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2010-10-22T11:18:00.000+05:30

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Learn Physics through physics notes and study material: 5 Secrets to Cracking Competitive Exams: "They’re not the kind of exams you’re used to in school, but you know you have to prepare hard for them if you want to go to college. Almost ..."

Waves in continous medium

2010-10-18T11:42:00.001+05:30

There are essentially two ways of transporting energy from one place to another (a) Actual transport of matter for example a fired bullet and (b) Waves carry energy but there is no transport of matter for example sound waves carry energy so thay can move diphagram of the ear.
Here we will consider the oscillations of open or unbounded systems i.e., systems having no outer boundaries.
If such system is disturbed , waves travel in the system with a speed determined by the properties of the system.
Waves are not reflected back in such a system.
The waves generated by driving force are called travelling waves ; these waves travel from the point where the driving force produces the disturbance.
If the driving force produces a harmonic disturbance the travelling wave it produces are called harmonic travelling waves.
In the steady state, all moving parts of the system oscillates with simple harmonic motion at the driving frequency.
Waves where the displacements or oscillations are transverse (i.e., perpandicular) to the direction of wave propagation is called transverse wave.
The wavelength (denoted by λ) of the wave is defined as the distance, measured along the direction of the propagation of the wave, between two nearest points which are in the same state of viberation.
Wavelength λ is just the distance travelled by the wave during one time period T of particle oscillation. Thus wave velocity
v=λ/T=λν
where ν=1/T - is the frequency of the wave.
This relation between wave velocity, frequency and wavelength also holds for longitudinal waves in which the displacements or oscillation in the medium are parallel to the direction of wave propagation.
Waves in spring and sound waves are longitudinal waves.
Wavelength for longitudinal waves is the distance between two successive compressions or rarefactions.
Sound waves are also compressional.
Assumptions that are made while obtaining wave equation are:-
1. Amplitude A of particle oscillations does not change in course of the propagation of wave.
2. The medium is isotopic and homogeneous so that velocity of wave does not chance from place to place
Displacement of particle at x at any time t is
Ψ(x,t) = A sin{2π(t-x/v)/T)}
The function Ψ(x,t) repeats itself in a distance λ . Wavelength of a wave is also known as spatial periodicity of the wave.
The wave is thus doubly periodic. It has temporal periodicity T and spatial periodicity λ.
Let us define quantities
k=2π/λ and
ω=2π/T
then wave function can be written as
Ψ(x,t) = A sin{ωt-kx}
where quantity k is known as wave number of the wave and ω is called angular frequency of particle oscillations in wave.
Harmonic wave travelling in
+ x direction : Ψ(x,t) = A sin{ωt-kx}
- x direction : Ψ(x,t) = A sin{ωt+kx}
above equations can also be equally well described by cosine function.
Classicsl wave equation is

Important inferences from above wave equation
1. Whenever second order time derivative of any physical quantity is related to second order space derivative as in above equation , a wave of some sort must travell in the medium.
2. Velocity of that wave is given by the square root of the coefficent of second order space derivative.
Individual derivatives which makes up the medium do not propagate through the medium with the wave; they merely oscillates ( transversly or longitudinally) about there equilibrium positions.
It is their phase relationship which we observe as wave.
Wave velocity is also called phase velocity with which crest or troughs in case of transverse wave and compressions or rarefactions in case of longitudinal waves travell through the medium.
The phase velocity is given by
v=λ/T=λν
or,
v=ω/k
Ψ(x,t)=f(vt-x) is the solution of the above given wave equation.

Uncertainity Principle

2010-10-07T12:12:00.072+05:30

Uncertainity principle says that "If a measurement of position is made with accuracy Δx and if the measurement of momentum is made simultaneously with accuracy Δp , then the product of two errors can never be smaller than a number of order h
ΔpΔx≥(∼h)                                  (1)
Similarly if the energy of the syatem is measured to accuracy ΔE , then time to which this measurement refers must have a minimum uncertainity given by
ΔEΔt≥(∼h)                                    (2)
In generalised sence we can say that if Δq is the error in the measurement of any co-ordinate and Δp is the error in its canonically conjugate momentum then we have,
ΔpΔq≥(∼h)                                  (3)
Consider the relation between the range of position Δx and range of wave number Δk appearing in a wave packet then
ΔxΔk≥1                                                     (4)
and this is a general property not restricted to quantum mechanics. Uncertainity principle is obtaines when the following quantum mechanical interpretation of quantities appearing in above equation are taken into account.
(1) The de-Brogli equation p=hk creates a relationship between wave number and momentum , which is not present in classical mechanics.
(2) Whenever either the momentum or the position of an electron is measured , the result is always some definite number. A classical wave packet always covers a range of positions and range of wave numbers.
Δx is a measure of minimum uncertainity or lack of complete determination of the position that can be ascribed to the electron. and Δk is the measure of minimum uncertainity or lack of complete determination of the momentum that can be ascribed to it.

Relation of spreading wave packet to uncertainity principle
Narrower the wave packet to begin with , the more rapidly it spreads. Because of the confinement of the packet within the region Δx₀ the fourier analysis contains many waves of length of order of Δx₀ , hence momenta p≅h/Δx₀
therefore
Δv≅p/m≅h/mΔx₀
Although average velocity of the packet is equal to the group velocity , there is still a strong chance that the actual velocity will fluctuate about this average by the same amount. The distance covered by the particle is not completely determined but it may vary as much as
Δx≅tΔv≅ht/mΔx₀
The spread of the wave packet may therefore be regarded as one of the manifestations of the lack of complete determination of initial velocity necesarily associated with the narrow wave packet.

Relation of stability of atom to uncertainity principle
From uncertainity principle if an electron is localized it must have on an average a high momentum and have high kinetic energy as it takes energy to localize a particle. According to uncertainity principle it takes a momentum  Δp≅h/Δx and an energy nearly equal to h²/2m(Δx)² to keep an electron localised within a region Δx. Momentum creates a pressure which tends to oppose localization of the electron. In an atom the pressure is opposed by the force attracting the electron back to the nucleus. Thus the electron will come to equilibrium when the attractive forces balances the effective pressure and, this way , the mean radius of the lowest quantum state is determined. This point of balance can be found from the condition that total energy must be minimum. Thus we have
W≅  (h²/2m(Δx)²) - (e²/Δx)
Differentiating both the sides w.r.t. Δx and making ∂W/∂(Δx) = 0 we get
Δx≅h²/me²
THis result is just the radius of first Bohr orbit although not exact but qualitative.. The limitation of the localizability of the electron is inherent in the wave-particle nature of matter. In order to have an electron in very small space , we must have very high fourier components in its wave function and therefore the possibility of very high moments. There is no way to force an electron to occpy a well defined position and still remain at rest.

Force on a conductor

2010-10-06T10:11:00.000+05:30

We have already learned in our previous discussion that field inside a conductor is zero and the field immidiately outside is
E_n=n(σ/ε₀)                               (1)
where n is the unit normal vector to the surface of the conductor. We also know that any charge a conductor may carry is distributed on the surface of the conductor.
In presence of an electric field this surface charge will experience a force. If we consider a small area element ΔS of the surface of the conductor then force acting on area element is given by
ΔF=(σΔS).E₀       (2)
where σ is the surface charge density of the conductor , (σΔS) is the amount of charge on the area element ΔS and E₀ is the field in the region where charge element (σΔS) is located.
Now there are two fields present  E_σ and E₀ and the resultant field both inside and outside the conductor near area element  ΔS would be equal to the superposition of both the fields  E_σ and E₀ . Figure below shows the directions of both the fields inside and outside the conductor

Now field E₀ has same value both inside and outside the conductor and surface element ΔS suffers discontinuty because of the charge on the surface and this makes field E_σon either side pointing away from the surfaceas shown in the figure given above. Since E=0 inside the conductor

E<sub>in=E₀+E_σ=0
E_in=E₀=E_σ
Since direction of  E_σ and E₀ are opposite to each other and outside the conductor near its surface
E_out=E₀+E_σ=2E₀
Thus , E₀ =E/2                      (3)
Equation (2) thus becomes,regardless of the of ΔF=½(σΔS).E                  (4)
From equation 4 , force acting per unit area of the surface of the conductor is
f=½σ.E                                       (5)
Here is the  E_σ electric field intensity created by charge on area element ΔS at the point very close to this area element. In this region this area element behaves as infinite uniformly charged sheet hence we have,
E_σ=σ/2ε₀                                                  (6)
Now,
E=2E₀=2E_σ=(σ/ε₀)n=E_n
which is in accordance with equation 1. Hence from equation 5
f=σ²/2ε₀ = (ε₀E²/2)n                (7)
This quantity f is known as surface density of force. From equation 7 we can conclude that regardless of the sign of σ and hence direction of E , f is always directed in outward direction of the conductor.

Electric field due to charged conductor

2010-09-28T11:03:00.000+05:30

In our previous post we have discussed that electric field inside a conductor is zero and any charge the conductor may carry shall be distributed on the surface of the conductor. For our discussion consider a conductor carrying charge on its surface again consider a small surface element ds over which we can consider surface charge density σ to be approximately constant.
For positive charge distributed over the surface of the conductor , electric field E would be directed at right angels to the surface pointing in outwards direction. Now E due to charge carrying conductor can be calculated using Gauss's law. For this draw a Gaussin cylendrical surface as shown below in the figure

Now S is the area of cross-section of the surface. The flux due to cylendrical surface is zero because electric field and the normal to the surface are perpandicular to each other. Since electric field inside the conductor is zero hence only contribution to the flux is due to the chare on area S lying outside the surface of the conductor. So total flux through the surface would be

From Gauss's law,
ES=q/ε₀=σS/ε₀
or,
E=σ/ε₀
and this is the required relation for the field of charged conductor

How conductors behave in the presence of electrostatic field

2010-09-16T11:27:00.000+05:30

We know that conductors like copper , silver, aluminium etc. , have very large number of free and movable charge carriers , usually one free electron per atom. These free electrons are not bound to its atom and moves freely in the space between the atoms. These free electrons can move under the action of electric field present inside the conductor.
Consider an arbitrary shaped conductor placed inside an electric field such that the field in the conductor is directed from left to right. As a result of this electric field positive charge in the conductor moves from left to right and negative charge moves from right to left. As a result there is a surplus negative charge on the left side of the conductor and a surplus positive charge on the right side of the conductor. This induced surplus chare on both the sides of the conductor acts as a source of an induced electric field which is directed from right to left i.e., in the direction opposite to the initial electric field.
Now with the increase in the amount of induced electric charges, magnitude of induced electric field also increases which cancels out the original electric field having direction opposite to it. This results in a progressive decrease in total field inside the conductor. In the end induced electric field cancels out all the initial electric field thus reaching an electrostatic equilibrium where there is zero electric field at each and every point inside the conductor. Hence we can conclude thet,
E=0 inside the conductor
Now if we apply Gauss's law to any arbitrary surface inside the conductor then total charge enclosed by the gaussian surface equals zero as vector E=0 at all points inside the gaussian surface. From this we conclude that
Al the excess charge (if any) is distributed on the surface of the conductor
We have established the fact that there is no E inside the conductor so tangential component of E is zero on the surface of the conductor hence the potential difference between any two points on the surface of the conductor would also be zero. This indicates that the surface of conductor in electrostatics is equipotential one. Since there is no E inside the conductor so all the points in the conductor are at the same potential.
This is almost all I intended to write in this topic however if you have any doubts then let me know and also you can tell me about the topic you want me to write next in the blog.

Electrostatics

2010-09-15T11:20:00.000+05:30

You all know physicsgoeasy is my blog and i had already given a good amount of notes on basic electrostatics in this blog so I did not intend to write further on basic electrostatics topics. For detailed basic notes on electristatics you can visit my website physicscstalyst.com and links to my blog postes of electrostatic related material are given below.
Learn Physics for IITJEE , AIEEE , PMT and CBSE Board Examination: Electricity Index: "Electric Charge and Coulumb Law Electric Field Charge Density And Conductor Electricty Test Series -1 Electric Potential Electric Flux and G..."
However I will discuss complicated and higher level electrostatic related topics in this blog.

Vector Differentiation 2

2010-01-14T18:54:00.000+05:30

In this post we'll discuss about Divergence and curl of vector fields.
Divergence of vector fields
We already know about dot product of two vectors. Now consider a vector field φ(x,y,z) and we now find out the dot product of operator ∇ and vector field φ(x,y,z). So,

or,

Thus we see that dot product of del operator with a vector field is a scalar quantity. This scalar quantity ∇⋅φ(x,y,z) is known as divergence of the vector function φ(x,y,z). Thus,
∇⋅φ = div φ='divergence of φ'
Physically we can say that divergence of any vector field is the measure of how much the vector diverges (spreads out) or converges at a given point. Divergence of a vector at any point can be positive, negative and zero illustrated below in the figures.

Consider the example of a fluid, if the Divergence is positive at any point in a fluid , then either the fluid is expanding and its density at the point is falling with time or the point is a source of the fluid. Again if the divergence is negative , then either the fluid is contracting and the density is rising at the point or the point is at the sink. If the flux entering field space is equal to that leaving it then divergence of the field is zero i.e. there is no source or sink and its density is also not changing which means that liquid is incompressible. Any vector having its divergence equal to zero is known as solenoidal vector.
Curl of the vector fields
We have already discussed the dot product of del operator (∇) with a vector which is known as divergence of vector. Now let us find the cross product of operator ∇ with the vector field φ(x,y,z). Thus,

or

Thus we have seen that curl of a vector function is a vector whose components can be written by usual rule of cross products. Any vector field φ(x,y,z) for which ∇xφ(x,y,z) = 0 is said to be irrotational.The curl of a scalar field makes no sense.The curl of a vector field φ(x,y,z) at a point P may be regarded as a measure of the circulation or how much the field curls around a point.
In our next discussion we'll learn about second derivatives of vector fields and we'll discuss more about divergence and curl while discussing vector integral calculas.

Vector Differentiation 1

2009-10-03T03:27:00.000+05:30

Here in this post we will revise our concept of Vector Calculas (differentiation of vectors). This mathematical tool would help us in expressing certain basic ideas with a great convenience while studying electrodynamics.

DIFFERENTIATION OF VECTORS
Consider a vector function f(u) such that
f(u) = f_x(u)i + f_y(u)j + f_z(u)k
where f_x, f_y(u) and f_z(u) are scalar functions of u and are components of vector f(u) along x, y, and z directions.If we want to find the derivative of f(u) with respect to u we will have to proceed in the similar manner we used to do with ordinary derivatives thus

where df(u)/du is also a vecor.
Thus in cartesian coordinated derivative of vector f(u) is given by

SCALAR AND VECTOR FIELDS
When we talk about fields then in this case a particular scalar or vector quantity is defined not just at a point in the space but it is defined continously throughout some region in space or maybe the entire region in the space. Now a scalar field φ(x,y,z) assocites a scalar with each point in the region of space under consideration and a vector field f(x,y,z) associates a vector with each point.
In electrodynamics we will come across the cases where variation in scalar and vector fields from one point to is continous and is also differentiable in the particular region of space under consideration.

GRADIENT OF A SCALAR FIELD
Consider a scalar field φ(x,y,z). This function depends on three variables. Now how would we find the derivative of such functions. If we infinitesimal change dx, dy and dz along x, y, and z axis simultaneously then total differential Dφ of function φ(x,y,z) is given as

above expression comes from our previous knowledge of partial differentiation.
If we closely examine above equation this could be a result of dot product of two vectors thus,

or,
dφ=(∇φ)•(dr)
where

is gradient of φ(x,y,z) and gradient of a scalar function is a vector quantity as it is the multiplication of a vector by a scalar.
Thus we see that gradient of any scalar field has both magnitude and direction. Again consider the function φ(x,y,z) then from ordinary calculas any change in this function as discussed above is given by

thus
dφ=(∇φ)•(dr) = |∇φ| |dr|cosθ
FRom this we see that dφ(x,y,z) will be maximum when cosθ=1 which would be the case when dr would be parallel to ∇φ. Thus function φ changes maximally when one moves in the direction same as that of gradient. So we can say that the direction of ∇φ is along the greatest increase of φ and the mahnitude of |∇φ| gives the slope along that direction.
CONCLUSION: The gradient ∇φ points in the direction of the maximum increase of function φ(x,y,z) and the magnitude |∇φ| gives the slope or rate of increase along the maximal direction.

THE OPERATOR ∇
While discussing gradient of a scalar function we find that gradient of any function is given by

or,

where the term in parentheses is called "del"

Del is an vector derivative or vector operator and this operator acts on everything to its right in an expression, until the end of the expression or a closing bracket is reached. There are three ways in which ∇ can act or operate on a scalar or vector function
1. On a scalar function φ : ∇φ (the gradient);
2. On a vector function f, via the dot product: ∇ • v (the divergence);
3. On a vector function f, via the cross product: ∇ x v (the curl).
Out of these three ways of operation of ∇ on any function we have already discussed gradient of a scalar function.

In this post we learned about scalar and vector fields, gradient of scalar fields and ∇ operator. In the next post we'll lern more about vector differential calculas i.e, in particular we'll discuss divergance and curl of vector fields.
Yes, your comments and querries are heartly invited please do give us a reaponce and comment on our work so that we could further improve it and your responce is more then enough to encourage us.

Vector Algebra 2

2009-08-07T01:56:00.000+05:30

In this post we'll lern Vector algebra in component form.
Component of any vector is the projection of that vector along the three coordinate axis X, Y, Z.

VECTOR ADDITION
In component form addition of two vectors is
C = (A_x+ B_x)i + (A_y+ B_y)j + (A_y+ B_y)k
where,
A = (A_x, A_y, A_z) and B = (B_x, B_y, B_z)
Thus in component form resultant vector C becomes,
C_x = A_x+ B_x
C_y = A_y+ B_y
C_z = A_z+ B_z

SUBTRACTION OF TWO VECTORS
In component form subtraction of two vectors is
D = (A_x- B_x)i + (A_y- B_y)j + (A_y- B_y)k
where,
A = (A_x, A_y, A_z) and B = (B_x, B_y, B_z)
Thus in component form resultant vector D becomes,
D_x = A_x - B_x
D_y = A_y- B_y
D_z = A_z- B_z

NOTE:- Two vectors add or subtract like components.

DOT PRODUCT OF TWO VECTORS
A.B = (A_xi + A_yj + A_zk) . (B_xi + B_yj + B_zk)
= A_xB_x + A_yB_y + A_zB_z.
Thus for calculating the dot product of two vectors, first multiply like components, and then add.

CROSS PRODUCT OF TWO VECTORS

A x B = (A_xi + A_yj + A_zk) x (B_xi + B_yj + B_zk)
= (A_yB_z - A_zB_y)i + (A_zB_x - A_xB_z)j + ( A_xB_y - A_yB_x)k.

Cross product of two vectors is itself a vector.
To calculate the cross product, form the determinantwhose first row is x, y, z, whose second row is A (in component form), and whose third row is B.

VECTOR TRIPPLE PRODUCT

Vector product of two vectors can be made to undergo dot or cross product with any third vector.

(a) Scalar tripple product:-
For three vectors A, B, and C, their scalar triple product is defined as
A . (B x C) = B . (C x A) = C . (A x B)
obtained in cyclic permutation. If A = (A_x, A_y, A_z) , B = (B_x, B_y, B_z) , and C = (C_x, C_y, C_z) then A . (B x C) is the volume of a parallelepiped having A, B, and C as edges and can easily obtained by finding the determinant of the 3 x 3 matrix formed by A, B, and C.

(b) Vector Triple Product:-
For vectors A, B, and C, we define the vector tiple product as
A x (B x C) = B(A . C) - C(A - B)
Note that
(A . B)C ≠ A(B . C)
but
(A . B)C = C(A . B).

Vector Algebra 1

2009-07-22T19:06:00.001+05:30

Here in this post we will go through a quick recap of vector algebra keeping in mind that reader already had detail knowledge and problem solving skills related to the topic being discussed. Here we are briefing Vector Algebra because concepts of electrostatics , electromagnetism and many more physical phenomenon can best be conveniently expressed using this tool.

A vector is a quantity that requires both a magnitude (= 0) and a direction in space it can be represented by an arrow in space for example electrostatic force, electrostatic field etc. In symbolic form we will represent vectors by bold letters. In component form vector A is written as
A = A_xi+ A_yj+A_zk

ADDITION OF VECTORS
Two vectors A and B can be added together to give another resultant vector C.
C = A + B

SUBTRACTION OF VECTORS
Two vectors A and B can be subtracted to give another resultant vector D.
D = A - B = A + (-B)

SCALAR MULTIPLICATION OF VECTOR
When we multiply any vector A with any scalar quantity 'n' then it's direction remains unchanged and magnitude gets multiplied by 'n'. Thus,
n(A) = nA
Scalar multiplication of vectors is distributive i.e.,
n(A + B) = nA +nB

DOT PRODUCT OF VECTORS
Dot product of two vectors A and B is defined as the product of the magnitudes of vectors A and B and the cosine of the angle between them when both te vectors are placed tail to tail. Dot product is represented as A.B thus,
A.B = |A| |B| cosθ
where θ is the angle between two vectors.
Result of dot product of two vectors is a scalar quantity.
Dot product is commutative : A.B = B.A
Dot product is distributive : A . (B+C) = A.B + A.C also A.A = |A|²

CROSS PRODUCT OF TWO VECTORS
Cross product or vector product of two vectors A and B is defined as
A x B = |A| |B| sinθ nˆ
where nˆ is the unit vector pointing in the direction perpandicular to the plane of both A and B. Result of vector product is also a vector quantity.
Cross product is distributive i.e., A x (B + C) = (A x B) + (A x C) but not commutative and the cross product of two parallel vectors is zero.

In our next post we'll study vector algebra in component form and also lern about vector triple products.

Overview of electrostatics and electricity

2009-07-18T01:44:00.001+05:30

Electrostatics involves electric charges namely positive and negative charges, the forces between them which is known as electric force , the field that surrounds them, and their behavior in materials. Coulumb's law is the simple relation that governs electrostatic interactions and the field around the charges is described using the concept of electric field. Coulumb's law is a inverse square law which gives the force between two charges kept at some distance (say r ) apart from each other. Like Coulumb's law, law of gravitation is also a inverse square law but gravitational interactions are only attractive in nature and electrical interactions are attractve as well as repulsive depending on the nature if interacting charges. Charges of same kind repel each other and charges different kinds , i.e. one charge positive and other negative , attract each other. One more thing electric interactions are much more stronger then gravitational interactions and gravitational force are almost negligible in comparison to the forces of electric origin. This is always true when we study the interactions of atomic and subatomic particles. But when we study objects very large in size say a person ,a planet or satellites, the net governing force in this case is gravitational force not electric.
Now coming to the properties of electric charges we know that electric charge is quantized and it also obeys law of conservation means total charge remains conserved. In what we say electrostatic interactions electric charges are at rest in our frame of refrence. Now think what happens when we are at rest in our frame of reference and charge under consideration is moving with velocity v with respect to us such a miving charge leads to the origin of magnetism which we will discuss in later section.
Again a question what is electricity. Electricity, deals with stationary and moving electric charges, the actions of force between these charges, and the electric and magnetic fields generated by them. Electrostatics is simply the electricity at rest. Electricity is the backbone of the modern society in which we use various instruments which depends on electric current for their functioning and without it we would not have telephones, television, household appliances and many more gadgets which are now part of our daily life. In electricity we study the motion of electric charges, or electric currents and also the voltages that produce currents and the ways to control currents.
We have learned about what electrostatics is and what we study in it. Now we will discuss why we study electrostatics and where it finds its applications.Electric interactions are of immence importance in chemistry and biology and have many technological applications. Concepts of electricity proved to be of basic importance for studying atomic physics, nuclear physics and solid state physics. It also find importance in studying advanced level physic.

About Us

2009-07-18T01:34:00.000+05:30

This blog has been started keeping in mind to help graduate level students learn physics. Here you would find material you need for preparing your annual exams and various competetive exams. This blog will help you gain knowledge of physics in an easy and regular manner. Here in this blog i'll be providing you brief and short notes on various topics being tought in graduate lavel physics. Apart from notes i'll try my best to help you solve your problems related to physics and try to come up with numerical , conceptual and derivative related problems along with their solutions. Problems and discussions from your side are invited with the hope that we will come out with solutions.
All the material given in the blog would be designed by the team having qualifications like MSc physics and Engineering.