
\begin{align*}
\epsilon = \frac{e}{l}
\end{align*}
$$
Hook's Law
$$
\begin{align*}
\gamma = \frac{Stres}{Strain} &= \frac{\sigma}{\epsilon}\\\
\frac{F}{A} &= \gamma\frac{e}{l}
\end{align*}
$$
Elastic Strain Energy
$$
\begin{align*}
E_o &= \frac{1}{2} * Tension * Extension\\\
E_o &= \frac{1}{2} * Stress * Strain\\
\end{align*}
$$
Force on a rod when expansion or compression is prevented
$$
\begin{align*}
F = \gamma A \alpha (\delta\theta)
\end{align*}
$$
Newtons Law of viscosity (viscose b/w 2 liquid layers moving at v1 and v2)
$$
\begin{align*}
F &= \eta A \frac{(v_1 - v_2)}{d}\\\
\eta &\to coefficient\ of\ viscosity\\
\end{align*}
$$
Stock's Law (viscose force on a spherical object with radius r and speed v)
$$
\begin{align*}
F = 6\pi r \eta v
\end{align*}
$$
Acceleration of a small spherical object falling in a liquid medium
$$
\begin{align*}
&\frac{(\sigma -\rho)g}{\sigma} - \frac{9\eta v}{2r^2 \sigma} = a\\\
r &\to radius\ of\ the\ sphere\\
\sigma &\to density\ of\ the\ sphere\\
\rho &\to density\ of\ the\ liquid\\
v &\to velocity\ of\ the\ sphere\\
\end{align*}
$$
Terminal Velocity
$$
\begin{align*}
v = \frac{2r^2g(\sigma - \rho)}{9\eta}
\end{align*}
$$
Poiseuille's Equation (Rate of volume flow through a steady flow of fluid)
$$
\begin{align*}
\frac{v}{t} = \frac{\pi}{8}\frac{\delta P}{l}\frac{r^4}{\eta}
\end{align*}
$$