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Formulas and Concepts

Physics pocket diary of

concepts and formulas

Formulas and Concepts

Kinematics

Note: - Bold letter are used to denote vector quantity i,j,z are the unit vector along x,y and z axis

Quick review of Kinematics formulas

S.No. Type of

Motion

Formula

1 Motion in one dimension

r=xi v=(dx/dt)i a=dv/dt=(d^2 x/dt^2 )i and a=vdv/dr v=u+at s=ut+1/2at^2 v^2 =u^2 +2as In integral form r=∫vdt v=∫adt 2 Motion in two dimension

r=xi +yj v=dr/dt=(dx/dt)i + (dy/dt)j a=dv/dt=(d^2 x/dt^2 )i+(d^2 x/dt^2 )j and a=vdv/dr Constant accelerated equation same as above 3 Motion in three dimension

r=xi +yj+zk v=dr/dt=(dx/dt)i + (dy/dt)j+(dz/dt)k a=dv/dt=(d^2 x/dt^2 )i+(d^2 x/dt^2 )j+(d^2 x/dt^2 )k a=vdv/dr Constant accelerated equation same as above 4 Projectile Motion x=(v 0 cosθ 0 )t y=(v 0 sinθ 0 )t-gt^2 / vx= v 0 cosθ 0 and vy= v 0 sinθ 0 t-gt , where θ 0 is the angle initial velocity makes with the positive x axis. 5 Uniform circular motion

a=v^2 /R , where a is centripetal acceleration whose direction of is always along radius of the circle towards the centre and a=4π^2 R/T^2 acceleration in uniform circular motion in terms of time period T

Concept of relative velocity For two objects A and B moving with the uniform velocities VA and VB. Relative velocity is defined as VBA=VB-VA where VBA is relative velocity of B relative to A Similarly relative velocity of A relative to B VAB=VA-VB

Special cases: -

S.No. Case Description

1 For straight line motion

If the objects are moving in the same direction, relative velocity can be get by subtracting other. If they are moving in opposite direction ,relative velocity will be get by adding the velocities example like train problems 2 For two dimensions motion

if va=vxai + vyaj vb=vxbi + vybj Relative velocity of B relative to A =vxbi + vybj -(vxai + vyaj) =i(vxb-vxa) + j(vyb-vya) 3 For three dimensions motion

va=vxai + vyaj +vzaz vb=vxbi + vybj + vzbz Relative velocity of B relative to A =vxbi + vybj + vzbz -(vxai + vyaj +vzaz) =i(vxb-vxa) + j(vyb-vya)+z(vyb-vya)

Free fall acceleration

S.No. Point 1 Freely falling motion of any body under the effect of gravity is an example of uniformly accelerated motion. 2 Kinematics equation of motion under gravity can be obtained by replacing acceleration 'a' in equations of motion by acceleration due to gravity 'g'. 3 Thus kinematics equations of motion under gravity are v = v 0 + gt , x = v 0 t + ½ ( gt^2 ) and v^2 = (v 0 )^2 + 2gx 4 Value of g is equal to 9.8 m.s-2.The value of g is taken positive when the body falls vertically downwards and negative when the body is projected up against gravity.

Formulas and Concepts

Work, Energy and Power

S.No. Term Description

1 Work 1. Work done by the force is defined as dot product of force and displacement vector. For constant Force W=F.s where F is the force vector and s is displacement Vector

1. For variable Force dW=F.ds or W=∫F.ds It is a scalar quantity

2 Conservative And Non Conservative Forces

1. If the work done by the force in a closed path is zero, then it is called conservative Force
2. If the work done by the force in a closed path is not zero, then it is called non conservative Force
3. Gravitational ,electrical force are Conservative Forces and Non Conservative Forces are frictional forces

3 Kinetic Energy 1. It is the energy possessed by the body in motion. It is defined as K.E=(1/2)mv^2

1. Net work done by the external force is equal to the change in the kinetic energy of the system W=Kf-Ki

4 Potential Energy 1. It is the kind of energy possessed due to configuration of the system. It is due to conservative force. It is defined as dU=-F.dr Uf-Ui=-∫F.dr Where F is the conservative force F=-(∂U/∂x)i-(∂U/∂y)j-(∂U/∂z)k For gravtitional Force

1. Change in Potential Energy =mgh where h is the height between the two points
2. Mechanical Energy is defined as E=K.E+P.E

5 Law Of conservation of Energy

In absence of external forces, internal forces being conservative, total energy of the system remains constant. K.E 1 +P.E 1 =K.E 2 +P.E 2

6 Power Power is rate of doing work i.e., P=work/Time. Unit of power is Watt. 1W=1Js-1. In terms of force P= F.v and it is a scalar quantity.

Momentum and Collision

S.No. Term Description

1 Linear Momentum The linear momentum p of an object of mass m moving with velocity v is defined as p=mv Impulse of a constant force delivered to an object is equal to the change in momentum of the object F∆t = ∆p = mvf - mvi Momentum of system of particles is the vector sum of individual momentum of the particle ptotal=∑viMi 2 Conservation of momentum

When no net external force acts on an isolated system, the total momentum of the system is constant. This principle is called conservation of momentum. if ∑Fext=0 then ∑viMi=constant

3 Collision Inelastic collision - the momentum of the system is conserved, but kinetic energy is not. Perfectly inelastic collision - the colliding objects stick together. Elastic collision - both the momentum and the kinetic energy of the system are conserved. 4 Inelastic collision While colliding if two bodies stick together then speed of the

Formulas and Concepts

composite body is

1 2

11 2 2

m m

mu mu

v

Kinetic energy of the system after collision is less then that before collison 5 Elastic collision in one dimension

Final velocities of bodies after collision are

2 1 2

2 1 1 2

1 2 (^1) m m u

2 m u m m

m m v (^)  

 



 + 

 



−

2 1 2

2 1 1 1 2

1 (^2) m m u

m m u m m

2 m v (^)  

also (^) u 1 −u 2 =v 2 −v 1

Special cases of Elastic Collision

S.No. Case Description

1 m 1 =m 2 v 1 =u 2 and v 2 =u 1 2 When one of the bodies is at rest say u 2 =

1 1 2

1 2 1 u m m

m m v (^)  

   



−

and 1 1 2

1 2 u m m

2 m v (^)  

   



=

3 When m 1 =m 2 and u 2 =0 i.e., m 2 is at rest v 0

1 =^ and^ v^2 =u^1 4 When body in motion has negligible mass i.e. m 1 <>m 2 1 1

v = u and^ v 2 = 2 u 2

Mechanics of system of particles

S.No. Term Description 1 Centre of mass It is that point where entire mass of the system is imagined to be concentrated, for consideration of its translational motion. 2 position vector of centre of mass

Rcm=∑riMi/∑Mi where ri is the coordinate of element i and Mi is mass of element i 3 In coordinate system

xcm=∑xiMi/∑Mi ycm=∑yiMi/∑Mi zcm=∑ziMi/∑Mi 4 Velocity of CM vCM=∑viMi/∑Mi The total momentum of a system of particles is equal the total mass times the velocity of the centre of mass 5 Force When Newton’s second law of motion is applied to the system of particles we find Ftot=MaCM with aCM=d^2 RCM/dt^2 Thus centre of mass of the system moves as if all the mass of the system were concentrated at the centre of mass and external force were applied to that point. 6 Momentum conservation in COM motion

P=MvCM which means that total linear momentum of system of particles is equal to the product of the total mass of the system and the velocity of its centre of mass.

Rigid body dynamics

S.No. Term Description

1 Angular Displacement -When a rigid body rotates about a fixed axis, the angular displacement is the angle ∆θ swept out by a line passing through any point on the body and intersecting the axis of rotation perpendicularly -It can be positive (counter clockwise) or negative (clockwise). -Analogous to a component of the displacement vector. -SI unit: radian (rad). Other units: degree, revolution. 2 Angular velocity -Average angular velocity, is equal to ∆θ/∆t. Instantaneous Angular Velocity ω=dθ/dt -Angular velocity can be positive or negative.

Formulas and Concepts

16 Equilibrium of a rigid body

Fnet=0 and τext=

17 Angular Impulse ∫τdt term is called angular impulse. It is basically the change in angular momentum 18 Pure rolling motion of sphere/cylinder/disc

-Relative velocity of the point of contact between the body and platform is zero -Friction is responsible for pure rolling motion -If friction is non dissipative in nature E = (1/2)mvcm^2 +(1/2)Iω^2 +mgh

Gravitation

S.No. Term Description

1 Newton’s Law of gravitation 2

12 r

Gm m F =

where G is the universal gravitational constant

G=6.67 × 10 -11Nm^2 Kg-

2 Acceleration due to gravity

g=GM/R^2 where M is the mass of the earth and R is the radius of the earth

3 Gravitational potential energy

PE of mass m at point h above surface of earth is

( R h )

GmM PE

4 Gravitational potential (^) ( R h )

GM

V

Law of orbits

Each planet revolves round the sun in an elliptical orbit with sun at one of the foci of elliptical orbit. Law of areas

The straight line joining the sun and the planet sweeps equal area in equal interval of time.

5 Kepler’s Law of planetary motion

Law of periods

The squares of the periods of the planet are proportional to the cubes of their mean distance from sun i.e., T^2 VR^3 6 Escape velocity Escape velocity is the minimum velocity with which a body must be projected in order that it may escape earth’s gravitational pull. Its magnitude is ve=√(2MG/R) and in terms of g ve=√(2gR)

Orbital Velocity

The velocity which is imparted to an artificial satellite few hundred Km above the earth’s surface so that it may start orbiting the earth v 0 =√(gR)

7 Satellites

Periodic Time

T=2π√[(R+h)^3 /gR^2 ]

With altitude 

R

h gh g

With

depth 

= ^ −

R

d

gd g 1

8 Variation of g

With latitude

g φ= g − 0. 037 cos^2 φ

Elasticity

S.No. Term Description 1 Elasticity The ability of a body to regain its original shape and size when deforming force is withdrawn 2 Stress Stress=F/A where F is applied force and A is area over which it acts. 3 Strain It is the ratio of the change in size or shape to the original size or shape. Longitudinal strain = ∆l/l volume strain = ∆V/V and shear strain is due to change in shape of the body. 4 Hook’s Law Hook's law is the fundamental law of elasticity and is stated as “for small deformations stress is proportional to strain".

Formulas and Concepts

Thus, stress proportional to strain or, stress/strain = constant This constant is known as modulus of elasticity of a given material Young's Modulus of Elasticity Y=Fl/A∆l Bulk Modulus of Elasticity K=-V∆P/∆V

5 Elastic Modulus Modulus of Rigidity η=F/Aθ 6 Poisson's Ratio

The ratio of lateral strain to the longitudinal strain is called Poisson’s ratio which is constant for material of that body. σ=l∆D/D∆l 7 Strain energy

Energy stored per unit volume in a strained wire is E=½(stress)x(strain)

Hydrostatics

S.No. Term Description 1 Fluid pressure It is force exerted normally on a unit area of surface of fluid P=F/A. Its unit is Pascal 1Pa=1Nm-2. 2 Pascal’s Law Pressure in a fluid in equilibrium is same everywhere. 3 Density Density of a substance is defined as the mass per unit volume. 4 Atmospheric pressure

Weight of all the air above the earth causes atmospheric pressure which exerts pressure on the surface of earth. Atmospheric pressure at sea level is P 0 =1.01x10^5 Pa 5 Hydrostatic pressure

At depth h below the surface of the fluid is P=ρgh where ρ is the density of the fluid and g is acceleration due to gravity. 6 Gauge pressure P=P 0 + ρgh , pressure at any point in fluid is sum of atmospheric pressure and pressure due to all the fluid above that point. 7 Archimedes principle

When a solid body is fully or partly immersed in a fluid it experience a buoyant force equal to the weight of fluid displaced by it. 8 Upthrust It is the weight of the displaced liquid. 9 Boyle’s law PV=constant 10 Charle’s law V/T=constant

Hydrodynamics

S.No. Term Description 1 Streamline flow

In such a flow of liquid in a tube each particle follows the path of its preceding particle. 2 Turbulent flow

It is irregular flow which does not obey above condition.

3 Bernoulli’s principle p^ u gh cons tan t 2

(^1 )

• ρ + ρ =

4 Continuity of flow

A 1 (^) v 1 = A 2 v 2 where A 1 and A 2 are the area of cross section of tube of variable cross section and v 1 and v 2 are the velocity of flow of liquids crossing these areas. 5 Viscosity Viscous force between two layers of fluid of area A and velocity gradient

dv/dx is dx

dv F = − η A where η is the coefficient of viscosity.

6 Stokes’ law Viscous force on a spherical body of radius r falling through a liquid of viscosity η is F = 6 πη rv where v is the velocity of the sphere.

7 Poiseuilli’s equation

Volume of a liquid flowing per second through a capillary tube of radius r when its end are maintained at a pressure difference P is given by

l

Q η

π 8

Pr 4 = where l is the length of the tube.

Simple Harmonic Motion

S.No. Term Description 1 SHM In SHM the restoring force is proportional to the displacement from the mean position and opposes its increase. Restoring force is F=-Kx Where K=Force constant , x=displacement of the system from its mean or equilibrium position

Formulas and Concepts

-Examples are sound waves travelling through an intervening medium, water waves, light waves etc. 2 Mechanical waves

Waves requiring material medium for their propagation are MECHANICAL WAVES. These are governed by Newton's law of motion. -Sound waves are mechanical waves in atmosphere between source and the listener and require medium for their propagation. 3 Non mechanical waves

-Those waves which does not require material medium for their propagation are called NON MECHANICAL WAVES. -Examples are waves associated with light or light waves , radio waves, X-rays, micro waves, UV light, visible light and many more. 4 Transverse waves

These are such waves where the displacements or oscillations are perpendicular to the direction of propagation of wave. 5 Longitudinal waves

Longitudinal waves are those waves in which displacement or oscillations in medium are parallel to the direction of propagation of wave for example sound waves 6 Equation of harmonic wave

-At any time t , displacement y of the particle from it's equilibrium position as a function of the coordinate x of the particle is y(x,t)=A sin(ωt-kx) where, A is the amplitude of the wave k is the wave number ω is angular frequency of the wave and (ωt-kx) is the phase. 7 Wave number Wavelength λ and wave number k are related by the relation k=2π/λ 8 Frequency Time period T and frequency f of the wave are related to ω by ω/2π = f = 1/T 9 Speed of wave speed of the wave is given by v = ω/k = λ/T = λf 10 Speed of a transverse wave

Speed of a transverse wave on a stretched string depends on tension and the linear mass density of the string not on frequency of the wave i.e, v=√T/μ T=Tension in the string μ=Linear mass density of the string. 11 Speed of longitudinal waves

Speed of longitudinal waves in a medium is given by v=√B/ρ B=bulk modulus ρ=Density of the medium Speed of longitudinal waves in ideal gas is v=√γP/ρ P=Pressure of the gas , ρ=Density of the gas and γ=Cp/CV

12 Principle of superposition

When two or more waves traverse thrugh the same medium,the displacement of any particle of the medium is the sum of the displacement that the individual waves would give it. y=Σyi(x,t) 13 Interference of waves

If two sinusoidal waves of the same amplitude and wavelength travel in the same direction they interfere to produce a resultant sinusoidal wave travelling in that direction with resultant wave given by the relation y′(x,t)=[2Amcos(υ/2)]sin(ωt-kx+υ/2)where υ is the phase difference between two waves. -If υ=0 then interference would be fully constructive. -If υ=π then waves would be out of phase and there interference would be destructive. 14 Reflection of waves

When a pulse or travelling wave encounters any boundary it gets reflected. If an incident wave is represented by yi(x,t)=A sin(ωt-kx)then reflected wave at rigid boundary is yr(x,t)=A sin(ωt+kx+π) =-Asin(ωt+kx) and for reflections at open boundary reflected wave is given by yr(x,t)=Asin(ωt+kx) 15 Standing waves The interference of two identical waves moving in opposite directions produces standing waves. The particle displacement in standing wave is given by y(x,t)=[2Acos(kx)]sin(ωt) In standing waves amplitude of waves is different at different points i.e., at nodes amplitude is zero and at antinodes amplitude is maximum which is equal to sum of amplitudes of constituting waves.

Formulas and Concepts

Normal modes of stretched string

Frequency of transverse motion of stretched string of length L fixed at both the ends is given by f=nv/2L where n=1,2,3,4,....... -The set of frequencies given by above relation are called normal modes of oscillation of the system. Mode n=1 is called the fundamental mode with frequency f 1 =v/2L. Second harmonic is the oscillation mode with n=2 and so on. -Thus the string has infinite number of possible frequency of vibration which are harmonics of fundamental frequency f 1 such that fn=nf 1 17 Beats Thus beats arises when two waves having slightly differing frequencies ν 1 and ν 2 and comparable amplitude are superposed. -Here interfering waves have slightly differing frequencies ν 1 and ν 2 such that |ν 1 -ν 2 |<<(ν 1 +ν 2 )/ The beat frequency is νbeat= ν 1 V ν 2 18 Doppler effect -Doppler effect is a change in the observed frequency of the wave when the source S and the observer O move relative to the medium. -There are three different ways where we can analyse this change in frequency. (1) When observer is stationary and source is approaching observer is ν = ν 0 (1+Vs/V) where, vs=velocity of source relative to the medium v=velocity of wave relative to the medium ν =observed frequency of sound waves in term of source frequency ν 0 =source frequency -Change in frequency when source recedes from stationary observer is ν=ν 0 (1-Vs/V) -Observer at rest measures higher frequency when source approaches it and it measures lower frequency when source reseeds from the observer. (2) Observer is moving with a velocity Vo towards source and the source is at rest is ν=ν 0 (1+Vo/V) (3) Both source and observer are moving then frequency observed by observer is ν=ν 0 (V+Vo)/(V+Vs) and all the symbols have respective meanings as told earlier

Thermal expansion

S.No. Term Description 1 Coefficient of linear

expansion lt

lt l

0

α =^ −^0 where α=coefficient of linear expansion,^ lt =

length at t^0 C and l 0 is length at 0^0 C. 2 Length at temperature t^0 C lt=l 0 (1+αt) 3 Coefficient of superficial expansion At

At A

0

β=^ −^0

4 Area at temperature t^0 C At=A 0 (1+βt) 5 Coefficient of volume

expansion Vt

Vt V

0

γ=^ −^0

6 Volume at temperature t^0 C Vt=V 0 (1+γt) 7 Coefficient of apparent expansion of a liquid Vt

Va V

0

− 0 γ (^) α= whereγ (^) α=coefficient of apparent expansion,

V 0 = volume at 0^0 C and Vα = apparent volume at t^0 C 8 Coefficient of real expansion of a liquid

V t

Vr V

r 0

γ =^ −^0 where^ γ^ r =coefficient of real expansion,

V 0 = volume at 0^0 C and Vr = real volume at t^0 C 9 Density variation with

temperature t

d

dt

0 where^ dt =density at temperature t

(^0) C , d 0 = density

at 0^0 C. 10 Pressure coefficient of gas

P t

Pt P

P 0

γ = −^0

Formulas and Concepts

15 Coefficient of performance of refrigerator

β= Q 2 /W =Q 2 /(Q 1 -Q 2 )

16 Efficiency of carnot engine

β= Q 2 /W =Q 2 /(Q 1 -Q 2 )

17 Carnot Theorem Carnot’s theorem consists of two parts (i) No engine working between two given temperatures can be more efficent than a reversible Carnot engine working between same source and sink. (ii) All reversible engines working between same source and sink (same limits or temperature) have the same efficiency irrespective the working substance.

Heat transfer

S.No. Term description 1 Thermal Conductivity L

kAT T H

( 2 − 1 ) = Where H is the quantity of heat flowing through the slab

and k is the constant called thermal conductivity of material of slab. 2 Convection Convection is transfer of heat by actual motion of matter 3 Radiation Radiation process does not need any material medium for heat transfer 4 Stefan Boltzmann law

The rate urad at which an object emits energy via EM radiation depends on objects surface area A and temperature T in kelvin of that area and is given by urad = σεAT^4 Where σ= 5.6703×10-8^ W/m^2 K^4 is Stefan boltzmann constant and ε is emissivity of object's surface with value between 0 and 1 5 Wein's displacement law

λmT = b Where b=0.2896×10-2^ mk for black body and is known as Wien's constant

Emissive Power

It is the energy radiated per unit area per unit solid angle normal to the area. E = ∆u/ [(∆A) (∆ω) (∆t)] where, ∆u is the energy radiated by area ∆A of surface in solid angle ∆ω in time ∆t. Absorptive Power

is defined as the fraction of the incident radiation that is absorbedby the body a(absorptive power) = energy absorbed / energy incident

6 Kirchoff's law

Kirchoff's Law

"It status that at any given temperature the ratio of emissive power to the absorptive power is constant for all bodies and this constant is equal to the emissive power of perfect B.B. at the same temperature. E/abody=EB.B 7 Newton's Law of Cooling

For small temperature difference between the body and surrounding rate of cooling is directly proportional to the temperature difference and surface area exposed i.e., dT/dt = - bA (T 1 - T 2 ). This is known a Newton's law of cooling

Kinetic theory of gases

S.No. Term Description

Boyle's law At constant temperature, the volume of a given mass of gas is inversely proportional to pressure. Thus PV = constant Charle's Law When pressure of a gas is constant the volume of a given mass of gas is directly proportional to its absolute temperature. V/T = Constant

1 Gas laws

Dalton's law of partial pressures

The total pressure of mixture of ideal gases is sum of partial pressures of individual gases of which mixture is made of 2 Ideal gas equation

PV = nRT where n is number of moles of gas

3 Pressure of gas P = (1/3)ρvmq^2 or PV = (1/3) Nmvmq^2 where vmq^2 known as mean

Formulas and Concepts

square speed 4 rms speed vrms = √(3P/ρ) = √(3PV/M) =√(3RT/M) 5 Law of Equipartition of energy

each velocity component has, on the average, an associated kinetic energy (1/2)KT

Monatomic gases

CV= (3/2)R , CP = 5/2 R and γmono = CP/CV= 5/

6 Specific Heat Capacity Diatomic gases CV =(5/2)R , CP=(9/2)R and γ=CP/CV=9/

7 Specific heat Capacity of Solids

C=3R this is Dulang and Petit law

8 Mean free path If v is the distance traversed by molecule in one second then mean free path is given by λ = total distance traversed in one second /no. of collision suffered by the molecules =v/πσ^2 vn =1/πσ^2 n

Electric Charge, Force and Field

S.No. Term Description 1 Charge Charges are of two types (a) positive charge (b) negative charge like charges repel each other and unlike charges attract each other. 2 Properties of charge

1. Quantisation: - q=ne where n=0,1, 2........ and e is charge of an electron.
2. Additivety: - qnet=Σq
3. conservation :- total charge of an isolated system is constant 3 Coulomb’s law The mutual electrostatic force between the charges q 1 and q 2 separated by a distance r is given by Force on the charge q 1 F 1 =Kq 1 q 2 r 12 /r^2 where r 12 is the unit vector in the direction from q 2 to q 1 For more then two charges in the system the force acting on any charges is vector sum of the coulomb force from each of the other charges. This is called principle of superposition for q 1 ,q 2 ,q 3 ......qn charges are present in the system. Force on charge q 1 F=Kq 1 q 2 r 12 /r 122 +Kq 1 q 3 r 13 /r 132 +Kq 1 q 4 r 14 /r 142 ....Kq 1 qnr1n/r1n^2 Similarly for the other charges... 4 Electric field -The region around a particular charge in which its electrical effects can be observed is called the electric field of the charge -Electric field has its own existence and is present even if there is no charge to experience the electric force. 5 Electric field Intensity

E=F/q 0 Where F is the electric force experience by the test charge q 0 at this point.It is the vector quantity Some point to note on this

1. Electric field lines extend away from the positive charge and towards the negative charges
2. Electric field produces the force so if a charge q is placed in the electric field E the force experience by the charge is F=qE 3 Principle of superposition also applies to electric field so E=E 1 +E 2 +E 3 +E 4 +...... Electric field intensity due to point charge E=KQr/r^2 Where r is the distance from the point charge and r is the unit vector along the direction from source to point. Electric field for the Uniformly charged ring

E=KQx/(r^2 +x^2 )3/ Where x is the distance from the centre of the ring. At x= E= Electric Field due to uniformly charged disc

E= (σ/2ε 0 )(1- x/(√R^2 +x^2 )) σ=Surface charge density of the disc. At x=0 E=σ/2ε 0

6 Some useful Formula

Electric Field Intensity due to Infinite sheet of the charge

E=σ/2ε 0

7 Charge density Linear charge density λ=Q/L=dQ/dL Surface charge density σ=Q/A=dQ/dA

Formulas and Concepts

=(p/4πεr^3 )(√(3cos^2 θ+1)) Torque on dipole=pXE Potential Energy U=-p.E 8 Few more points

1. ∫E.dl over closed path is zero 2.Electric potential in the spherical charge conductor is Q/4πεR where R is the radius of the shell and the potential is same everywhere in the conductor
2. Conductor surface is a equipotential surface

Capacitance

S.No. Term Description 1 Capacitance C of the capacitor

C=q/V or q=CV -Unit of capacitance is Farads or CV-1^ capacitance of a capacitor is constant and depends on shape, size and separation of the two conductors and also on insulating medium being used for making capacitor. 2 Capacitance of parallel plate cap

C=(ε 0 A)/d where, C= capacitance of capacitor A= area of conducting plate d= distance between plates of the capacitor ε 0 =8.854× 10-12^ and is known as electric permittivity in vacuum. 3 parallel plate air capacitor in presence of dielectric medium

C=εA/d

4 Capacitance of spherical capacitor having radii a, b (b>a)

(a) air as dielectric between them C=(4πε 0 ab)/(b-a) (b) dielectric with relative permittivity ε C=(4πεab)/(b-a) 5 Parallel combination of capacitors

C=Q/V= C 1 +C 2 +C 3 , resultant capacitance C is greater then the capacitance of greatest individual one. 6 Series combination of capacitors

1/C=1/C 1 +1/C 2 +1/C 3 , resultant capacitance C is less then the capacitance of smallest individual capacitor. 7 Energy stored in capacitor

Energy stored in capacitor is E=QV/ or E=CV^2 / or E=Q^2 /2C factor 1/2 is due to average potential difference across the capacitor while it is charged. 8 Force between plates

of capacitor K A

Q

F

2 ε 0

9 Force per unit area of plates 0

2

σ

K

f =

Where σ is charge per unit area.

Electric current and D.C. circuits

S.No. Term Description 1 Electromotive force EMF of a cell is the total energy per unit charge when the cell is on an open circuit i.e., when the current through the cell is zero. 2 Electric current I=q/t it is the rate of flow of electric charge. Unit of current is ampere. 3 Drift speed of electron in a conductor (^) neA

I

vd =

Where I is the current, n is the number of electrons

per unit volume and A is the area of cross section of conductor. 4 Resistivity of conductor

τ

ρ (^2) ne

m = where m is the mass of electron and^ τ^ is the

relaxation time 5 Ohm’s law V=IR where R is the resistance of the given conductor and unit of resistance is ohm(Ω)

Formulas and Concepts

6 Electrical resistivity Ρ=RA/l where l is the length of the wire and A is its area of cross-section 7 Resistors in series R=R 1 +R 2 +R 3 +…. 8 Resistors in parallel (^1111) ..... 1 2 3

= + + + R R R R

9 Terminal voltage It is equal to emf of battery minus potential drop across internal resistance r across the battery. Terminal voltage = E-Ir 10 Kirchoff’s first law The algebraic sum of current at any junction in a circuit is zero. 11 Kirchoff’s second law The algebraic sum of the products of the current and resistances and the emf in a closed loop is zero. 12 Heating effect of current Heat energy delivered by current when it flows through resistance of R ohm for t sec. maintained at potential difference V is H=V^2 t/R 13 Electrical power P = VI = I^2 R = V^2 /R 14 Variation of resistance with temperature

R=R 0 (1+α(T-T 0 ))

15 Variation of resistivity with temperature

ρ=ρ 0 (1+α(T-T 0 ))

Magnetic effect of current

S.No. Term Description 1 Biot-Savart law Magnetic fiels dB at any point whose position vector is r wrt current elemant dl is given by 3

4 r

I d

d

l r)

B 0

×

π

μ

2 Magnetic field due to long current carrying conductor r

I B π

μ 4

02

3 Magnetic field at centre of a circular loop

r

I

B

=^ μ^0

4 Magnetic field at centre of coil of n turns

r

In B 2

=^ μ^0

5 Magnetic field on the axis of a circular loop 2 23 /^2

2 0 2 ( r x )

Ir B

μ if there are n turns in the coil then

2 2 3 / 2

2 0 2 ( r x )

Inr B

=

μ

6 Ampere’s circuital law d I

∫ B ⋅ I =^ μ 0

7 Field due to toroidal solenoid

B = μ 0 nI

8 Field inside straight solenoid

B = μ 0 nI and direction of field is parallel to the axis of solenoid

9 Force on moving charge in magnetic field

F = q( v × B ) Direction of force is perpendicular to both^ v^ and^ B

10 Force on current carrying conductor in the magnetic field

F = I ( l × B ) where l is the length of the conductor in the direction of current in it

11 Force between two parallel wires carrying current

R

II

F

=^012

12 Torque on a current carrying loop

τ = ( m × B ) Where^ m^ is the magnetic moment of the dipole and

magnitude of magnetic moment is m=NIB where A is the area of the loop and N is the number of turns in the loop. 13 Lorentz force Force on electron moving with velocity v in presence of both uniform electric and magnetic field is (^) F = − e ( E + v × B )

14 Magnetic dipole moment of bar magnet

m=q(2a) where q is the pole strength and (2a) is the length of the bar magnet. It is the vector pointing from south to north pole of the magnet. 15 torque on the bar magnet

τ = m × B

16 Potential energy of a magnetic dipole

U=-mBcosθ

Formulas and Concepts

2 Decay of charge in CR circuit CR

t q q e

− = 0 and current in CR circuit is^ CR

t

i i e

−

3 Capacitive time constant

CR has dimensions of time and is called capacitive time constant for circuit 4 Energy stored in inductor

U =^1 2 ( Li^2 )

5 Energy stored in capacitor U = 1 2 ( CE 2 ) =^12 ( q 0 E ) Where E is the maximum value of potential

difference set up across the plates. 6 LC oscillations Frequency of oscillations is

LC

f

Alternating Current

S.No. Term Description 1 Alternating current

It is current whose magnitude changes with time and direction reverses periodically. I=I 0 sinωt where I 0 is the peak value of a.c. and ω=2π/T is the frequency 2 Mean value of a.c.

Im=2I 0 /π = 0.636I 0

3 RMS value Irms=I 0 /√ 4 a.c. through resistor

Alternating emf is in phase with current

5 a.c. through inductor

Emf leads the current by an phase angle π/

6 a.c. through capacitor

Emf lags behind the current by an phase angle π/

7 Inductive reactance

Opposition offered by inductor to the flow of current mathematically, X (^) L = ω L = 2 π fL

8 Capacitive reactance

Opposition offered by capacitor to the flow of current mathematically,

C fC

X C

9 a.c. through series LR circuit

Emf leads the current by an phase angle φ given by R

ω L tan φ = and

impedance of circuit is (^) Z = R^2 +( ω 2 L^2 )

10 a.c. through series CR circuit

Emf lags behind the current by an phase angle φ given by R

φ^ ω^ C

tan = and impedance of circuit is  

2 2

C

Z R

ω

11 a.c. through series LCR circuit

Emf leads/lags behind the current by an phase angle φ given by

R

C

L

tan

emf leads the current when

C

L

and lags behind

when C

L ω

ω

1 < and impedance of circuit is^2 1  

  

= +^ − C Z R L ω ω

12 Average power of an a.c. circuit cos^ φ

2

Pavg = IrmsR = ErmsI rms

Where φ is called power factor of the circuit.

13 Transformer It is a device used to change low alternating voltage at high current into high voltage at low current and vice-versa. Primary and secondary voltage for a transformer are related as P

S S P

V = V

and current through

the coils is related as S

P S P

I = I

Electromagnetic waves

S.No. Term Description

1 Conduction current It is the current due to the flow of electrons through the connecting wires in an electric circuit

Formulas and Concepts

2 Displacement current It arises due to time rate of change of electric flux in some part of circuit

dt

d

I D

ϕ

3 Modified ampere’s circuital law.^ ( ) (^0 dt )

d I (^) C ID IC E

∫ B^ dl =μ^^0 + =μ^0 +^ ε

where IC is the conduction

current. Gauss’s law in electrostatics 0

q Eds

∫ ⋅^ =

Gauss’s law in magnetism (^) ⋅ = 0 ∫ Bds Faraday’s law of EM induction (^) dt

d φ B

∫ E ⋅ dl =^ −

4 Maxwell’s Equations

Ampere-Maxwell’s circuital

law.^ ( 0 )

dt

d

I C E

ϕ

∫ Bdl ==μ^ 0 +^ ε

5 Velocity of EM waves in free space

v^1 3108 m / s 00

= = × ε μ

Huygens’ Principle and Interference of Light

S.No. Term Description 1 Wave front It is the locus of points in the medium which at any instant are vibrating in the same phase. 1 Each point on the given primary Wavefront acts as a source of secondary wavelets spreading out disturbance in all direction.

2 Huygens’ Principle

2 The tangential plane to these secondary wavelets constitutes the new wave front 3 Interference It is the phenomenon of non uniform distribution of energy in the medium due to superposition of two light waves. 4 Condition of maximum intensity

φ = 2 n π or x=nλ where n=0,1,2,3,….

6 Condition of minimum intensity

φ = ( 2 n + 1 ) π or x=(2n+1)λ/2 where n=0,1,2,3,….

7 Ratio of maximum and minimum intensity 1 22

2 1 2 min

max ( )

a a

a a I

I

8 Distance of nth bright fringe from centre of the screen d

nD yn

λ = where d is the separation distance between two

coherent source of light, D is the distance between screen and slit, λ is the wavelength of light used. 9 Angular position of nth bright fringe (^) d

n D

yn n

10 Distance of nth dark fringe from centre of the screen

d

n D

yn

( 2 + 1 ) λ

11 Angular position of nth dark fringe d

n D

yn n 2 θ ′= ′=(^2 +^1 )^ λ

12 Fringe width d

D λ β =

Diffraction and polarisation of light

S.No. Term Description 1 Diffraction It is the phenomenon of bending of light waves round the sharp corners and spreading into the regions of geometrical shadow of the object. 2 Single slit diffraction Condition for dark fringes is

a

n λ

sinθ = where n=^ ±^ 1,^ ±^ 2,^ ±^ 3,^ ±^ 4……. ,

a is the width of slit and θ is angle of diffraction. Condition for bright fringes is a

n 2

sin λ θ

3 Width of central maximum is (^) a

λ D θ

0 =^

where D is the distance between slit and screen.

4 Diffraction The arrangement of large number of narrow rectangular slits of equal