Kinetic energy is the energy due to motion and is given by 1/2 mv2 where m is the mass and v is the speed.. Potential energy due to position is given by mgh and potential energy due to s
Trang 1
DIRECT THEORY NOTES
1 UNITS AND DIMENSIONS
Physics is the science of natural phenomena It is the science of observation and measurement It is the science of interpretation of results To measure a quantity we need units To define and establish units, we require physical standards
The important requisites of a physical standard are accessibility, accuracy and invariance Such requisites are possessed by modern standards An example of physical standard is the wavelength of light, which is used to define metre
In the international system used today in science and engineering, we have seven base units and three supplementary units The system is called SI Table 1.1 gives ten fundamental quantities, i.e seven base units and three supplementary units, their symbols and what they represent
TABLE 1.1 FUNDAMENTAL QUANTITIES Quantity symbol What it represents
Conventions Followed in Using SI units
1 Symbols should not be used with plural, e.g 10 m is correct, but 10 ms in wrong
2 A void using bars in writing units, e.g write m/s2 as ms-2
3 As far as possible multiples of 103n of base units are to be used Each multiple has a prefix E.g when n
= 1, the prefix is kilo Symbol is ‘k’ When n = 2, prefix is mega; symbol M When n = -1, prefix is ‘milli’ and symbol m n = -2, micro etc
Dimensions:
Physical quantities are classified into fundamental and derived quantities The five fundamental quantities
in physics are mass (M), length (L), time (T), temperature (θ) and current (I) All others are derived quantities
Dimensions are the relation between fundamental and derived quantities If we write any derived quantity
as Ma Lb Tc θd Ie, then this is called a dimensional formula and the powers a,b,c,d,e are called dimensions The dimension of the quantities will be given along with their definitions
2 MOTION IN ONE AND TWO DIMENSIONS
Scalars and Vectors
Physical quantities are classified into scalars and vectors
A scalar is fully defined and understood, when its magnitude or value is given, e.g mass, length, distance, speed, etc
A vector quantity is fully defined only when both its magnitude and direction are given.e.g., force, acceleration, magnetic field etc All quantities having magnitude and direction will not be vectors They must also obey the rules of geometric addition
Vector Algebra
Two vectors can be added by (i) triangle method, (ii) parallelogram method, and (iii) analytical method The sum of two vectors A and B at an angle α is the vector C, where the magnitude of C is given by the analytical method as
C = A2+ B2+ 2AB cos α
and the direction of C with vector A, θ is given by
Trang 2tan θ =
α+
αcosBA
sinB
To subtract a vector we reverse the vector to be subtracted and add with the other, e.g A-B = A+(-B)
To add a number of vectors we use the law of polygon of vectors If we represent the vectors as the sides
of a polygon, the side that completes the polygon in the opposite direction is the sum
Multiplication of vectors : Two vectors A and B can be multiplied in two ways If the product is scalar, it
is scalar multiplication or dot product
The dot product of two vectors a and B is defined as A.B = A B Cos α where α is the angle between them
If the product is a vector, it is called a vector product or cross product The cross product of A and B is A x
B = AB sin α nˆ is unit vector perpendicular to the plane of A and B
Resolution of a vector : A vector F inclined at angle α with a given direction can be resolved as two
rectangular components, as F cos α along the direction and F sin α perpendicular to it
Unit vectors : A vector F can be written F = are known as unit vectors along the three axes of coordinates The magnitude of each is one unit F
kˆandjˆiˆwhereFkˆFjˆF
Displacement is the change in the position of a body in a given direction It is a vector
The rate of displacement is velocity The rate of travelling a distance is speed
v = u + at
t 2
v u
s=⎢⎣⎡ + ⎥⎦⎤
2
at 2
1 ut
=
2
1 n a u
sn
Equations of motion under gravity (bodies moving vertically up and down under gracvity) are got by substituting a = + or – g depending on whether the body is travelling downwards or upwards g is usually taken as 9.8 ms-2 For rough calculations it can be taken as 10
Projectile motion : A projectile is a body thrown at an angle so that it moves in a vertical plane under the
action of gravity If h is the maximum height, R is the horizontal range, T is the time of flight, Rm is the maximum range, u is the initial velocity of projection and α is the angle of projection of the body
g
u
R
g 2 sin
u
h
2 m
2
2 2
Uniform circular motion : A body in uniform circular motion has constant speed and varying velocity
The acceleration towards the centre of the body is
r
v2 or ω2 r, where v is its velocity and ω angular
velocity The force towards centre acting on the body is the centripetal force =
r
mv2 or mω2r
Trang 3Physics at a Glance 3
Centrifugal force: An observer in a circular motion feels a radially outward force This force is called the centrifugal force Its value is equal to the centripetal force It is pseudo a force The properties of pseudo force are given in indirect theory notes
In a non-uniform circular motion, speed and velocity change An example such a motion is a stone tied to the end of a string and whirled in a vertical circle At the highest point, when the string has minimum tension (T1)
Two frames having relative acceleration are non-inertial frames Newton’s second law is obeyed in such frames
Newton’s Laws of Motion :
First law: A body at rest or moving with uniform speed in a straight line continues so until an external force acts on it
Second law: The rate of change of momentum is directly proportional to the unbalanced (resultant) force acting on the body
Third law: For every action there is an equal and opposite reaction
Newton’s first law defines force while the second law measures it as f = ma, where m is mass and a the acceleration
Unit of force: newton (N) Dimensions of force: MLT-2
Weight of a body is a force exerted by the earth on the body: W = mg It is measured by the reaction, resistance or tension
In a freely falling body, only the force of gravity acts That is, the reaction or resistance is zero Hence a freely falling body experiences weightlessness
Law of Conservation of Momentum :
A closed system is one, in which no external force acts The total momentum of a closed system remains constant
Examples: Rocket propulsion, recoil of a gun, explosion of a shell and collision
For recoil of a gun, mv = MV, where m is the mass of the shot, M is the mass of the gun, v is the velocity
of the shot and V is the recoil velocity of the gun More accurate formula will be
mv = (M-m)V For m << M, it can be written as
mv = MV
For collision in a straight line, by the law of conservation of momentum, we have
m1 u1 + m2 u2 = m1 v1 + m2 v2
where m1, m2 are the two colliding masses, u1, u2, their initial velocities and v1 v2 their final velocities
Coefficient of restitution : The ratio of the relative velocity of two bodies after impact to that before
impact is defined as the coefficient of restitution e
2 1
2 1
uu
vv
Trang 4where h1 is the height of rebound and h2 is the height of fall
If e = 1, the collision is perfectly elastic If e = 0, the collision is perfectly inelastic If e is between 0 and 1 the collision is inelastic Ordinary mechanical collisions are inelastic In an elastic collision, both momentum and kinetic energy are conserved In an inelastic collision, momentum is conserved but kinetic energy is lost
The kinetic energy lost in an inelastic collision is
∆E = [ ] [ ]2
2 1 2 2 1 2
m m m m 2
where u1, u2 are their initial velocities of masses m1, m2 respectively and ‘e’ coefficient of restitution
Friction: This is a force, which always opposes relative motion between two surfaces The value of
frictional force is µa mg, where µs is called coefficient of static friction When the body is in motion, the frictional force acting on it is the dynamic frictional force µk mg The coefficient of static friction is the ratio of force of static friction to normal reaction i.e., Fs /R The coefficient of dynamic friction does not depend on the speed of motion as long as the speed is constant The angle of friction λ is the angle between the resultant reaction and normal reaction It is related to µs by the equation
tan λ = µs
A body placed on a rough inclined plane is in equilibrium until the angle of plane θ is equal to the angle of friction λ Thus the equilibrium condition on a rough inclined plane is
µs = tanθ = tan λ
4 WORK, POWER AND ENERGY
The work done by a force = f s cosθ, where f is the force, s the displacement, θ the angle between the force
and the displacement W = f s
Unit of work : joule (J) Dimensions of work : ML2T-2
The rate of doing work is called power P =
dt
dW time
work = Unit: watt (W) Dimensions : ML2T-3
A body has energy if it can do work Energy is measured by the work a body can do Kinetic energy is the energy due to motion and is given by 1/2 mv2 where m is the mass and v is the speed Potential energy is the energy due to position or state of strain Potential energy due to position is given by mgh and potential energy due to state of strain of a spring = 1/2 kx2 where k is the force constant and x is the extension of the spring
Work done = energy
f x s =
2
1mv2 or f x s = mgh
Power = force x velocity, i.e., P = f.v
5 CENTRE OF MASS AND ROTATIONAL MOTION
Centre of mass is a point where the whole mass of the body is supposed to be concentrated To find centre
of mass of a system of two particles of masses m1, m2 at distances x1 and x2 from the origin (which we can choose conveniently)
xcm =
2 1
2 2 1 1
mm
xmxm
++
The centre of mass in terms of vector distance r is given by
rcm =
2 1 2 2 1
m m r m r m
+
+ , where r1 and r2 are position vectors of the two masses
Velocity of centre of mass is given by the equation
vcm =
2 1
2 2 1 1
mm
vmvm
+
+
, where v1 and v2 are velocity vectors of the two masses
Acceleration of centre of mass is given by the equation
acm =
2 1 2 2 1 1
m m a m a m
+
+
, where a1 and a2 are acceleration vectors of two masses
In the absence of external force, the centre of mass is at rest or moves with the uniform speed The above equations can be extended to find the centre of mass of a system of more than two particles
The angular momentum of a particle about the origin is
Trang 5Physics at a Glance 5
L = r x p
where r is the radius vector and p is the momentum vector
The torque acting on a particle about the origin is
where r is the radius vector and F is the force vector
The magnitude of the torque = rF sin θ where θ is an angle between the radius vector and the force vector The moment of inertia of a particle about an axis is the product of mass and square of distance from the axis (mr2) The moment of inertia of a rigid body is I = MK2 where K is the radius of gyration
The theorem of parallel axis states that
where Ix, Iy are moments of inertia about two mutually perpendicular axes lying in the plane of a lamina, Iz
is the moment of inertia about an axis perpendicular to its plane
The angular momentum of a rigid body
L = Iω , where ω is the angular velocity
The torque acting on a rigid body
τ = Iα, where α is the angular acceleration
The law of conservation of angular momentum states that the total angular momentum of a body remains constant in the absence of external torque
The kinetic energy of rotation of a rigid body is given by
Erot =
2
1Iω2, where I is moment of inertia and ω is angular velocity
For a body having translatory and rotatory motion, the total kinetic energy is given by
sing
where k is radius of gyration and θ is the angle of the plane
Kepler’s Laws of planetary Motion :
First law : Each planet moves around the sun in an elliptical orbit, with the sun at one of the foci
Second law: The areal velocity of the planet is a constant, i.e the radius vector joining the sun to the planet sweeps out equal areas in equal intervals of time
Third law: The square of the period of the planet around the sun is proportional to the cube of the mean distance from sun i.e T2 ∝ R3
6 GRAVITATION AND SATELLITES
Newton’s law of gravitation states that the gravitational force between two particles of masses m1 and m2 at
a distance r acts along the line joining them, and has a magnitude
F = 21 2
r
mm
Unit: Nkg-1 Dimensions: LT-2 (that of acceleration)
The gravitational potential at a point due to mass m at distance r is given by
r
m G
−
Trang 6where M is mass of the earth and R is the radius of the earth
The relation between field and potential is that E =
-dr
dV or simply
r V
The mean density of earth is given by
RG
4
gπ
hR
(
GM
+
=+The value of g at a depth x from the surface of earth is
g2 =
R
)xR(
where R is the radius of the earth
The velocity of an earth satellite going very close to the earth is given by
R
GM =
This velocity is known as the first cosmic velocity
The velocity of an earth satellite orbiting at a height h from the surface of the earth is given by
hR
GM
+The period of a satellite at a distance r from the centre of the earth is given by
This is known as the second cosmic velocity It does not depend on the direction of projection
The period of geo-stationary satellites is one day (24 hours) The distance of such a satellite from the centre of earth is nearly 42,000 km or from the surface of earth 36,000 km
7 WAVE MOTION AND OSCILLATIONS
A wave is a disturbance set up in a given medium from equilibrium condition
A progressive wave is one which progresses or travels in a given direction unobstructed It is a travelling wave, which carries energy
Different forms of an equation of a progressive wave:
Trang 7x - T
t 2 sin
a
y
v
x t f 2 sin
where y is the transverse displacement, a amplitude, f frequency, T period, λ wavelength and x is position
of the particle at the time t
A stationary wave is superposition of two progressive waves in opposite directions Equation of a stationary wave:
Wave frequency (f): Wave frequency is the number of vibrations per second made by the source or particle
in the medium
Period (T) The period T of wave motion is the time for one complete oscillation
Wavelength (λ): Wavelength is the distance between two particles in the medium which are in the same state of vibration
Wave speed (v): Wave speed is the distance travelled by the wave in one second
Wave number (v): Wave number is the number of waves in unit length It is reciprocal of wavelength Wave vector (k): Wave vector or propagation constant is 2π/ λ or 2πv
When two waves of the same frequency and amplitudes a1 and a2 and phase difference φ superpose the resultant amplitude A is given by the equation
A = a12+a22+ a1a2cosφ
The physical effect of superposition is interference
Simple Harmonic Motion
A body is in S.H.M if a restoring force proportional to displacement acts on it
The equation of simple harmonic motion is
f = -kx
where k is a constant k is the restoring force for unit displacement
Acceleration is proportional to displacement of S.H.M
a = -ω2 x where ω is the angular frequency of motion
Period when two springs are in series =
2 1 2 1
k k k + k ( m 2π , where k1, k2 are force constants of the springs
Period when two springs are in parallel =
2
1 + k k
m 2π
Period of a simple pendulum =
g
1 2π for small angular amplitudes Here l is length of the pendulum
Period of a test tube floating in a liquid =
Adg
m 2π , where, m is mass of the test-tube, A corss sectional area, d density of liquid
Period of a liquid oscillating in a U-tube =
g
L 2π , where, L is total length of the liquid in the U-tube
Period of a body dropped in a tunnel diametrically dug across earth =
g
R2π , where, R is radius of the earth
Trang 8Period of the longest pendulum suspended in the vicinity of earth =
g
R2π
Table 1.2 gives all properties of S.H.M
Here a is the amplitude of S.H.M, x is the displacement, ω is the angular frequency i.e ω = 2πf, f is the frequency of S.H.M., T is the period of S.H.M., v is velocity of S.H.M, and a is the acceleration of S.H.M
TABLE 1.2 Properties of Simple Harmonic Motion
Position of the particle Quantity
At equilibrium position At extreme position Intermediate position
Transverse Vibration of Strings
Velocity of transverse waves in a stretched string is given by
µ
T
where T is stretching tension, µ is linear density, i.e., mass per unit length
The laws of transverse vibration are relations amongst various parameters of a vibrating string If f is frequency, µ linear density, l vibrating length, T stretching tension, the laws are
tstanconsareTandlwhen1
f
tstanconsarelandwhenT
f
tstanconsareTandwhen
µα
The frequency of transverse vibration is given by the equation
Tl
2
n
2
π
The fundamental mode corresponds to n=1
Vibration of Air Column and pipes
A closed pipe is one which is closed at one end An open pipe is one which is open at both ends The fundamental frequency of a closed pipe is given by
Trang 9Physics at a Glance 9
where L is length of the pipe The closed pipes will produce only odd harmonics f, 3f, 5f…etc., while the open pipe will produce all harmonics i.e odd and even f, 2f, 3f ,etc
Velocity of Sound in Air
Sound waves are basically longitudinal Velocity of sound waves in a medium depends on elasticity and inertia of the medium Velocity of sound in a solid is given by
v = ρE
where E is Young’s modulus and ρ density of the solid
Velocity of sound in liquid is given by
ρ
K
where, K is Bulk modulus and ρ density of the solid
Velocity of sound in a gas is given by
v =
M RT
ρ
γ
where γ is ratio of specific heats , T temperature and M molecular mass
Velocity of sound does not depend on pressure It is directly proportional to square root of Kelvin temperature Velocity of sound increases with humidity Velocity of sound also changes with motion of the medium, i.e wind
Doppler Effect
The apparent change in frequency of light or sound wave due to relative motion of source or the observer
is called the Doppler effect
Equation for apparent frequency when the listener and source approach each other with velocity UL and Us respectively is given by
where V is velocity of sound in air
In this equation, if listener is at rest, we put UL = 0 If the source is at rest, we put US = 0 If the listener is receding, we put UL = -Us
The apparent frequency of light wave of speed ‘c’ from a source of speed ‘v’ given by
where ∆λ is the change in wavelength and ∆f change in frequency
The increase in frequency (or decrease in wavelength) is violet shift The decrease in frequency (or
increase in wavelength) is red shift
8 PROPERTIES OF MATTER
Stress in force per unit area Unit: Nm-2 Dimensions : ML-1T-2
Strain is the ratio of change in dimension to original dimension It has no unit, no dimensions
Hook’s law states that stress is proportional to strain
stressalLongitudin
Bulk modulus K ( or B) is given by
K =
strainVolumetric
stressVolumetric
Shear modulus n (or G) is given by
n =
shearofAngle
stressShearing
Trang 10Poisson’s ratio σ is given by
σ =
strainlextensionaal
Longitudin
strainnalcontractioLateral
Work done during stretching or Energy of the wire =
2
1force x extension Energy density or work done per unit volume =
2
1 stress x strain Thrust is the normal force acting on a fluid surface
Pressure at a point is thrust on unit area around the point
Unit: Nm-2 (Pa) Dimensions: ML-1T-2
The pressure due to a liquid column of height h is given by
P = h ρ g
where ρ is density of liquid
Viscosity arises due to tendency of liquid layers to resist relative motion The viscous force is given by Newton’s formula
r v
where η is coefficient of viscosity of the liquid, A area of cross section and
r
vis velocity gradient
Unit of coefficient of viscosity: Nsm-2 Dimensions: ML-1T-1
Poiseuille’s formula gives the rate of flow of liquid through a tube The volume V of the liquid flowing through the tube in t seconds is given by
t
V =
l 8
Pr4η π
where, P = pressure difference at the ends of the tube, r = radius of the tube, l = length of the tube, and η = coefficient of viscosity of the liquid
This formula is valid only when the motion is streamlined and the liquid has low viscosity
The viscous force acting on a sphere of radius a, in a highly viscous medium of viscosity η is given by
F = 6 π η a v
where, v is the terminal velocity of the sphere
The terminal velocity of a sphere in terms of density of the sphere ρ and viscosity η
v = ( )
9
g a
kinetic energy + potential energy + pressure energy = constant
ρ++gh P
v
2
1 2 = constant for unit mass of liquid
The velocity of efflux is the horizontal velocity with which the liquid flows through a narrow orifice in a vessel This is given by
Trang 11S x area of contact between the film and glass plate
The work done W against surface tension to increase the area is given by
W = increase in area x surface tension
9 HEAT AND THERMODYNAMICS
The linear expansivity α of a substance is defined as the increase in length per unit length per degree rise in temperature
where ∆ l is the increase in length l original length, ∆T rise in temperature
The areal expansivity β of a substance is defined as the increase in area per unit area per degree rise in temperature
T A
where ∆ A is the increase in area, A original area and ∆T rise in temperature
The cubical (volume) expansivity γ of a substance is the increase in volume per unit volume per degree rise
in temperature
T V
V
∆
γ
where ∆V is increase in volume, V original volume and ∆T rise in temperature
It can be shown that
β = 2 α and γ = 3α approximately
A liquid has two volume expansivities They are apparent and real If γ (real) is real expansivity and γ(app) is apparent expansivity
γ (real) = γ (app) + γ (cont)
where γ (cont) is cubical expansivity of the container
The density of a liquids d1 and d2 at two temperatures T1 and T2 respectively are related by the equation (T2
PP
where Pt is pressure at t0C, Po is pressure at 00C at constant volume
Volume coefficient of a gas is given by
t x V
V V
where Vt is volume at t0 C and V0 volume 00C at constant pressure
For any two temperatures t1 and t2
1 2 2 1 1 2 1
2
1
1 2
t P - t P P - P and
t V - t
V
V -
=
α
Trang 12Gas Laws
If P is the pressure of a gas, V its volume and T its temperature and ρ its density, then
PV = constant at constant T (Boyle’s law)
constant at constant pressure (Charles’ laws)
The ideal gas equation is a relation when all the above three parameters vary For one mol of gas,
PV = RT and PV = nRT for n mol of gas
The heat energy required to raise the temperature of a body of mass ‘m’ and specific heat ‘c’ through ∆T is given by
H = mc ∆T
A gas has three variables They are pressure P, volume V, and temperature T When a gas is heated, both its pressure and volume change For convenience, one of them is kept constant So a gas has two specific heats They are specific heat at constant pressure (Cp) and specific heat at constant volume (CV)
Specific heat at constant volume CV is the heat required to raise the temperature of unit mass ( 1 mol) through 1 K
Specific heat at constant pressure (Cp) is the heat required to raise the temperature of 1 mol of gas through 1
K at constant pressure
Units : J mol-1 K-1 Dimensions: L 2T-2K-1
Cp exceeds CV by a factor equal to external work done by the gas
(m is molecular mass and r is gas constant for 1 kg)
From the kinetic theory, root mean square velocity of gas molecules is
Vrms =
ρ
P3
where P is the pressure of the gas and ρ its density
The rms velocity does not depend on pressure
It is directly proportional to square root of temperature
Crms =
M
RT3
( T- temperature, M- molar mass) The average kinetic energy of a gas molecule per degree of freedom is given by
2
1
B
where kB is Boltzmann’s constant
The average kinetic energy of one mole of monatomic gas is given by
2
3
For one molecule, we divide the right hand side by N, the Avogadro number
Mean free path is the average distance travelled by a gas molecule between two successive collisions Mean free path λ is given by Maxwell’s equation,
2
n2
1
=
σπ
where σ is the diameter of the molecule and n is the number of molecules in unit volume
Thermodynamics
It deals with study of interaction of heat and other forms of energy, especially mechanical energy
Zeroeth law of thermodynamics states that ‘if two systems A and B are in thermal equilibrium independently with a third system C, then A and B will be in thermal equilibrium with themselves
The zeroeth law defines temperature It says without a third system C (which could be thermometer), it will not be possible to compare the temperature of two systems A and B
First law of thermodynamics states that if we supply heat ∆ Q to a system, work done by the system is ∆W and the increase in internal energy is ∆U, then
Trang 13Physics at a Glance 13
∆ Q = ∆ U + ∆ W
Internal energy of a thermodynamic system is the sum of potential and kinetic energies
Work done by a thermodynamic system is the area of the P-V graph
Critical constants: Critical temperature (Tc) is that temperature above which a gas cannot be liquefied by compression
Critical pressure (Pc) is the pressure required to liquefy a gas at critical temperature
Critical volume (Vc) is the volume of unit mass of gas at critical pressure and critical temperature These three are called critical constants
Triple point is the temperature at which the three states of matter (solid, liquid and gas) remain in equilibrium
Triple point of water is 273.16 K at a pressure of 610 Pa (4.58 mm of Hg or 0.006 atmosphere)
Work done during an isothermal process at a temperature T, when a gas changes its volume from V1 to V2
=
−γ
=
)T-T(1R
)VP-VP(11
2 1
2 2 1 1
for 1 mol For n mol multiply the RHS by ‘n’
Equation to the process
First law applied
to it (∆ Q = ∆ U + ∆W)
Isothermal Constant temperature PV= constant ∆ U = 0, ∆Q = ∆ W
Isobaric Constant pressure P = constant V/T = constant ∆W = p dV ∆Q = ∆ U + P dv
Isochoric Constant volume P/T = constant dV = 0 ∆ Q = ∆ W
Adiabatic Total heat
Second law of thermodynamics states that heat flows only from a body at higher temperature to that at
lower temperature External work has to be done to transfer heat from a cold body to a hot body
Units of thermal conductivity: Wm-1 K-1 Dimensions: MLT-3K-1
Convection is the mode of transmission of heat by actual motion of molecules It is possible only in fluids Radiation is the transmission of heat as electromagnetic waves Radiant heat travels with the speed of light
It requires no medium
Blackbody is one which absorbs all the heat falling on it and emits all radiation it has when heated top a suitable temperature
Blackbody radiation is the radiation emitted by a black body The laws of Black body radiation are:
1 Wein’s displacement law states that λmT = constant, where λ m is the wavelength corresponding to maximum energy The constant is known as Wein’s constant
Trang 142 Stefan’s law of radiation: The energy emitted by a black body per unit area per second is directly proportional to the fourth power of its Kelvin temperature
E = σ T4 , where the constant σ is known as Stefan’s constant
Units of Stefan’s constant: Wm-2K-4 Dimensions: MT-3K-4
Relative emittance of a body is the ratio of heat emitted by the body per unit area in one second to that emitted by a perfectly black body under identical conditions
The radiation emitted from any body of relative emittance (emissivity) e, having an area A, at a temperature
T during a time t, into surroundings of temperature T0 is given by
E = σ Ate (T4-T0)
Heat Engines
A heat engine is a device, which converts heat energy into mechanical work A heat engine absorbs a quantity of heat Q1, from the source and rejects a heat Q2 to the sink, and thereby converts Q1-Q2 into useful work Its efficiency η is given by
η =
1
2 1
Q
Q-Q
The efficiency of an ideal heat engine (Carnot’s engine) working between two temperature T1 (source) and T2 (sink) is given by
η =
1
2 1
Q
Q-Q
=
1
2 1
T
T-T
A Carnot’s refrigerator is a reversible heat engine, which operates at the reverse cycle It absorbs a quantity of heat Q2 from the sink and rejects a quantity of heat Q1 to the source (surroundings) If W is the external energy supplied, then Q1 = Q2 + W The ratio
2
Q
− = 1 2
2
TT
T
T -
1
2 1
r
qAq
Electrostatic field at a point is the force acting on a unit charge placed at that point
Two units of electrostatic field: N C-1 and Vm-1 Dimensions: MLT-3 I-1
Field due to a point charge q at a distance ris given by
r
Aq
Electrostatic potential at a point is the potential energy of a unit charge placed at the point
Unit: JC-1 = volt (V) Dimensions: ML2T-3I-1
The electrostatic potential energy between two point charges q1, q2 at a distance r is
Trang 15Physics at a Glance 15
Unit: C m Dimensions: ITL
The field due to an electric dipole at a point along the axial line at a distance r from its centre is
2 2 3
2 2
r
pA)a
r
Ap )
a
-r
The potential energy of a dipole in a uniform electric field E is
U = -p.E = -pE cos θ
where θ is the angle between field and axis of the dipole
The torque acting on a dipole in a uniform field is
τ = p x E
The magnitude of torque = pEsin θ In a non-uniform field a dipole experiences both force and torque Potential difference between two points a and B in an electric field is VB – VA = line integral of electric field It is given by
VB – VA = -B∫
A
dl
.
E
If there is more than one charge, the algebraic sum of charges should be taken on the right hand side
By applying this theorem we can evaluate the electric field E due to the following:
1 A thin sheet of charge at a distance r
E=
0
2ε
σ
(σ charge per unit area)
2 A line of charge at a distance r ( A long thin needle)
E=
r
2πε0
λ (λ charge per unit length)
3 Two plane sheets of charge per unit area + σ1 and + σ 2
E= 0, at a point inside
0
2 0
1
2
2 ε
σ+ε
2ε
σ+σ
This is the equation for the field between the parallel plate condenser
Capacitance is the ratio of charge q to potential V
Trang 16The capacitance of a parallel condenser is
2 1 r 0
rr
rr4
+
επε
where r1,r2 are the radii of inner and outer concentric spheres the condenser is made of
The capacitance of a cylindrical condenser made from two concentric cylinders of radii r1 and r2 and of length L is
)r/rlogx3
2
L2
1 2 10
r
0επε
where εr is relative permittivity of the medium between the cylinders
Capacitance of a parallel plate condenser with two dielectric media of thickness t1 and t2 and dielectric constants k1 and k2 respectively is
1 2 2 1
2 1 0
tk+t
k
Akkε
Three capacitors of capacitance C1, C2, C3 joined in series have an effective value C given by the equation
3 2
1 C
1 C
1
C
When ‘n’ identical capacitors each of capacitance C are joined in parallel, the effective value of capacitance
is nC When n identical capacitors are in series will have an effective value
n
C
The energy of a charged conductor of capacitance C, charge q and potential V is given by
V and q of terms in qV
2
1
E
q and C of terms in C
2
1
E
2 2
Unit of R: ohm (Ω) Dimensions of R : ML2T-3I-1
The resistance of the conductor of length and cross-sectional area A is given by
A
L
ρ
where ρ is called resistivity (specific resistance) Unit: Ω m
Three resistors R1, R2, R3 in series have an effective value R, given by
R = R1 + R2 + R3
Two resistors R1, R2 in parallel have an effective value R given by
2 1 2 1 2
R R R or R
1 R
=
Three resistors R1, R2, R3 in parallel have an effective value R given by
3 2
1 R
1 R
1
R
3 1 3 2 2 1
3 2 1
R R R R R R
R R R R
+ +
=