Part 1

1: Physical Quantities -+

prefix table.png|center|600

1 Dyne = $10^{- 5}$ Newton; dyne is from CGS system and Newton from SI
Uncertainty error affects precision, not accuracy
Least Count measures precision
Precision: How detailed a measurement is
Accuracy: How correct a measurement is
Young's Modulus, Pressure, Stress and Energy Density have same dimensions

Quantity	Definition	Dimensions
Young's Modulus	Measure of stiffness (stress/strain)	M L⁻¹ T⁻²
Pressure	Force per unit area (F/A)	M L⁻¹ T⁻²
Stress	Force applied per unit area (F/A)	M L⁻¹ T⁻²
Energy Density	Energy per unit volume (E/V)	M L⁻¹ T⁻²

Shape	Volume	Perimeter	Area
Sphere	(4/3) π r³	N/A	4 π r²
Circle	N/A	2 π r	π r²
Square	N/A	4a	a²

2: Vectors -

3d axes.png|center|400

resolving = splitting into components
minimum no. components = 2
maximum no. components = $\infty$
there can be a maximum of 3 rectangular components
Modulus = Magnitude = $\sqrt{A_{x}^{2} + A_{y}^{2} + A_{z}^{2}}$
angle with $x = θ = T a n^{- 1} \frac{A y}{A x}$
west is on the left, east is the right

Operation	Trigonometric Formula	Component Formula	Perpendicular	Parallel
Dot Product	$\vec{A} \times \vec{B} = A B \cos θ$	$A \cdot B = A_{x} B_{x} + A_{y} B_{y} + A_{z} B_{z}$	$0$ if $θ = 90^{\circ}$	Max if $θ = 0^{\circ}$
Cross Product	$\vec{A} \cdot \vec{B} = A B \sin θ \hat{n}$	$A \times B = \| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ A_{x} & A_{y} & A_{z} \\ B_{x} & B_{y} & B_{z} \end{matrix} \|$	Max if $θ = 90^{\circ}$	$0$ if $θ = 0^{\circ}$

if sum of two forces of same magnitude is equal to the individual force, the angel $θ = 120^{\circ}$
projection of B on A where $θ$ is the angle b/w them is: = component of B along A = $| B | \cos θ$ = $\frac{\vec{A} \vec{B}}{| A |} = \vec{B} \hat{A}$
Resultant of two vectors $\vec{A} and \vec{B}$ with same initial at an angle $θ$ is: $R = \sqrt{| A |^{2} + | B |^{2} - 2 | A | | B | \cos θ}$
among unit vectors, $i, j, k$
- $i \times j = k; j \times i = - k$
- $j \times k = i; k \times j = - i$
- $k \times i = j; i \times k = - j$
  if either of any among these pairs is -ve then simply separate it and solve
  e.g. $i \times (- j) = - (i \times j) = - (k) = - k$

\hat{i} . (\hat{j} \times \hat{k})

equals?

a)0b)3c)1d)-1

3: Motion and Forces+

state of motion and rest are relative
acceleration of a free-falling object remains constant, unless obstructed by drag force of medium
Laws of Motion show relation between a body, the force acting on it and the resulting motion
net force on a body at rest or in uniform motion is zero
in circular motion speed can be constant while acceleration $\neq 0$

formulae	formulae	formulae
$V_{f} = V_{i} + a t$	$P = m v$	$τ = F \times r$
$S = V_{i} t + \frac{1}{2} a t^{2}$	$F = m a$	$\cos θ \hat{i} + \sin θ \hat{j}$
$2 a S = V_{f}^{2} - V_{i}^{2}$	$Impulse = F \times t = Δ P$

a baseball is hit straight up and is caught 2 seconds later, the maximum height reached is?

a)4.9mb)9.8mc)12.6md)19.6m

it took 1 sec to go up and 1 sec to come down, $S = V_{i} t + \frac{1}{2} g t^{2}$

gravity is stronger at poles than at equator
projectile formulae

Height	Range	Total time	time to reach maximum height
$H = \frac{V_{\circ}^{2} S i n^{2} θ}{2 g}$	$R = \frac{V_{\circ}^{2} S i n 2 θ}{g}$	$T = \frac{2 V_{\circ} S i n θ}{g}$	$T = \frac{V_{\circ} S i n θ}{g}$

Height and Range of a projectile become equal at angle $θ = \tan^{- 1} (4) = {75.96}^{o}$
The range of project at $θ$ and at $ϕ$ are same if $θ + ϕ = 90,$ e.g. at $60^{o}$ or $30^{o}$
at maximum height, the horizontal acceleration becomes zero in projectile motion

4: Work and Energy-+

Work = Energy $= J = k W h$
$P_{o w e r} = \frac{W}{Δ t} = F \times v$
Work = Change in energy = $\frac{1}{2} m v_{o}^{2} - \frac{1}{2} m v^{2}$

Under the influence of a force, an object of mass 4 kg accelerates from 3 m/s to 6 m/s in 8s. How much work was done on the object during this time?

a)54Jb)96Jc)72Jd)27J

Torque $= τ = r \times F \sin (θ)$

If torque produced is

300 N m

by force

250 N

and moment arm

4 m

, then

θ

is:

a)73b)17c)43d)45

Power of a motor torque: $P = τ \times ω; ω$ is angular velocity
$F \times t = Δ P = Δ (m v)$ =(impulse = change in momentum)
centripetal force does no work (angle between force and displacement is 90, $c o s 90 = 0$ )
1 horse power = 746 watt
$1 k W h = 3.6 \times 10 ⁶$ joules
K.E = $\frac{P^{2}}{2 m}$ relation of K.E and momentum

Energy	formula
P.E	$m_{} g h$
K.E	$\frac{1}{2} m v^{2}$
Work	$F \times d \times C o s θ$

Energy units: Joules, Kilowatt-hour, electron volt
mass of earth: $m = 6.0 \times 10^{24} k g$
radius of earth: $r = 6.4 \times 10^{6} m$
$V_{e s c} = \sqrt{\frac{2 \times G \times M}{R}} = \sqrt{2 g \times R} = 11.2 k m s^{- 1}$ : to escape from gravitational field
$V_{o r b i t a l} = \sqrt{\frac{G \times M}{R}} = \sqrt{g \times R} = 78 k m s^{- 1}$ : to enter an orbit
$V_{e s c} = \sqrt{2} \times V_{o r b i t a l}$
work done for the same mass of object is dependent on the height reached or distance covered

A man lifts a body to a height of 1m in 30s. An other man lifts the same mass to same height in 60s. The work done my them is in the ratio:

a)1:2b)1:1c)2:1d)4:1

Friction: opposition force
- $F = μ N;$ where $μ$ is the coefficient of friction(unitless) and $N$ is the normal force between the surfaces
- Friction has unit of newton

5: Circular Motion-+

quantity	Translational	Rotational	relation
displacment	$\vec{d}$	$θ$	$θ = \frac{S}{r}$
velocity	$v = \frac{\vec{d}}{t}$	$ω = \frac{Δ θ}{Δ t}$	$v = ω r$
acceleration	$a = \frac{Δ v}{Δ t}$	$α = \frac{Δ ω}{Δ t}$	$a = r α$
mass/moment of inertia	$m$	$I$	$I = m r^{2}$
momentum	$\vec{p} = m \vec{v}$	$L = I ω$	$\vec{L} = \vec{r} \times \vec{p}$
force/torque	$F = m a$	$τ = \vec{r} \times \vec{F} s i n (θ)$
Kinetic Energy	$\frac{1}{2} m v^{2}$	$\frac{1}{2} I ω^{2}$

If a spinning object halves its radius while its angular momentum is conserved, what happens to its angular velocity?

a)2 timesb)4 timesc)halvedd)remains same

If the radius of an object moving in a circle is halved, its moment of inertia about the center of the circle decreases by a factor of four $_{I = m r^{2}}$ . By conservation of angular momentum ( $L = I ω$ ), if $I$ becomes $\frac{I}{4}$ , then $ω$ must become $4 ω$ to keep $L$ constant.

$θ = \frac{S}{r}, S$ is length of the curve, $r$ is the radius
$θ = \frac{π}{180} r a d . = 0.00175 r a d .$ OR $1 rad. = {57.29}^{\circ} = {\frac{180}{π}}^{\circ}$
Centripetal Force $F_{c} = m a_{c} = \frac{m v^{2}}{r} = m r ω^{2}$
$a_{c} = \frac{v^{2}}{r}$
unit of linear momentum = $N s = K g m s^{- 1}$
unit of angular momentum = $J s = K g m^{2} s^{- 1}; P . S : same unit as planck’s constant$
- Angular momentum is constant in both speed and direction for: constant speed
- Angular momentum is constant in only direction for: varying speed

A figure skater spins with her arms pulled in. When she extends her arms, what happens to her angular velocity?

a) Her angular velocity decreases because her moment of inertia increases.
b)Her angular velocity remains constant as angular momentum is always conserved.
c)Her angular velocity decreases because her moment of inertia decreases.
d)Her angular velocity increases because her moment of inertia increases.

shape	moment of inertia
rod	$I = \frac{1}{12} m L^{2}$
Hoop/ring	$I = m r^{2}$
disc	$I = \frac{1}{2} m r^{2}$
sphere	$I = \frac{2}{5} m r^{2}$

pseudo force: arises without any contact e.g. Centrifugal force
Apparent weight
falling down: $T = m (g - a)$ = original weight - upward force
going up: $T = m (g + a)$ = original weight + extra force

6: Fluid Motion-

1 atm = 101325 Pa = 760 mmHg = 760 torr
Stroke's Law: $F = 6 π η r v$
Terminal Velocity: $V_{t} = \frac{2 g r^{2} p}{9 η} = \frac{m g}{6 π \times η \times r}$
Terminal velocity of a body is constant and its maximum velocity in that medium

Blood Pressure	Relatively	Flow	in Healthy person (in torr or mm Hg)
Systolic	High	Turbulent	120 mm Hg
Diastolic	Low	Laminar	80 mm Hg
written as 80/120

Equation of continuity: $flow rate = A_{1} V_{1} = A_{2} V_{2} = constant$
pressure of fluid in pipe is: $p r e s s u r e \propto d i a m t e r \propto \frac{1}{v e l o c i t y}$

Theorem	Derived from
Bernouli's theorem	conservation of energy
Equation of Continuity	conservation of mass

theorem	Formula	effect/definition
Stroke's Law	$F = 6 π η r v$	Drag force on a spherical body
Torricelli	$v = \sqrt{2 g (h_{2} - h_{1})}$	velocity of water(speed of efflux) from a hole, where Δh is the height from hole to surface
Venturi	$P_{1} - P_{2} = \frac{1}{2} ρ v_{2}^{2}$	Where space is narrow in a pipe speed of fluid is high, pressure will be low Speed and Pressure and
Bernoulli's Effect		Where speed of fluid is high, pressure will be low

7: Oscillations

in vibratory motion total energy remains constant i.e. sum of K.E and P.E
total energy in SHM = K.E + P.E = $\frac{1}{2} k A^{2}$ (A is amplitude)= constant in perfect system
Gravity of Moon = $\frac{1}{6} g_{e};$ its 1/6th times earth gravity i.e. $1.63 m s^{- 2}$

If the time period of a pendulum on earth is

\sqrt{6}

time period on moon will be

a)6 secb) $\frac{1}{6} s e c$ c)1 secd)36 sec

Apparent gravity
- descending : $g - a$ : decreased
- ascending : $g + a$ : increased

A simple pendulum suspended from the ceiling of a stationary lift has period

T_{o}

when the lift descends at steady speed the period is

T_{1}

and when it descends with constant downward acceleration the period is

T_{2}

, which one of the following is correct.

a) $T_{o} = T_{1} = T_{2}$ b) $T_{o} < T_{1} < T_{2}$ c) $T_{o} = T_{1} < T_{2}$ d) $T_{o} > T_{1} > T_{2}$

a lift is moving upwards with acceleration "a", the time period of a pendulum in the lift becomes:

a)increasedb)decreasedc)samed)zero

Extreme Position	Mean Position
K.E is minimum	K.E is maximum
P.E is maximum	P.E is minimum
Velocity is minimum	Velocity is maximum
Acceleration is maximum	Acceleration is minimum
Displacement is maxmimum	Displacement is minimum

system	Time period	P.E	K.E	accleration	Angular Freqeuncy	mass and amplitude
Spring	$T = 2 π \sqrt{\frac{m}{k}}$	$= \frac{1}{2} k x^{2}$	$= \frac{1}{2} k (x_{o}^{2} - x^{2})$	$a = - \frac{k}{m} x$	$2 π f = ω = \sqrt{\frac{k}{m}}$	mass does affect time period $\sqrt{m}$ times
Pendulum	$T = 2 π \sqrt{\frac{l}{g}}$			$a = - \frac{g}{l} x$	$2 π f = ω = \sqrt{\frac{g}{l}}$	do not affect time period

vibratory motion is simple harmonic if angular velocity $ω$ is uniform
projection of a body moving in a circle where $r$ is radius of circle and $x$ is current amplitude

displacement	velocity	acceleration
$x = x_{\circ} \sin ω t$ $x = x_{\circ} \cos ω t$	$v = ω \sqrt{r^{2} - x^{2}}$	$a = - ω^{2} x$

Initial position
- $x = x_{\circ} \sin ω t$ : mean position
- $x = x_{\circ} \cos ω t$ : extreme position

‘The displacement of a body executing SHM is given by the relation

x = x_{\circ} \sin ω t

, the intial phase is:

a)0⁰b)45⁰c)90⁰d)60⁰

Hook's law: $F = - k x;$ where $k$ is the force constant
- force constant has same unit as surface tension: $F m^{- 1}$
second pendulum: pendulum with a time period of 2 seconds or frequency of $\frac{1}{2} = 0.5$ hertz

Length of second pendulum on earth is

a)1mb)0.99mc)0.5md)none

Time period of second pendular on mars is:

a)1 secb)2 secc)0.66 secd)3 sec

System under forced vibrations is Driven Harmonic Oscillator

8: Waves

$v = λ f$

A source of sound frequency 600Hz is placed inside the water. The speed of sound in water is 1500m/s and in air it is 300ms. The frequency of sound recorded by am observer who is standing on air is

a) $t b$ b) $t b$ c) $t b$ d) $t b$

Doppler effect	Shift	Wavelength	Frequency
moving towards an observer	blue shift	decreases	increases
moving away from an observer	red shift	increases	decreases

\underset{important}{\underset{⏟}{speed of sound \propto \frac{1}{\sqrt{density}} \propto \sqrt{Temp}}} \propto rigidity \propto Modulus of elasticity \propto \frac{1}{compressibility}

speed of sound $\approx 340 m s^{- 1}$ and **does not directly depend on pressure
$\frac{V_{1}}{V_{2}} = \sqrt{\frac{T_{1}}{T_{2}}}; Temp is in Kelvin$
$T_{2} = n^{2} \times T_{1}; n = no. of times speed of sound$ and $T_{2} > T_{1}$

The temperature at which velocity of sound in air is two times of its velocity at 10°C

a)1132°Cb)859°Cc)658Kd)900°C

$T_{2} = 4 \times T_{1} = 4 \times 283 K = 1132 K = 859 ° C$

sound has double speed in air at 819°C than it does at 0°C
$\frac{V_{h}}{V_{o}} = \sqrt{\frac{M_{o}}{M_{h}}}$

Speed of sound in hydrogen, with respect to oxygen:

a)2 times**b)**4 timesc) $\frac{1}{4}$ timesd) $\frac{1}{2}$ times

taking molar mass instead of density as they are proportional, $M_{h} = 2, M_{o} = 32$

rarer to denser:
- reflection: 180 phase shift
- refraction: towards normal; $λ$ increases, $v$ increases
denser to rare:
- reflection: no change in phase angle
- refraction: away from normal; $λ$ decreases, $v$ decreases
Waves are Slow in Shallow(rarer) water and faster in deep (denser) water
- going from shallow to deep: $λ$ increases, $v$ increases
- going from deep to shallow : $λ$ decreases, $v$ decreases
Newton: movement of sound through air is isothermal (same temperature)
Laplace: movement of sound through air adiabatic(no heat transfer out of system) (correct hypothesis)
Sound intensity: power per unit area = $W m^{- 2}$ = Decibels
Decibels is a logarithmic scale
waves transport both energy and momentum
$\frac{C_{p}}{C_{v}} = γ = Gas Constant$
monoatomic gases have the highest $γ$ at 1.67
Constructive interference:
- In phase
- path difference = $n λ; n = 0, \pm 1, \pm 2$
Destructive interference:
- Out of phase
- path difference = $\underset{e . g . 0.5, 1.5, 2.5, \frac{3}{2}, \frac{5}{2}, e t c}{\underset{⏟}{(2 n + 1) \frac{λ}{2}}}; n = 0, \pm 1, \pm 2$
Entering from one medium to other
Frequency remains same and is independent of medium
Velocity and Wavelength are changed
in open pipe:
- at end anti-node is formed
- both odd and even harmonics can be made and hence contains more harmonics
in a closed pipe:
- in closed pipe: at end node is formed
- only odd harmonics are present and hence contains less harmonics
ratio b/w fundamental frequencies of open and closed end pipes: 1:2
whether you change the frequency or wavelength, the speed of sound in a medium remains same
speed of sound increases in air by $0.61 m s^{- 1}$ per 1 Kelvin increase
in any wave, crust and trough are separated by half a wave: $\frac{λ}{2}$
Ultrasonic: >20kHz
beats per second:
- difference of interfering waves= $f_{2} - f_{1} = b e a t s;$ where $f_{2}$ is greater

A tuning fork of frequency 50 Hz when sounded with another fork, 6 beats heard.

a)100Hzb)50Hzc)44Hzd)256Hz

Applications of super-position:
- Interference: $f =$ same; direction=same
- Beats: $f$ =small differences; direction=same
- Stationary waves: $f$ =same; direction=opposite
We can listen a maximum of 10 beats per second
$\begin{align}$

\text{ loudness }\propto \text{ amplitude }\end{align}$$

9: Optics

light rays directly from source are unpolarized and contain both magnetic and electric fields
no polarization in longitudinal waves
E.F and M.F are always perpendicular
$d \sin θ = m λ$
a slit produces a bright fringe at the center
in refraction: frequency remains unchanged
light rays from bulb are spherical
sky is blue due to light scattering

Refraction	wavelength	bend
denser to rarer	decreased	towards normal
rarer to denser	increased	away from normal

10: Optical Instruments

The final image produced by a compound microscope is virtual, inverted, and magnified

11: Heat and Thermodynamics

P V = n R T

Adiabatic: No heat exchange occurs
Isothermal: at constant temperature
Isobaric: at constant pressure
Work done on an isothermal gas with volume change: $W = n R T \ln (\frac{V_{1}}{V_{2}})$

gas expands isothermally from volume

V_{1} = 2 V_{1}

. The work done by the gas is:

a) $R T \ln 2$ b) $n R T \ln 2$ c) $P_{1} V_{1} \ln 2$ d)Zero

entropy is a measure of randomness/disorder
the more evenly/uniformly distributed heat is in the system, the greater the entropy
every natural process moves towards greater entropy(earth quake, floods, melting of ice, etc)
Efficiency of carnot engine: $(1 - \frac{T_{cold}}{T_{hot}}) \times 100$ ; temperature in kelvin
Laws of Thermodynamics:
- First Law: energy cannot be created or destroyed, only transformed from one form to another.
- Second Law: the entropy of an isolated system always increases over time, indicating a tendency towards disorder or randomness.
- Third Law: entropy of a perfect crystal at absolute zero (0 Kelvin) is zero, meaning a system at absolute zero is in its most ordered state

Part 2

12: Electrostatics+-

$vaccum of permitivity = ε_{\circ} = 8.85 \times 10^{- 12} \frac{C^{2}}{N m^{2}} = F a r a d / m e t e r$
$k = \frac{1}{4 π ε_{\circ}} = 9 \times 10^{9} \frac{N m^{2}}{C^{2}}$
if two similarly charged bodies are touched, charged is transferred so that both of them have equal charge e.g. after touching a 1C and 3C body they both have 2C
in a medium of permittivity $ε_{r}$ force between two charges becomes $\frac{F_{\circ}}{ε_{r}}$
a static charge possesses only electric field, but moving/flowing charge possesses both magnetic and electric fields

Quantity	effect upon addition of a dielectric in a capacitor
Voltage	decreases
Electric Intensity	decreases
Capacitence	increases
Charge	constant

in all equations charge is assumed to be positive, it maybe negative but it is preferred positive, if told to chose only one, choose positive
if both can be chosen, chose both
capacitor is also called the condenser
in a capacitor, if charge is 100%, then current is zero, current is 100% if charge is zero
$I % = 100 % - C %$
charge is quantized, multiple of e = $1.6 \times 10^{- 19}$


coulomb force b/w two charges	$F = \frac{1}{4 π ε_{\circ}} \frac{q_{1} q_{2}}{r^{2}}$
Capacitence	$C = \frac{Q}{V}$
Energy stored in capacitor	$W = \frac{1}{2} C V^{2} = \frac{1}{2} \frac{q^{2}}{C}$
Electric Intensity and Voltage	$E = - \frac{Δ V}{Δ r}$
Electric potential due to a charge	$V = \frac{1}{4 π ε_{\circ}} \frac{q}{r}$	$E = \frac{1}{4 π ε_{\circ}} \frac{q}{r^{2}}$
Electric Intensity	$\vec{E} = k \frac{q}{r^{2}} \hat{r}$	$\vec{E} = \frac{\vec{F}}{q}$
Electric Flux	$ϕ = \vec{E} . \vec{A_{r e a}}$

its $t = R C$ seconds for a capacitor to charge up to 63% of its capacitance
energy of capacitor in parallel is more than ones in series
capacitor holds charge due to electric fields and charges due to electrostatic induction

charged is quantized, any multiple of $1.6 \times 10^{- 19}$ (charge on one electron and minimum charge on a body)
Capacitors in parallel: $C_{e q} = C_{1} + C_{2}$
if same $= n C_{1}$
Capacitors in series: $C_{e q} = \frac{C_{1} \times C_{2}}{C_{1} + C_{2}}$
if same $= \frac{C_{1}}{n}$
if a capacitor has charge Q, then one plate has $Q_{+}$ and the other $Q_{-}$ i.e. they store
- equal and opposite charges
direction of electric field outside: ${Positive}_{high potential} ⟹ {Negative}_{low}$

13: Current Electricity+-

Quantity	Unit	relation
Conductance	mho/siemen	$\frac{1}{o h m} = \frac{1}{resistance}$
resistance	ohm
Conductivity	mho $m^{- 1}$	$\frac{1}{resistivity}$
resistivity	ohm $m$

drift velocity ≈ $10^{- 3} m s^{- 1}$
H = heat = $I^{2} R t$
Resistors in series: $R_{e q} = R_{1} + R_{2}$
if same $= n \times R_{1}$
Resistors in series: $R_{e q} = \frac{R_{1} \times R_{2}}{R_{1} + R_{2}}$
if same $= \frac{R_{1}}{n}$


$I = \frac{Δ Q}{Δ t}$	current and Charge through an area
$R = ρ \frac{L}{A}$	resistence, length, area and $ρ$ resistivity of a conductor

if wire is stretched to n times(its orignal length)then resistence becomes $$R'=n^2R$$
B B ~={red}R=~ O Y ~={green}G=~ ~={blue}B=~ ~={magenta}V=~ ~={yellow}Y=~ W
0 1 2 3 4 5 6 7 8 9

Power
for series	$P = I^{2} R$	Higher resistence consumes more power
for parallel	$P = \frac{V^{2}}{R}$	Lower resistence consumes more power
	$P = V I$
	$P = \frac{Δ W_{o r k}}{Δ T}$
	$P = F v$

If E is emf of cell, r is internel resistence and IR=V the terminal voltage$$\begin{align*}E = IR + Ir \I = \dfrac{E}{R+r}\end{align*}$$

14: ElectroMagnetism+-

current through two parallel wires
- if same direction: attract
- if opposite direction: repel
permeability of free space: $μ_{\circ} = 4 π \times 10^{- 7} \frac{W b}{A m}$
inductance of inductors is added similar to resistors
$I_{s e r i e s} = I_{1} + I_{2} + I_{3} \dots$
$I_{p a r a l l e l} = \frac{I_{1} \times I_{2} \times I_{3} \dots}{I_{1} + I_{2} + I_{3} \dots}$

thing	guy
magnetic effect of current	Oersted

quantity	units	Formula
Flux	$w b = T m^{2} = N A^{- 1} m^{-} 1$	$ϕ = B A C o s θ$
Flux Density Magnetic Field Strength Magnetic Field Induction	$\frac{w b}{m^{2}}$ = $T_{e l s a} = N A^{- 1} m$	$B$


Magnetic Field Inside a solenoid	$B = μ_{\circ} I n = μ_{\circ} I \frac{N}{l} = T e s l a$	not affected by radius(area of coil)
Magnetic Field at a distance r from wire	$B = \frac{μ_{\circ} I}{2 π r}$	$1 T e s l a = 10^{4} G a u s s$
Magnetic field at the center of a coil	$B = \frac{μ_{\circ} I}{2 R}$	$R = radius of wire$
Force on current carrying conductor	$F = I L B \times S i n θ$	direction using right hand palm rule
Torque on coil in magnetic field	$τ = N I A B \times \cos θ$	in radial magnetic field, its only NIAB
force on charge in magnetic field	$F = q (v \times b)$	no force on stationary charge, no work is done no change in K.E
force on charge in electric field	$F = q \times E$

A wire carrying current

I

is bent into a circular loop of radius

R

. The magnetic field at the center of the loop is

B

. If the radius is doubled while keeping the current constant, the new magnetic field will be:

a) $4 B$ b) $2 B$ c) $\frac{B}{2}$ d) $\frac{B}{4}$

Lorentz Force = Vector sum of magnetic and electric forces
= $\sqrt{[q (V \times B)]^{2} + (q E)^{2}} = \sqrt{{F_{e}}^{2} + {F_{m}}^{2}}$

law	based on law of
Electromagnetic Induction	Conservation of energy

a charge moving in a straight line perpendicular to a magnetic field moves in a circle
- a charge moving at an angle between 0 and 9 moves in an elliptical path
if field into the page charge(+ve) will move clockwise (using right hand rule)
a charge in an electric field moves towards the oppositely charged site
galvano meter only detects currents, not measure it

15: ElectroMagnetic Induction-+

Transformer only works for A.C, not for DC(dry cell, battery, etc)
ohm law(V=IR) does not apply on inductors only resistors
Induced emf depends on change of flux
but magnitude of induced emf depends on rate of change of flux
$\frac{N_{s}}{N_{p}} is the transformer ratio, not \frac{N_{p}}{N_{s}}$
$\frac{N_{s}}{N_{p}} = \frac{V_{s}}{V_{p}} = \frac{I_{p}}{I_{s}}$
Step up: $N_{s} > N_{p}$
Step down: $N_{s} < N_{p}$
A.C generator: Slip Rings
D.C generator: Split Rings
dynamo(A.C generator) converts: mechanical energy into electric
Induced emf sometimes acts as back emf
Induced emf does not depend on resistance of coil but induced current does
if there are x cycles in A.C of f hertz, then
current change direction 2x times
current reaches max. 2x times


Energy stored in Inductor/magnetic field	$E = \frac{1}{2} L \times I^{2}$
emf induced in rotating coil (A.C generator)	$ε = N ω A B \sin θ$	when flux( $ϕ$ ) is max. emf( $ε$ ) is min. and vice versa
Mutual Inductance Self Inductance	$M = L = - \frac{ε}{\frac{Δ t}{Δ I}} = \frac{emf induced}{change in current in coil}$	unit is henry

16: Alternating Current+-

$I_{r m s} = \frac{I_{\circ}}{\sqrt{2}} = \frac{I_{\circ}}{1.4} = 0.7 \times I_{\circ}$
$V_{r m s} = \frac{V_{\circ}}{\sqrt{2}} = \frac{V_{\circ}}{1.4} = 0.7 \times V_{\circ}$

V_{i n s} = V_{o} s i n (ω t)

wavelength( $λ$ ) of x-rays is in order of $10^{- 10} m = Angstorm$

Reactance( $Ω$ )	device	lead and lag( $θ$ )	Power Factor( $\cos θ$ )
$X_{L} = 2 π f L = ω L$	~={green}Inductor=~	~={green}current lags=~ by $\frac{π}{2}$	$\cos (\frac{π}{2}) = 0$
$X_{C} = \frac{1}{2 π f C} = \frac{1}{ω C}$	~={blue}Capacitor=~	~={blue}current leads=~ by $\frac{π}{2}$	$\cos (- \frac{π}{2}) = 0$
R	Resistor	current and voltage are in phase	$\cos 0 = 1$

CL circuit is an oscillatory or rejecting circuit

circuit	impedance( $Ω$ )	phase difference
RL	$Z = \sqrt{R^{2} + {R_{L}}^{2}}$	$θ = \tan^{- 1} (\frac{X_{L}}{R}) = \tan^{- 1} (\frac{L}{ω R})$
RC	$Z = \sqrt{R^{2} + {R_{C}}^{2}}$	$θ = \tan^{- 1} (\frac{X_{C}}{R}) = \tan^{- 1} (\frac{1}{ω C R})$
R	$Z = R$	$θ = θ^{'}$ ; no difference
RLC	$Z = \sqrt{R^{2} + (R_{L} - R_{C})^{2}}$	$θ = \tan^{- 1} (\frac{X_{L} - X_{C}}{R})$

$impedance = Z = \sqrt{R^{2} + (R_{L} - R_{C})^{2}}$

thing	definition
Carrier Wave	High frequency wave
Modulation Signal	Low frequency wave
modulated carrier wave	resulting wave from modulation

thing	definition	transmission freq range	use	flaw
Modulation	combining high freq radio wave and low freq signal
Amplitude Modulation	combining amplitude of high freq. and amplitude of low freq.	540 kHz to 1600 kHz	long range	easily distorted
Frequency Modulation	combining freq. of high freq. and amplitude of low freq.	88 MHz to 108 MHz	immunity to electrical disturbance	small range

RLC circuit behaves like an~={green} RC=~ circuit at ~={green}low frequency=~
RLC circuit behaves like an ~={blue}RL=~ circuit at ~={blue}high frequency=~
RLC circuit behaves like an R circuit at resonance frequency( power factor = 1)
at resonace frequency( $R_{L} = R_{C}$ ): $Z = R$
~={red}resonance frequency:=~ $f_{r} = \frac{1}{2 π \sqrt{L C}}$
In series at resonance freq.: impedance is min.,power is max
In parallel at resonance freq.: impedance is max., power is min
Radio waves have a prominent wave nature
Gamma rays have a prominent particle nature
as wavelength decreases/freq. increase particle nature increase
static charge: only E.F
moving charge: both E.F and M.F but no E.M wave
accelerating charge: changing E.F, M.F and E.M waves
choke is used to control flow of A.C
Maxwell's formula for speed of light

c = \frac{1}{\sqrt{μ_{0} ϵ_{0}}}

where:

$u_{0}$ is the permeability of free space
$ϵ_{0}$ is the permittivity of free space

17: Physics of Solids-+

hook's law is constant so wherever it is obeyed the graph between stress and strain will be a straight line
strain is : $\frac{L_{o}}{Δ L}$ so doubling the length will have a strain of 1
resistivity of conductors is in the order of $10^{- 8} o h m m$

thign	formula	unit	note
Stress	$\frac{F}{A}$	$N m^{- 2} = [M L^{- 1} T^{- 2}]$
Strain	$\frac{L_{o}}{Δ L}$	no unit	doubling the length will have a strain of 1

modulus	energy $U = \frac{1}{2} \times Stress \times strain = \frac{1}{2} \times Modulus \times {strain}^{2}$	type	formula	note
Young's	$U = \frac{1}{2} F \times l$	linear	$Y = \frac{F \times L_{o}}{A \times z Δ L} = \frac{m g L}{π r^{2} Δ L}$	Solid only and infinity for perfectly rigid body
Bulk's	$U = \frac{1}{2} σ ε V$	volumetric	$K = \frac{F \times V_{o}}{A \times Δ V}$	solids and gases, infinity for perfectly rigid body
Shear		angular		only solid

hence steel has more elasticity than rubber

Elasticity \propto Rigidity \propto \frac{1}{c o m p r e s s i b i l i t y}

Area of Stress-Strain curve has units of energy density
$Area = \frac{1}{2} x y = \frac{1}{2} \frac{F}{A} \times \frac{Δ L}{L_{o}} = \frac{1}{2} \frac{F \times Δ L}{A \times L} = \frac{1}{2} \frac{Work}{Volume} = Energy Density$

type	charge carrier	impurity
N-type	free electrons in conduction band	penta-valent
P-type	holes in valence band	tri-valent

material	in magnetic fields	behavior	examples
Dimagnetic	weakly repelled	no unpaired electron	copper, bismuth, water, antimony
Paramagnetic	weakly attracted	some unpaired electron so magnitizes momentarily	aluminum, platinum, oxygen
FerroMagnetic	strong attracted	many unpaired electron and retain magnitized state	iron, cobalt nickle

at 0K conduction band of semi-conductors is empty and valence band is filled and acts as an insulator
bond length of nickle = $0.91 \angstorm$
glass is a solid liquid bcz of irregular(liquid) but fixed(solid) relative position of molecules
Modulus of elasticity relates inversly with temperature
$Elasticity \propto Rigidity \propto Modulus of elasticity \propto speed of sound in solid \propto \frac{1}{Compressibility} \propto \frac{1}{Change in length per unit force}$

18: Electronics +

a pn-junction is biased as long as a potential is applied whether current flows or not

Type	minority charg	majority charge	Dopping	dopping example
P-Type	free electrons	holes	Trivalent/Acceptor
N-Type	holes	free electrons	Pentavalent/Donor

bias	conventional current/potential)	depletion region
forward	from P to N	is shortened
reverse	from N to P	is widened

recitification	diodes	conducting at once	$f_{output}$	output
half wave	1	1	$f_{Output} = f_{Input}$
full wave	4	2	$f_{Output} = 2 \times f_{Input}$	pulsating DC

depletion regions increases(reverse bias) and decreases both due to majority charge carriers
Emitter current> collector current> base current
$I_{e} = I_{c} + I_{b}$
AND can be represented by two switches in series
OR can be represented by two switches in parallel

identification	gates	symbolic
true only when both are zero	NOR
false only when both are one	NAND

true only on different inputs	XOR	$\overset{―}{A} . B + A . \overset{―}{B}$
true only same inputs	XNOR	$\overset{―}{\overset{―}{A} . B + A . \overset{―}{B}}$

OP-AMP can amplify both AC and DC

19: Dawn of Modern Physics

motion alters space and time

constants	value	symbol
Stefan's	$5.67 \times 10^{- 8} W m^{- 2} K^{- 4}$	$σ$
Wien's	$2.9 \times 10^{- 3} m K$
Planck's	$6.63 \times 10^{- 31} J s$

Relation between energy of emitted photons and temperature: $\begin{align}$

\text{Frequency of emitted photon} \qquad \propto \qquad & \dfrac{1}{ \text{Wavelength}} \propto \qquad\text{Temperature}
\end

- $ 1 e V = 1.6 \times 10^{- 19} J $ $ $ eV \overset{\times 1.6 \times 10^{- 19}}{\to} Joule

at high(relativistic) speeds of observer and as speed increases	at speed of light	become double when v=
time increases	stops	$\frac{2}{\sqrt{3}} c$
length decreases	becomes zero	$\frac{2}{\sqrt{3}} c$
mass increases	becomes infinity	$\frac{\sqrt{3}}{2} c$

no object can travel faster than $c \approx 3.0 \times 10^{8}$ and all forms of radiation/photons travel at $c$
there is no absolute standard of motion
$E = \underset{mass-energy}{\underset{⏟}{m c^{2}}} = \overset{frequency of photon}{\overset{⏞}{h f}} = \underset{wavelength of photons}{\underset{⏟}{h \frac{c}{λ}}} = \overset{momentum of photon}{\overset{⏞}{p c}}$
$E = \frac{h c}{λ} = \frac{1240 e V . n m}{λ}$
$h f - ϕ = K . E$

A photon with wavelength 400 nm strikes a metal surface with work function 2.0 eV. The maximum kinetic energy of the emitted electron is closest to:

a)1.1 eVb)3.1eVc)2.1eVd)4.1eV

Energy relation	def	Relation
$E = m c^{2}$	Mass-energy conversion
$E = h f$	Energy of photon with $f$ frequency
$E = h \frac{c}{λ}$	Energy of photon with $λ$ wavelength
$E = p c$	Momentum of photons with $E$ energy

rest mass of photon is zero

thing	formula
momentum of photon	$ρ = \frac{h f}{c}$
debroglie wavelength	$λ = \frac{h}{m v}$

Gamma and X-rays: prominent particle nature
IR, Micro, radio waves: prominent wave nature

λ \propto wave nature \propto \frac{1}{Momentum of Photon} \propto \frac{1}{P a r t i c l e N a t u r e}

Photo Electric Effect

\begin{aligned} E - ϕ & = K.E \\ h f - h f_{o} & = \frac{1}{2} m v^{2} \end{aligned}

\begin{array}{r} intensity of light \propto current \propto no. of electrons emitted \\ frequency of light \propto energy of emitted electron \propto stopping potential \end{array}

20: Atomic Spectra:

Wavelength for X-rays in orders of: $10^{- 10} m$
ionization energy of hydrogen: $13.6 e V$
any color shown on an object is the color not absorbed

a red colored cloth__ red light

a)transmitb)reflectc)refractd)absorb

21: Nuclear Physics

muons have same charge as electrons
mesons : one quark and one anti quark
baryon: three quarks
electrons: a fundamental particle
neutrons are made of three quarks: 2 up, 1 down
up-quarks have $\frac{2}{3} e$ charge
there are 6 leptons
penetration power:

Alpha < Beta < Gamma

humans emit IR
decay calculation: $\underset{remaining}{\underset{⏟}{N}} = \frac{\overset{orignal amount}{\overset{⏞}{N_{o}}}}{2^{n}}, n = \frac{\overset{time elapsed}{\overset{⏞}{t}}}{\underset{half life}{\underset{⏟}{T}}}$

Half-life of radioactive isotope with initial mass 1 kg is 19 minutes. How much mass is decayed in 57 minutes?

a)125gb)875gc)250gd)750g

decayed = $N_{o} - N$

rate of decay: activity: $\underset{remaining}{\underset{⏟}{A}} = \frac{\overset{orignal amount}{\overset{⏞}{A_{o}}}}{2^{n}}, n = \frac{\overset{time elapsed}{\overset{⏞}{t}}}{\underset{half life}{\underset{⏟}{T}}}$

In 420 days, the activity of a sample of polonium (Po) fell to

\frac{1}{8}

th of its initial value. The half life of polonium is

a)280 daysb)70 daysc)140 daysd)210 days

$\frac{A_{o}}{8} = \frac{A_{o}}{2^{\frac{t}{T}}}; 2^{3} = 2^{\frac{420}{T}}; T = \frac{420}{3} = 140$

only noble gases are used in GM counters
forces in nature occur in pairs
fission of two deuterons, releases 1 neutron
Fast moving Neutron: K.E> 1KeV
X-rays are unaffected by magnetic field
if $α$ -alpha is release:

X^{A} ⟶ Y_{Z - 2}^{A - 4} + H e_{2}^{4}; a helium nuclei is released

if $β^{-}$ beta is released:

X^{A} ⟶ Y_{Z + 1}^{A} + β_{- 1}^{0};

A nucleus

X_{Z}^{A}

emits an

α

-particle. The resultant nucleus emits a

β^{+}

particle. The respective atomic and mass number of the final nucleus will be:

a)Z-3, A-4b)Z-1, A-4c)Z-2, A-4d)Z, A-2

$β^{+}$ was emitted not $β^{-}$

if $γ$ -gamma is released

X^{*} ⟶ X^{} + photon

X_{Z}^{A} \to X_{Z + 1}^{A} \to X_{Z - 1}^{* A - 4} \to X_{Z - 1}^{A - 4}

a) $α, β, γ$ b) $β, α, γ$ c) $γ, β, α$ d) $β, γ, α$

at least one alpha and two beta emissions are required for an isotope

thing	definition	formation	examplpe
isotope	same atomic number, different mass number	$1 α, 2 β$	$H_{1}^{1}, H_{1}^{2}$
isobars	same mass number, different atomic number	$1 β$	$H_{2}^{3}, H_{1}^{3}$
isotons	same mass number, same proton number, different neutron	no>:(
isodiapers	different mass, atomic number, and neutrons	$1 α$

thing	unit	old/other unit
Strength of radiations	Becquerel(Bq) one disintegration per second	Curie(Ci) $3.7 \times 10^{10}$ disintegrations per second
Effect of radiation Dose (D) = $\frac{E_{nergy}}{M_{ass}}$	Gray (Gy) one joule per kilogram	Radiation Absorbed Dose (Rad) 1 rad = 0.01 Gy 100 rad = 1 Gy
Biological Effectiveness RBE

Miscellaneous

$1 m = 39.3 inch; 1 inch = 2.54 c m$