Advantages of potential formulation in electrostatics

We already know about electric field and electric potential. We also know that electrostatic field is completely characterized by vector function E(r). The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding. Now a question arises why do we need introduction of electric potential when we already have electric field for the description of electric force between charges.
Firstly, the concept of electric potential is very useful not only in physics but as well as in engineering .This is because if we know the potential we can easily calculate the work  done by field forces when a charge is displaced from point 1 to point 2 that is
\[{W_{12}} = q({\varphi _1} - {\varphi _2})\]
where \({\varphi _1}\)  and \({\varphi _2}\) are the potentials at points 1 and 2. This means that required work is equal to the decrease in the potential energy of charge q when it is displaced from point 1 to 2.
Calculation of the work of field forces with the help of above mentioned formula is not just simple but the only possible method in some cases.
Secondly in some cases of electrostatic field calculation it is often easier to first calculate the potential and then find the gradient of potential  \({\varphi}\) to calculate the value of electric field intensity E. Also for calculating \({\varphi}\) we only need to evaluate one integral but for calculation of E we must take three integrals all for x, y, and z directions since E is a vector quantity.
But we must note that for problems with high symmetry we must directly calculate E using Gauss's Theorem which is much simpler way to find electric field intensity when charge distribution is symmetrical.

Relative velocity

Before discussing relative velocity we must know that all the motion is relative. Everything moves-even things that appear to be at rest. They move relative to the Sun and stars. When we discuss the motion of something, we describe the motion relative to something else. When we say a racing car reaches a speed of 200 kilometers per hour, we mean relative to the track. Unless stated otherwise, when we discuss the speeds of things in our environment, we mean relative to the surface of Earth. Motion is relative.
Now we come to relative velocity
Before studying further I am assuming that you are familiar with the concept of frame of reference and velocity.
Observations made in different frames of reference are related to each other to know how, consider this example of two trains approaching one another, each with a speed of 90 km/h with respect to the Earth. Observers on the Earth beside the train tracks will measure 90 km/hr for the speed of each of the trains. Observers on either one of the trains (a different frame of reference) will measure a speed of 180 km/h for the train approaching them. When the velocities are along the same line, simple addition or subtraction is sufficient to obtain the relative velocity. But if they are not along the same line, we must make use of vector addition. And it is also necessary that when specifying a velocity, we specify what the reference frame is.
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What is entropy?

Entropy is a state function of a system, it depends only on the equilibrium state of the system. The chenge in entropy between initial and final equilibrium states is
$\Delta S=\int_{i}^{f}\frac{dQ}{T}$
where dQ is the infinitesimal heat transfer that takes place reversibly. The change in entropy for process, including irreversible one between given initial and final equilibrium state, is the same.The second law of thermodynamics may be expressed in terms of entropy, $\Delta S \geqslant 0$.
For reversible process  $\Delta S = 0$.
For irreversible process  $\Delta S > 0$.
Entropy is a measure of the disorder in a system. The second law states that the natural (irreversible) process tend to evolve to state of greater disorder, or from states of low probability to states of high probability.

H.C Verma Concepts of physics OR NCERT physics books

Many times i have had users ask me a question which book to opt between H.C Verma Concepts of physics and NCERT physics books. My answer is you can keep both the books set for your preparation as NCERT is nice book to learn a good concepts and is according to the exam level of class 11 and class 12. But what i think about  HC verma is that it is the best book for problems ,go for solved as well asboth unsolved questions in HC verma and do practice them without looking at the solutions. MCQ and conceptual problems given in the book are good. So the book is also good for objective purpose but first do subjective questions to get a grip over the concepts you have learned. You can also first solve the questions given in the NCERT book as they are bit easy in comparison to the questions given  in the HC Verma book. For further reference you can consult books like resnik halliday to get and in depth knowledge of  theory.

Class 9 Physics Notes : Motion (concept map 1)

Learn the basic concept behind motion just by looking at the following picture. From this figure learn about motion , distance and displacement.

CSIR-NET Physics and GATE physics: Fourier Series

CSIR-NET Physics and GATE physics: Fourier Series: Fourier series is an expansion of a periodic function of period $2\pi$ which is representation of a function in a series of sine or cosine...

Mathematics revision sheet for class 11 and class 12 physics

We have two quantities x and y such that y=f(x) where f(x) is some function of x.We may be interested in finding followings things
1. \frac{dy}{dx}
2. Maximum and Minimum values of y.It can be find with the method of Maxima and Minima
\frac{dy}{dx} is the called the derivative of y w.r.t to x
It is defined as
\frac{dy}{dx}=\lim_{\Delta x \to 0}\left ( \frac{\Delta y}{\Delta x} \right )
Some commonly known functions and their derivatives are:-
y=secx\frac{dy}{dx}=secx tanx
y=ln x\frac{dy}{dx}=\frac{1}{x}
Some important and useful rules for finding derivatives of composite functions
1. \frac{d}{dx}(cy)=c\frac{dy}{dx} where c is constant
2. \frac{d}{dx}(a+b)=\frac{da}{dx} + \frac{da}{dx} where a and b are function of x
3. \frac{d}{dx}(ab)=a\frac{db}{dx}+b\frac{da}{dx}
4. \frac{d}{dx}(\frac{a}{b})=\frac{[b\frac{da}{dx}-a\frac{db}{dx}]}{b^{2}}
5. \frac{dy}{dx}=(\frac{dy}{da})(\frac{da}{dx})
Maximum and Minimum values of y
Step 1:
fine the derivative of y w.r.t x
Solve the equation to find out the values of x
find the second derivative of y w.r.t x and calculate the values of
for the values of x from step2
if \frac{d^{2}y}{dx^{2}}>0 then the value of x corresponds to mimina of y then y_{min} can be find out by putting this value of x
if \frac{d^{2}y}{dx^{2}}<0 then the value of x corresponds to maxima of y then y_{max} can be find out by putting this value of x
It reads as integration of function f(x) w.r.t. x within the limits from x=a to x=b.
Integration of some important functions are
\int sinx dx=-cosx
\int cosx dx=sinx
\int sec^{x}dx=tanx
\int cosec^{x}dx=-cotx
\int \frac{1}{x}dx=lnx
\int x^{n}dx=\frac{x^{n+1}}{n+1}
\int e^x dx=e^x
Useful rules for integration are
\int cf(x)dx=c\int f(x)dx
\int[f(x)+h(x)]=\int f(x)dx+\int h(x)dx
\int f(x)g(x)dx=f(x)\int g(x)dx -\int\left ( f'(x)\int g(x)dx \right ) dx
Properties of trigonometric functions
1. Pythagorean identity
sin^2 A +cos^2 A=1
1+tan^ A=sec^2 A
1+cot^2 A=cosec^2 A
2. Periodic function
3.Even-Odd Identities
4. Quotient identities
tan(A)=\frac {sin A}{cos A}
cot(A)=\frac{cos A}{sin A}

5. Co-function identities

sin\left ( \frac{\pi}{2}-A \right )=cos(A)
cos\left ( \frac{\pi}{2}-A \right )=sin(A)
tan \left ( \frac{\pi}{2}-A \right )=cot(A)
cosec \left( \frac{\pi}{2}-A \right )=sec(A)
sec\left ( \frac{\pi}{2}-A \right )=cosec(A)
cot\left ( \frac{\pi}{2}-A \right )=tan(A)

6. Sum difference formulas

sin(A\pm B)=sin(A)cos(B) \pm sin(B)cos(A)
cos(A \pm B)=cos(A)cos(B) \mp sin(A)sin(B)
tan(A \pm B)=\frac {tan(A) \pm tan(B)}{1 \mp tan(A) tan (B)}

7. Double angle formulas

cos(2A)=cos^2(A)-sin^2(A)=2cos^2 (A)-1=1-2sin^2(A)

8. Product to sum formulas
sin(A)cos(B)=\frac {1}{2}[cos(A-B)-cos[A+B]
cos(A)cos(B)=\frac {1}{2}[cos(A-B)+cos[A+B]
sin(A)cos(B)=\frac {1}{2}[sin(A+B)+sin[A-B]
cos(A)sin(B)=\frac {1}{2}[sin(A+B)-sin[A-B]

9. Power reducing formulas

sin^2 A=\frac{1-cos(2A)}{2}
cos^2 A=\frac{1+cos(2A)}{2}
tan^2 A=\frac{1-cos(2A)}{1+cos(2A)}

10. reciprocal identities
Binomial Theorem
From the binomial formula, if we let a = 1 and b = x, we can also obtain the binomial series which is valid for any real number n if |x| < 1.
Geometric Series
a,aq,aq^2,aq^3,aq^^{n-1} where q is not equal to 0, q is the common ratio and a is a scale factor.Formula for the sum of the first n numbers of geometric progression
Infinite geometric series where |q| < 1
If |q| < 1 then a_{n} \to 0, when n -> infinity So the sum S of such a infinite geometric progression is:
S=\frac{1}{(1-x)} which is valid only for |x|
Arithmetic Progression
The sum S of the first n values of a finite sequence is given by the formula:
S=\frac{n}{2}[(2a + d(n-1)]
Quadratic Formula

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