Electrostatic Charge

Last minute preparation for board exams

• There are less then two weeks remaining before your board exams start and it is the high time students should concentrate and revise what they have studied over the past year.
• Students should manage their time so that they can divide their time for revising different subjects.
• Students should avoid being stressed and they should be very disciplined and should manage their time well for the  remaining  days before the exams.
• Students should avoid late night studies and get a sound sleep during the night. this way their body clock would not go erratic and students will remain relaxed and stress free.
• Students should not study anything new at this point of time and must revise the previously studied topics as studying new topics might confuse them.
• It is advisable to students that they should avoid hangout on social networking websites as their each and every second before exams is precious.

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Time management tips for students

1) Prepare a Schedule/Calendar for the studies.You only have 24 hours a day, 168 hours a week . Effective study time management means that you use your given time in the most efficient way. Therefore it is important to prepare a regularly updated schedule/calendar, where you:

a) write down all your scheduled tests

b) write down all tasks, assignments and duties,

c) record your planned leisure-activities, such as gym, etc.

d) Write an ordered list of what you need to get done.Prioritize the items on your list.

2) Study time management: Take yourself some minutes after coming home from school and evaluate the amount of time you can invest for your studies. This is where you start to manage your study time, so I would recommend you to take up to 15 minutes for this! Make sure to include time for studying, completing tasks and assignments, projects, etc.

3)One of the best things you can do to manage your time (then be managed by it) is to learn to say “no.” Say “no” to people asking you to play pool when you should study or when someone wants help with something at an inconvenient time for you. Suggest a different time to help when it’s more convenient for you.

Remember that there’s a time for everything. Sure, studying for a long time is good, but take breaks. Take power naps when your brain starts frying. If the weather’s nice, a good game of ultimate Frisbee also refreshes the tired mind.

4) Another every important study time management technique is to schedule your time dependent on your personal energy level. Your energy level reaches its peak at approximately 7am, but is very low during the time from 3pm to 5pm.A lot of students are very productive in the morning, so it could be very useful to rise up early . Others prefer the evening hours as their most productive part of the day. The only important thing is that you know for yourself on which time-period you can work most efficient and adapt your schedule accordingly to your high-energy periods.

5) Procrastination: One of the biggest obstacles you might face while applying effective study time management is procrastination, which is correlated to discipline. Every student is affected by procrastination once in a while or continuously. The desire to avoid specific tasks comes within boredom, fear or similar concerns. Not getting started will lead to stress and anxiety, which are very negative emotions during your studies. Luckily you can overcome procrastination by applying the following study time management technique: create smaller sub goals and sub tasks. You will notice that a sub task will help you to get started with your studies and will get you motivated to do the next tasks as well.

6) Get the most out of class. Go to class and be on time. Be well-prepared and ready for discussion.

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Electric and Magnetic Fields

• We will now discuss electric and magnetic field vectors (E and B)at a point in the absence of charge.
• Now let us place a charge q at point (x,y,z) in space. If this charge experiences a force as given by Lorentz force equation then we can associate vectors E and B with this point (x,y,z) in space.
• Thus at any time t vectors E(x,y,z,t) and B(x,y,z,t) gives forces experienced by any charge q at point (x,y,z) with a condition that placing this charge at point (x,y,z) in space does not disturb the position or motion of all other charges responsible for the generation of the field.
• So, every point in space is associated with vector E and B which are functions of x,y,z and t.
• Since E(or B) can be specified at every point in space , we call it a field.
• A field is that physicsl quantity which takes on different values at different points in space for example velocity field of a flowing liquid.
• Electromagnetic fields as we know are produced by complex formulas but the relationships between values of the fields at one point and the values of the feld at neighbour points are vary simple and can form differential equations which can completely describe the field.
• To understand and visualize the behaviour of field we can consider the field as a function of position and
• time. We can also create a mental picture of field by drawing the vectors at many points in space each of which gives strength and direction of field at that point.
• Flux is one property of field and flux of a vector field through a surface is defined as the average value of normal component of the vector times the area of the surface.
• Another property is the circulation of the vector field and for any vector field circulation around any imagined closed curve is defined as the average tangential component of the vector multiplied by the circumfrance of the loop.
• With just the idea of flux and circulation we can define all the laws of electricity and magnetism.

Refrence:- The Feynman lectures on physics Vol 2

Galilean Transformation

Motion is a relative concept . You are sitting on a train. A object on trains seems to be at rest from your perspective but it is in motion from the person standing on the ground.So it is tied to particular frame of refrence choosen by observer.

Now Different observer may use different frame of refrence , And velocity ,acceleration may be different from these different frame of refrence

It is important to know how these measurement in different frame of refrence are related.

For two observer S₁ and S₂ who move relative to each other with rectilinear motion, The velocity,displacement and acceleration of an object measured from these two observer are related as

r₂=r₁ -R

Where r₁ is the position vector of object from observer S₁ and r₂ is the position vector of object from observer S₂. R is the position of observer S₂ from S₁

v₂=v₁ -v

Where v₁ is the velocity of object relative to observer S₁ and v₂ is the velocity of object relative to observer S₂. v is the velocity S₂ relative to S₁

a₂=a₁ -a

Where a₁ is the acceleration of object relative to observer S₁ and a₂ is the acceleration of object relative to observer S₂. a is the acceleration S₂relative to S₁

When the two observer are moving in uniform relative translation motion with velocity v and t=0 when the observer were coincident,these equation becomes

r₂=r₁ -vt

v₂=v₁ -v

a₂=a₁

t₁=t₂

The above equation are called Galilean transformation of coordinates ,velocities and accelerations

So acceleration of the object remains same for all observer in uniform relative translational motion

Law of Radioactive decay

• Radioactivity is a nuclear phenomenon
• When a nucleus disintegrates by emitting a particle ( α and β) or by capturing an electron from the atomic shell( K-shell) ,the process is called radioactive decay. This decay process is spontaneous.
• Let us take a radioactive sample containing N₀ at time t=0 i.e, at the beginning. We wish to calculate the number N of these nuclei left after time t.
• The number of nuclei of a given radioactive sample disintegrating per sec is called the activity of that sample is
  dN/dt=rate of decrease of nuclei with time=Activity of sample at time t                              --(1)
• Experimentally it is found that the activity at any instant of time t is directly proportional to the number N of parent type nuclei present at that time
  
  
  Where λ > 0 is proportionality constant and negative sign indicates that N decreases as t increases
• From equation (2) we get
  
  i.e. ,λ is fractional change in N per sec
  => λ is not merely a proportionality constant ,but it gives us the probability of decay per unit interval of time
• Hence λ is called the probability constant or decay constant or disintegration constant
• dN is the no of parent nuclei that decay between t and t+dt and we have taken N as continuous variable 
• From (2)
  
  N₀=No of radioactive nuclei at t=0
• From (4) we see that law of radioactive decay is exponential in character
  
  
  
• From figure it can be noted that only half the amount of radon present initially after 3.83 days and 1/4 after 7.66 days and so on
• Plot shows that in a fixed time interval a fixed fraction of the amount of radioactive substance at the beginning of interval decays
• This faction is independent of the amount of radioactive substance and depends only on the interval of the time
• The decay constant λ is a characteristics of radioactive substance and it depends in no way on the amount of the substance present

What are black holes

According to special theory of relativity, the maximum velocity that can exist is the velocity of light
c=3X10⁸ m/s
Now we also Know that escape velocity which is the velocity needed for an object to become essentially free of the gravitational effect of another object is given
$v=\sqrt{\frac{2\mathrm{GM}}{R}}$
or
$v^2=\frac{2\mathrm{GM}}{R}$
We may speculate on the mass and radius of a steller body that has an escape velocity of c.
Then
$c^2=\frac{2\mathrm{GM}}{R}$
or
$R=\frac{2\mathrm{GM}}{c^2}$
This quantity R is called the Schwarzschild Radius and is usually designated by R_S
Substituting the values of G and c,we have
R_S=1.485 X10^-2 M
The above equation gives us the relation between M and R_S. It states that A body of Mass M in kg and radius R_S in m or smaller produces such a strong gravtitional at its surface that no particle on its surface can escape.This even applied to electromagnetic radiation ( photons) including light.
So Even light cannot escape from such planet or body. That is the reason such body bodies are termed as Black holes
The most common way for a black hole to form is probably in a supernova, an exploding star. When a star with about 25 times the mass of the Sun ends its life, it explodes. The outer part of the star screams outward at high speed, but the inner part of the star, its core, collapses down. If there is enough mass, the gravity of the collapsing core will compress it so much that it can become a black hole. When it’s all over, the black hole will have a few times the mass of the Sun. This is called a “stellar-mass black hole”, what many astronomers think of as a “regular” black hole.

Fictitous Force and Free falling Elevator

A non inertial Frame of Refrence is the frame of refrence which is having acceleration(a₀) Example Free Falling elevator

Newtons law are valid in inertial frame of refrence only. We cannot apply them as it is in non inertial frame of refrence as it left out the acceleration of the object because of the frame of refrence.

Let me provide simple example to illustrate this.

You are moving in car which is accelerating. From your frame of refrence,a block on the road would be accelerating in opposite direction.

Now if we apply newton's law, we will find acceleration to be zero as no net force is acting. But the block has acceleration from car frame of refrence.

So to fix this issue,we will add one fictitous quantity in the Newton law equation like

F+F₀=Ma

Where F₀ is the fictitous force and it is in opposite direction of accleration frame

F₀=-Ma₀

Where a₀ is the acceleration of Non inertial frame of refrence

Now let me further illustrate this with example of free falling elevator

A free falling elevator is an non inertial frame of refrence.The acceleration of the elevator as defined from the observer on the ground will be

g=-gk

Where k is the unit vector along vertical axis in the upward direction

Let us take an unattached object of mass M in the elevator.

From the frame of refrence of free falling elevator,Two forces are acting on the mass

Fictitous Force due to non inertail frame of refrence

F_fic=Mgk It is opposite to the acceleration of the elevator

Gravitional Force on the Mass

F_G=-Mgk

So the total Force on the body from elevator frame of refrence

F_T=F_fic+F_G=0

Which mean body is unacclerated in the elevator frame of refrence. The body will appear to be suspended in the air ,if it has no initial relative velocity to the elevator

Few more important things about Fictitous force

• Fictitous force are called pseudo force.
• It simple represent the effect of the acceleration of the non inertial frame of refrence
• They dont have any physical significance
• Whenever we work the problem in non inertial frame of refrence,always make sure to include fictitous force in the newton law

Different type of Force and their origin

A) Normal Force: When we put a book on a table ,the molecules of the book exert downward forces on the molecules of the table.The molecules composing the upper layer of the tabletop move downward until the repulsion of the molecules below balances the forces applied by the book.
This compression always occur but it is too slight to notice
The following things are good to notice about normal force
a) For a body resting on a surface,Normal force is equal and opposite to the resultant of all the other forces which acts on the body in the direction perpendicular to surface
b) Weight and Normal force are not action reaction pair as they act on same body whereas action -reaction acts on different body

B) Tension: Tension usually comes under picture for string and rope. The string can be think of small short sections interacting by contact forces
Each section pulls the section to either side of it and by Newton third law,it is pulled by the adjacent section. The magnitude of the forces acting between the section is called Tension
The Following things are good to notice about tension
a) If the tension is uniform,the net string force on each small section is zero and section remains at rest
b) if the string is accelerating ,the tension generally varies along the string
c) If there is external force on the string then also tension varies along the string

C) Friction Force: Friction force generally comes into picture when one body moves or tries to move along the surface of the second body
Friction always opposes the motion which would occur in its absence. Friction occurs becuase of interatomic forces at the actual area of contact on atomic scale.
a) When we try to push a box on the table, if we push the box gently ,the box remains at rest. Force of friction assumes a value equal to the force applied by us. The force of friction cannot increase indefinately. If we push the box hard,the box start to slide

D) Viscous Forces: A body moving through liquid or gas is retarted by the force of viscosity exerted on it by fluid. It arised because a body moving through the medium exerts forces which set the nearby fluid into motion.By newton third law,the fluid exerts a reaction force on the body
The Following things are good to notice about viscosity
a)Viscous forces are proportional to velocity
b) It always retard the motion

Polarization by reflection

• This simple method of obtaining plane polarized light by reflection was discovered by malus in 1808
  
• We found that when a beam of light is reflected from the surface of a transparent medium like glass or water, the reflected light is partially polarized and degree of the polarization varies with angle of incidence
  
• The percentage of polarized light is greatest in reflected beam when light beam is incident on the transparent medium with an incident angle equal to the angle of polarization
  
• For ordinary glass with refractive index =1.52 ,angle of polarization is 57.5⁰
  
• Figure below shows the polarization of light by reflection
  
  
   
• we can use a Polaroid as an analyzer to show that reflected light is plane polarized . we rather say that reflected light is partially plane polarized
  
• the examination of transmitted light shows the variation in intensity indicating that the light is partially polarized
  
• The vibrations of this plane polarized reflected light are found to be perpendicular to the plane of incidence and therefore ,the reflected light is said to be plane polarized in the plane of incidence

Kinematics Question

Question:

A ball of mass 100 g is projected vertically upward from the ground with avelocity of 49m/s. At the same time another identical ball is dropped from aheight of 98 m to fall freely along the same path as followed by the firstball. After some time ,the two balls collide and stick together and finallyfall together .

g=9.8m/s²

i is the unit vector along upward direction

1) Find the time at which balls collide in air

a) 2.1 sec

b) 1.2 sec

c)1 sec

d) 2 sec

2) Find the velocity vector of the combined mass just after collision

a) 4.8i

b) 3i

c)-4.9i

d) 4.9i

3) Find the time of the flight of masses

a) 3sec

b) 4 sec

c) 6.53 sec

d) 7 sec

Solution:

a) We will first find where and when the two balls collide. Let us assumethat the balls collide at time t after they have been set into motion. At thisinstant t when two balls collide they are at the same height h from the groundas shown below in the figure.

The height of the first ball after t seconds = 49t-0.5(9.8t²) =4.9t(20-t)

Height of second ball after t secinds = 98 - downwards distance moved by itin t seconds

=98-0.5t²=4.9(20-t²)

therefore, 4.9t(20-t)=4.9(20-t²)

or, 10t-t²=20-t² or t=2s

b) The ball thus collides 2s after the start of their motion. Theirvelocities at this instance are

ball 1 : v₁= (49-98×2)m/s = 29.4 m/s directed upwards=29.4 i

ball 2 : v₂=(0+9.8×2)m/s = 19.6 m/s directed downwards=-19.6 i

If v is the velocity of the combined mass of two balls after theystick togather due to their collision then from law of conservation of momentum

200×v=100×29.4-100×19.6

v=4.9m/s upward direction =4.9 i

c) The joint mass thus moves upwards , after collision with a velocity of4.9 m/s. Its height above the ground at this instant is (consider the positionof either of the balls)

(98-0.5×9.8×2²)m=78.4m

We can now find the time t' taken by this joined mass of the balls to reachthe ground. For this joined mass we have

u=4.9m/s , s=78.4m , a=-g = -9.8m/s²

-78.4=4.9t'+0.5(-9.8)t²

t'²-t'-16=0

Solving the equation for t' using formula for quadratic equations and leaving out the negative solution we get t'=4.532 s

The joint mass thus takes 4.53 s to fall to the ground. Since the balls collide 2s after they started their motion ,

the total time of flight is (2+4.53) s = 6.53 s

Question on Bolr atom model

Question :

The ionization energy of a hydrogen like Bohr atom is 4 rydbergs.

(a) What is the wavelength of radiation emitted when the electron jumps from the first excited state to the ground state.

(b) Whet is the radius of the first orbit of this atom ? Given that Bohr radius of hydrogen atom = 5×10^-11m and 1 rydberg = 2.2×10^-18J.

Solution :

In terms of rydberg constant R , the energy of electron in the n'th orbit of hydrogen like atom is

$E_n=-\frac{R{Z}^2}{n^2}$

where R=2.2×10^-18J. The ionization energy of the atom is

${\mathrm{\Delta E}=E}_{\infty }-E_1=-R{Z}^2(\frac{1}{\infty }-\frac{1}{1^2})=R{Z}^2$

Given that ΔE=4R. Therefore , 4R=RZ² or, Z=2

(a) The energy of the radiation emitted when the electron jumps from the first excited state (n=2) to the ground state (n=1) is

${E=E}_2-E_1=-R{Z}^2(\frac{1}{2^2}-\frac{1}{1^2})=\frac{3R{Z}^2}{4}=3R=3\times 2.2\times {10}^{-18}=6.6\times {10}^{-18}J$ (since Z=2)

Therefore wavelength of the radiation is given by

$\lambda =\frac{\mathrm{hc}}{E}=\frac{6.63\times {10}^{-34}\times 3\times {10}^8}{6.6\times {10}^{-18}}=301\AA$

(b) Radius of the first bohr orbit of the given atom is $=\frac{r_1}{Z}=2.5\times {10}^{-11}m$