Electric Potential

         I spend little time on thinking about Electric Potential while reading about Electrodynamics. It is really a great amazement the concept of Field and Potential is just created for the sake of Mathematical Simplicity. But it was realized later only that the field concept is a real Phenomenon.  

Electric Potential is defined as the work required per unit charge to move it in some linear path.

The definition and the sign of the potential functions are based only on our conventional sign systems. Eg. The potential of a positive charge is chosen to be a positive quantity with respect to the reference frame of point at infinity.

It is just our convention to use that the potential of positive charge is hill and the potential of a negative charge is a valley. But it can be chosen alternatively if we desire. For the sake of conventional mathematics we chose some sign conventions. These sign conventions also enter in Thermodynamics.

In mathematical form Potential difference between two points is defined as

Va – Vb = - ∫ E . dl

where E is Electric field vector

dl is linear integral path vector

V is called the Potential. The integral is from point “a” to point “b”.

The interesting relation between the potential function and the Electric field is that they are related in the following way

E = - ∂V/∂r

or                                            E  = - ΔV

From this equation we can simply deduce the following result that

if Electric field in any space is zero then it implies the potential at those points is const. And that is why equipotential surfaces are always perpendicular to the direction of the Electric field. If it is not then it will be contradictory to the above Equations.

But the great thing about the potential is, it is a scalar quantity. A scalar quantity in Nature simply turns out to be vector quantity in Nature by a simple mathematical formulation. Isn’t it just great?

It says that though the potential is scalar one, it always follows some rule from the Nature itself. And the rule is just expressed as the “gradient” operator “Δ” in mathematics.

This is the reason why Mathematics serves as the language for Physics.  

Faster Than Light? or Newton's Basic Laws are Wrong!

I was thinking about this problem in my first year of college. It is that we know.. No signal can travel faster than Light Speed and Newton’s Laws are valid in all reference frames i.e. When a force applied to a body, no matter how big is, from Newton’s first law the body will get an acceleration that is equal to a = F / m

Consider a body that is perfectly rigid i.e. neither incompressible nor stretchable. Assume that the body has a length of 5*10^8 meters which is more than one light year. It was also assumed that the experiment is done in vacuum so that there will not be any external effects. Now a force “F” is applied in “x” direction so that the body is displaced to a displacement of “dx” in x direction.

From Newton’s Laws of Motion,

Acceleration in x direction = Force / mass of the object (M)

                   a = F/M

The body is considered to be made of system of particles with infinitesimal mass “dm”. From the Centre of Mass Principle it is known that the path and the acceleration of the centre of mass is as same as the path of the point with mass “M” and acceleration “a”. Centre of Mass of an object that has uniformly distributed mass depends on the geometry of the object. We choose the object such that its centre of mass lies in that object.

From calculations we get,

Acceleration of Centre of Mass = F/M

So, it is sure that when it is applied an external force F to the object of mass M then the centre of mass of the object will get an acceleration of magnitude F/M.

From Electrostatics theory

When an external force is given to an object like rope, the motion of the rope is can be described as follows,

Consider that the rope has the molecules only arranged in linear order. Let’s say 5 molecules.

When the external force is applied, the first molecule will move in the direction of the force.

Let us take the case of pulling a rope and the Force Body Diagram of the first molecule is

Only if   “ Fext ” is greater than the net electrostatic attraction or repulsive force on the first molecule due to all other molecules namely -

“ Felectro-1 ”  , the first molecule will get an acceleration in the direction of “ Fext ” and will move distance "dx".

When the first molecule is moved through the distance “dx” in x direction, the second molecule will also move in the “x” direction to maintain the stable configuration. 

So the force on second molecule will increase to“ Fext ”  and it will be greater than the net attraction or repulsive force one second molecule due to all other molecules namely“ Felectro-2 ” . So, the second molecule will also move by distance “dx”.

Note: The body is considered to be perfectly rigid so that each molecule will be displaced by same distance “dx”. And so the external force will be transmitted to each particle.

In this way, throughout there will be electrostatic interaction between the molecules. And the interaction will make the last molecule that is at the other tip of the rope to move by the same displacement “dx”.

Thus we can explain the movement with Electrostatics Interaction.   

Here comes the question:

When the same situation is dealt with the object of such large length like 5*10^8 meters, it comes the problem.

The problem is,

From Special Theory of Relativity, “No kind of Electrical interaction can travel faster than Light Speed”.

But from Newton’s Laws of Motion, “The object that is the centre of mass of the object should instantaneously get acceleration

a = F/M and it is independent of time. The processes are instantaneous. ”. 

Which will happen in real life?

The question is whether the body will immediately move or not.

If it behaves by following the Special theory of relativity,

Then the other end of the object will move after some time interval of applied force. But it will be completely against the Newton’s laws of Motion. And it looks like a paradox. 

Signals faster than Light Speed

Light Speed cannot be achieved by any means of Physics. No signal can be transmitted faster than light Speed. I used to hear these statements often when studying relativity theory.

Keeping this in mind I started to think about signals. Why signals can’t be transmitted faster than light speed.  

I suddenly thought about an experiment with a rod.

Let us say I have a rod that is 5, 00,000 km - five lakhs kilometers of length. Assume that the stick is incompressible and rigid.

Now I am going to pull one of its ends to small displacement “dx” or small rotation “dθ”

Everyone knows from our experience that the other end of the rod spontaneously without any time gap would have been moved by the same displacement “dx” or rotation “dθ”.

I don’t think signals cannot be transmitted with this rod.

I will use the following way to transmit the signal to other end of the rod that is five lakh kilometers away from this end. It was already announced to my friend that if the rod moves below it indicates the number zero and if the rod moves above then it indicates the number one. In this way my friend on the other side can easily note down the numbers using a machine and can easily understand the messages as in the same way how a computer does.

You can also imagine two computers instead of two humans. The computers can interact in the following way though they are separated with the distance of five lakhs kilometers.

I am not arguing with Special theory of Relativity. I know the velocity defined in Special theory of relativity defines different signal velocity. Here I am not talking about signal velocity but I am just trying to get a way how we can transmit a signal for such a long distances. And I just want to simply show that it is not impossible for us to transmit signals to such large distances.

Using this method we can transmit signals between the stars and the galaxies by slowly spreading throughout the entire universe.

Here it was already assumed that my friend is there on the other side. It is easy to make him go on the other side by simply make him to sit on the other point of the rod. After making him to sit on the other side we can start to build the rod and move it in the space slowly.

This is just an idea but not the absolute way. I am just trying to prove the possibility of transmitting these signals. Using perfect Engineering we can find a more suitable method. But I don’t want go into that Engineering.    

Do you think there is anything wrong with this method?

Motion in a Resistive medium

In mechanics we used to study the motion of the objects without considering the resistive force they are present in out atmosphere. Any measurement we make in real life will not be accurate if we don’t consider the resistive forces in the Motion of an object.

          Hence studying about the motion with resistive forces gives us a clear understanding of what will really happen in nature. We will encounter some mathematics to get the understanding.

         

          It was experimentally understood that the resistive forces are proportional to the quadratic relation with velocity.

First to get a simple understanding we will make it a linear relation with velocity in atmosphere.

Note: All these Forces, velocities and acceleration are vector quantities. Dealing with vector quantities is quite complicated than the scalars. So we always used to split any motion into its components where we can deal it as the scalar quantities.

So we split a three dimensional motion into its fundamental components into two horizontal components and a vertical component. That is in a Right Hand Cartesian Coordinate system, we consider the x and y component as horizontal components since it is coplanar with the ground. And we took the z component as the vertical one as it is perpendicular to the plane of the ground. The ground is considered to be flat because we are taking only a small portion of earth that is a good approximation.  

Thus we are going to analyze the motion in three components but in two ways where the motion in x and y direction is similar.

Part – 1

Motion of the object is coplanar with the plane of the ground. So, the velocity will be considered as either the x or y component of velocity. Both components will give similar results.

When there is no other force in the horizontal direction using Newton’s Second Law,

m dv(x)/dt  (or) m dv(y)/dt = -m β v(x) or –m β v(y)

or simply,   we can write v(x) = v(y) = vx   

Hence it gives us.

                   m dvx  /dt = - m β vx   ,

β is constant that depends on the shape and size of the object and on the resistive medium. The negative sign indicates that the resistive force is in the opposite direction of the motion of the particle.

So,              dvx /dt = - β vx                        →               dvx / vx = - β dt

Integrating dvx  from “Vox” to “Vtx” and time from “0” to “t”

                   ∫ dvx / vx = -β  ∫ dt        

→               Vtx = Vox e^-βt   …..                eq.(1)

         

From this result we can know that,

·       If there is no resistive force that is β = 0 then   Vtx = Vox . The velocity will not change in that direction until there is an extra force other than the resistive force.

·       The velocity in the planar direction to the ground decreases exponentially with time.

  

Part – 2

Now we will talk about the vertical motion where we will some non-intuitive concepts.

Motion of the object is perpendicular to the plane of the ground. So, the velocity will be considered as the z component of velocity that is perpendicular to the ground. 

In the same way as we done before, Using Newton’s second law we can write the equation of motion in perpendicular direction to the plane of ground as follows,

but now we have to include gravitational force. Taking velocity perpendicular to plane as vz

                             m dvz / dt = - mg - β m vz

depending on the direction of the velocity, the resistive force will change its direction.

                            

                             dvz / dt = - g - β vz

                       

→               dvz /( g + β vz ) = - dt

Integrating vz  from “Voz” to “Vtz” and time from “0” to “t” gives

→               Vtz = Voz e^-βt + g/β (e^-βt ­– 1)  …………….                        eq.(2)

Thus we have our second equation.

From the equation (2) we can understand,

·       Velocity in perpendicular direction tends to a limiting value when time goes to infinite value. The limiting value is equal to  “-g/β”. 

·       And this limiting value is called the “The terminal velocity”.

·       From the equation it is known that if the initial velocity is lower than the terminal velocity, the object will tend to increase its speed until it reaches terminal velocity.

·       But if the initial velocity is higher than the terminal velocity the object will start to slow down until it attends the terminal velocity.

·       The terminal velocity of a skydiver in free fall is nearly equal to

200 kilometers per hour.

But to get more accurate results, the resistive force should be treated as quadratic relations with velocity. The mathematics to solve differential equation with quadratic relations is too much complicated than the linear relations. However we will give an attempt to that. 

Wonder of Surface Tension

When I was asked to talk about the uses of Surface tension, my mind collects the usual applications that we are seeing in our everyday life. That is by reducing the surface tension of water using soaps, we can easily wash our dresses. And I thought about capillary rise that is due to surface tension. But after some time, I thought about it so deep and I got a beautiful point about surface tension.

First of all Surface tension is the force on the surface of the molecules that tends to minimize the surface. It is the force per unit length on the surface.

I have seen many of them will treat adhesive, cohesive forces completely different from Surface tension. But it is not. We should not confuse with all these forces like Vander Waals force and London force. All these are simply the basic intermolecular forces.

In that way Surface tension is simply the cohesive force between the liquid molecules. It is a type of intermolecular forces.

To understand the use any of thing, first we should eliminate from the Nature and give a little thought to the Nature without that force.

I ended with this beautiful conclusion though it is a fundamental one.

Don’t think that the conclusion is capillary rise, because without capillary rise the plants can’t make food. Without capillary rise it will never make starch by photosynthesis. Though pressure difference in the water tubes of branches in the plants is a reason for the water flow, capillary rise plays the major role. But I am not going to talk about Capillary rise. I want to deeply analyze the fundamental importance of surface tension.

Do you think without Surface tension you can see water?

In more simple way, without Surface tension, do you think there will be a meaning for water?

The water will not be there if there is no surface tension. The water will not have any surface at all. It will become similar to gas.

Most simple example is sprayer. Using sprayers, the water drops are sprayed. It is the same case that will happen to water if there is no surface tension. The water itself will start to spray by itself. The water will not get a shape.

       

Without Surface tension the liquids will get transformed into gases.

       

You can ask, “okay.. What will happen if there is no surface tension that is if it is sprayed”?

The answer is “Humans will not be happened if there is no surface tension”.

Think how you will look, if your blood starts to lose it surface tension. The shape of the body will start to spread. Moreover you will become dead without surface tension.

The Plants, Animals, Humans and no living thing will be formed without surface tension.

Thank God for Surface tension.

So, I just started to wonder about Surface tension when I realized its fundamental importance in the creation of living things. 

Could you Count One Nano second ?

Yesterday I read my old ideas and notes. In the middle of those notes I find this note, that is,

When I was playing with my friend in a badminton game, one of another friend asked me to give him a chance to play. But I am not in a mood to give him a chance since I was playing just for 5 minutes.    

So, I asked him some extra time. I said him to count up to 100 and I assured that I will finish the game within that time. But I couldn’t finish it.

I know he would become angry if I ask him to count again 100 or asked any extra time. Suddenly I got this idea and turned to him saying,

“Ok.. I will give you your chance if and only if you count one Nano second in a perfect and exact way. It is just a small time so you can try it”.

He quickly replied that,

 “One nanosecond finished‼ Give me the Badminton Rocket‼”

I replied him,

“Aha‼ This is not one Nano second! This is way more than one second. You should perfectly count by your mouth exactly the timing of one Nano second.”

Now with an angry voice my friend,

“Hey One Nano second is already finished. Give me the rocket”

I made him mad by saying again,

 “But you never count it that time. You never count one Nano second‼ Only if you count one Nano second, then it is fair to give the chance. It is just a small time man! Please count it!”

Now he started to laugh and said,

“Hey! You are cheating‼ This is not far!”

In the meantime, I finished my game and gave his chance. But I didn’t forget that point. Isn’t it something rather a very good way to get a lot time?

But it is my best advice and I used it many times.   

If somebody like your boss or teacher or your parents asked to finish a difficult job within small amount of time, then the best way to get an excuse is..

Make them to count just one Nano Second.

It is a very good way and also a Scientific Way of confusing others.

Note:1

If they used an atomic clock and make it to stop at one Nano second, do you think they finished their task?

I think “No”. Though they made the clock to stop at one Nano second before they show it to you the time will be crossed more than two or more seconds. If he showed you the watch saying “See! One Nano second.”

But you can say, “Sir, but the time already crossed more than one Nano second. I want you show the time exactly at One Nano Second time interval from the start of the count” ‼‼

Note:2

If you make them to count infinity then also you will get a lot time. But you will also get fired from your job because it is a very big time. But One Nano second is a very small time. So that you can make your boss mad by saying,

“Oh! Boss.. Couldn’t you even count one Nano Second? What a terrible Calamity?”    

Note:3

But I thought this for a long time because there are so many physics and philosophical concepts behind this. 

Think about this and if you like share with me about your opinions.

Time travel to the Past

I used to watch science programs and sci-fi films related with time travel. In those programs they used to describe the time travel as people going into the past or the future using a portal.

Going to the future is possible with Einstein’s theory of Relativity if you are in a rocket that travels near to the speed of light.

But what about travelling back in time? It looks impossible using the basic concepts of Physics.

It is used to say that we can travel through time by Wormholes.

A Wormhole, also known as an Einstein–Rosen bridge, is a hypothetical topological feature of space-time that thought to be a "shortcut" through space-time. It is much like a tunnel with two ends each in separate points in space-time. We don’t know how we are going to travel in the past. But let us assume that we found a way to travel back in time. With a portal like worm hole, if one person can travel through time, then following questions arises in everybody’s mind.

Will there be the past of the man who is travelling?

I watched the Stephen Hawking’s program where he used to say that,

What if I killed myself by travelling back in time?  

But I am not going to ask the same question, but I want to ask questions rather something more related with the basic laws of physics i.e. The Conservation of Energy.

The Conservation of Energy is the fundamental tool we are using all over the Physics. Neither Classical Mechanics nor Quantum Mechanics can work without this Conservation of Energy. Energy can neither be created nor can be destroyed.

If it is assumed that with a time travel portal, if a person could travel through time, then it implies that he is suddenly vanishing from the Universe that we are living.

The human body is simply nothing but the Energy from Einstein’s E = mc^2 equation. That is we are just a form of energy.

If the person vanished through time then it means that Energy is destroyed in that frame of reference.  And if he come suddenly out of the portal by travelling through the past then,

In the reference frame of the Past time that is if somebody is doing experiments in the past and if he just happened to see the person who had been sent from the future, then it implies that in his reference frame Energy is created from nothing. In both of these reference frames Conservation Laws are violated.

May be you can argue that, if I see the Space and time as a unique one that is a reference frame outside the Space and time, then

The Conservation laws are not violated since the Energy disappeared in the Future time is immediately appears in the Past time. So energy is linked and so you will conclude that, “Neither the energy is destroyed nor created”.

But I strongly disagree this argument since from Einstein’s theory of Relativity; Newton’s laws are valid in all reference frames independent of Space and time.

So, Newton’s laws of Motion or Conservation laws should not be violated in both of the reference frames.

Any Energy or any force can’t be created out of nothing. Time travel in backwards that is to the past would violate all of the Conservation Laws.