Physics

Understanding Projectile Motion : Derivations

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Projectile Motion is one of the basic thing we encounter in our daily life. Studying it becomes important because the ‘understanding’ (not mugging up Formulae) which we take here comes handy while analyzing complex concepts and experiments 

“Formulae give you Marks but Derivations give you Understanding !

We already do have a video on our Channel about such a experiment on our channel where knowledge of Projectile is needed for the Analysis part :

Equation of Trajectory

We all have played ‘catch-catch’ and we know how the motion of a ball when thrown looks like. But how do you describe that curve mathematically. What’s the equation of that curve ?

Most Important Tip that fixes everything related to Projectile !

Divide this whole 2-D situation into separate 1-D problems (X and Y)

  • Note down all quantities in X separately. This will be your separate problem
  • Note down all quantities in Y separately. This will be your another separate problem
  • At last, combine them to get results for 2-D motion

Following the Tip :

Starting from origin (point where ball is throwed) till point P (x,y) :

The Projectile Trajectory is a Parabola !

Finding Range Expression

Range is basically the ‘horizontal’ distance which the ball covers (from origin to the point where it lands). It means that, we need to find R in the figure

  • This can be easily found since x = 0 and x = R are the two roots of the parabola.
  • To calculate the roots, simply put y = 0 in our Equation of Trajectory

Calculations :

Substituting y = 0 and taking ‘x’ common on RHS of Equation of Trajectory, we get : 

* In the simplifying process, you need to use sin2theta trigonometric identity

Finding Time of Flight Expression

Let’s find out for how much time does the body stay in the projectile motion.

Note that :

  • On landing, after completing the motion, the displacement in Y direction is zero (Pause and observe !), since it again came to the same Y-level (y = 0 here)
  • The time taken for this displacement in Y to become 0 is nothing but the Time of Flight (T)

Finding Maximum Height Covered :

If we look at the Equation of Continuity, in this case (simple throwing); the trajectory resembles downward parabola since a<0

Hence visualizing it graphically, 

Finishing the Calculations,

So, this is how, we get the expression for the Maximum Height reached, denoted by ‘H’

Conclusion

So, this is how we derive the expressions for Time of Flight, Range, Maximum Height and Equation of Trajectory. Note that – With this same approach, we can solve almost any kind of problem related to Projectile Motion. And, Why is that the case?

Because, whatever we discussed in this blog/article is not a trick or something, it’s a complete concept with complete understanding. This makes us equipped to solve any kind of problem (Throwing from a cliff, projectile on incline or just anything…)

Stress or Strain ? – Which comes first ?

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‘Stress‘ and ‘Strain‘ are the most encountered terms when it comes to studying Elasticity. Though, it might not have such a huge weightage in competitive exams like JEE/NEET, but trust me ladies and gentlemen : “There’s no Mechanical Engineering without these 2 terms” – Being a student pursuing my UG degree in Mechanical, I can confirm this. And if there’s no Mechanical, there are no cars, bridges, buildings, etc.

Through this article, We are going to find out which one of the two comes first – is it Stress or is it Strain ?

Table of Content :

  • Introduction to Stress
  • Introduction to Strain
  • Final Decision

1. Introduction to Stress :

Stress is defined as the Internal Restoring Force acting per unit area.

Now, What is this Internal Restoring Force ? Let’s understand the process to know what happens inside the material

The atoms inside the solid are arranged in a spring-ball system. So, when a load (external force) is applied, it disturbs the equilibrium state by making the springs deformed. This deformation is responsible for the Internal restoring force and we call it as Restoring, because it tends to bring the system back to its equilibrium.

Spring-ball arrangment in Solids

The following flow-chart explains the process :

Fig. Flowchart

2. Introduction to Strain :

Strain is defined as ‘Change in Dimensions / Original Dimensions’

Again, there are types of strain :

  • Longitudinal Strain – Change happens in the length
  • Shear Strain – There is a shift which leads to an angle change
  • Volumetric Strain – Change happens in the Volume
Fig. Longitudinal Strain (expansion)
Fig. Shear Strain (measured as angle)
Fig. Volumetric Strain (compression)

Important Note :
A careful observation of the Flowchart above would tell that : There is ‘Strain’ coming into the picture at the second step since on applying load, there is a deformation happening. This is exactly what we discuss in Strain

3. Final Decision

Now, It’s very much clear from the above discussion that : it’s the Strain which comes first ! All of this because of the Definition of Stress. Most of the times, we just memorize the formula of Stress as ‘Force/Area’ which is not complete.

  • Complete Answer is : Stress = Restoring Force/Area


Friction : Clearing the Misconception

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Misconception : Friction always opposes motion

Corrected Version : Friction always opposes ‘relative‘ motion. Relative is the word which most of the people miss and this creates the whole confusion. 

But, how can we explain this concept in a much clearer manner ? – We are going to do this in today’s short but important article

Explanation :

Analogy for better understanding :

You can imagine this scenario as a teacher controlling a small group of students on a picnic. The teacher strictly instructs that “No student should try going forward and no one should be left behind. Stay Together”

Teacher ordering students to stay together
  • In physics terms, what she means is : There should be no relative motion between any student i.e. all should move as a unit

Having an idea of this, we are in a postition to answer the following question below :

correction : ii) v1 < v2

Case 1 : if  (v1 > v2)

Then the direction of friction on the block will be forward while that on the surface will be backward. This is because :

  • As the surface moves faster, the block will say to surface, “Hey, be with me…you are too fast”. Hence it tries to oppose surface 
  • While, as the block moves slower relatively, the surface will say to block, “Hey, be with me…you are too slow…I will support you”. Hence, friction gets applied in forward direction for the block

Case 2 : if  (v1 < v2)

Then the direction of friction on the block will be backward while that on the surface will be forward.

Case 3 : if  (v1 = v2)

There is no friction between the block and the surface as there is no relative motion between the two

Perpendicular Axis Theorem + Mixed Problem

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In the last article, we got to know a very useful theorem called ‘Parallel Axis Theorem’. But as well know, it can be applied only when the 2 axes under consideration are parallel to each other. 

But what if we want to know MOI about an axis which is not the plane ? Perpendicular Axis Theorem comes to our rescue*

Topics Covered :

  • Perpendicular Axis Theorem
  • Example
  • Miscellaneous Activity (Parallel + Perpendicular)

1. Perpendicular Axis Theorem

Conditions :

  •  Applicable to only planar 2-D bodies
  • 3 axes to be considered
  • 2 axes in plane of the body and 3rd should be perpendicular to both
  • All 3 axes needs to be concurrent

Descriptive Statement :

‘The moment pf inertia of the planar body about an axis perpendicular to the plane is equal to the sum of moment of inertias of two perpendicular axis concurrent with perpendicular axis and lying in the plane of the body’

(Observe that all the points have been covered in ‘Conditions’ section above)

Mathematical Expression :

2. Example

Question : Find the moment of inertia Ip passing through center of mass C of the square plate having mass M and side length L.

Solution :

Would there be any difference in the answer you get in Figure 1 and Figure 2 ? (Remember, it’s a square plate)

Figure 1
Figure 2

The answer is : NO ! This is because of the beautiful symmetry that this square plate holds about axis shown in both the cases. There is the same mass distribution about the axis in both the case. That’s the reason, you can’t really make out the difference !

Step-1 : Remember, we need 3 axes : 2 in plane and 1 perpendicular to them and concurrent. In this we already got 2 planar axis (Combine figure 1 and figure 2). These will be our Ix and Iy. Hence, Ix = Iy = Ip. We get

Step-2 : Now, we also know the standard MOI for square plate about an axis passing through C and perpendicular to plane of square plate. i.e. ML^2/6. This is our Iz

Step-3 : Applying perpendicular axis theorem

Solve it yourself !

Now that we have learnt about Parallel and Perpendicular Axis theorem, we are in a good state to apply this to a problem which requires both these theorems. (This itself is a good hint)

Question : Find the moment of inertia Ip for the uniform disc of mass M and radius R


Parallel Axis Theorem (in detail)

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To calculate moment of inertia about an unknown axis, we often take help of 2 Theorems namely :

  • Parallel Axis Theorem
  • Perpendicular Axis Theorem

There’s one thing common in both : which is you need to know atleast one moment of inertia about an axis. This will act as a reference for you while calculating the unknown MOI

Topics Covered :

  • Parallel Axis Theorem
  • Important Observation
  • Example

1. Parallel Axis Theorem

Figure 1

1.1 Conditions to apply

  • Applicable on all types of bodies
  • Axis through COM || Axis about which MOI is to be found out

This is an Alert !
There are infinite axes passing through center of mass C. Don’t just choose any axis passing through C. Choose only that axis passing through C which is parallel to required axis

1.2 Theorem

The mathematical equation for this theorem can be given as :

Where,

Ip is the MOI about the required axis

Icom is the MOI about the axis passing through COM

h is the distance between the parallel axes

M is the mass of the body

1.3 Important Observation

Hence, we can say that among all the parallel axes in the plane (shown in Figure 1), the moment of inertia of the body about the axis passing through COM is the least. We also know the expression for torque :

Therefore, we can say that, for rotations in a given plane, choosing an axis through the center of mass gives the greatest angular acceleration for a given torque

DOWNLOAD

Proof for the statement (1)

2. Example

Question : Find the moment of inertia of the rod about the axis passing through P. The rod has mass M and length L

Solution :

Step-1 : Choose an axis parallel to the required one and it must be passing through COM of the body

Step-2 : Apply Parallel axis theorem

  • And we are already aware of standard MOI about axis passing through COM for rod
  • h = L/2


How to write Coefficient of Restitution expression ? – Part 1

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(Use Desktop/Laptop for better Experience)

Topic Under Chapter – Center of Mass and Collisions

Coefficient of Restitution also called in short as COR is one of the important concepts to be taken into consideration when it comes 

in designing sports equipments like badminton racquet, tennis racquet, several types of balls like baseball, basketball, cricket

ball ,etc. 

  • This is mainly because of the fact that these sports like basketball, badminton, tennis, etc involve collisions which makes terms like collision energy and rebound energy come into picture. To relate how good the bounce will happen, we have a ratio known as ‘Coefficient of Restitution’. As simple as that !

1. Rules to Remember

There are 2 cases :

  • Before collision (Deals with the Approach of the bodies)
  • After Collision (Deals with the Separation of the bodies)


Remember This !
For Before Collision (Vapp) : The velocity component which supports the approach i.e. that component feels as if bodies should approach each other, is positive

Remember This !
For After Collision (Vsep) : The velocity component which supports the separation i.e. that component feels as if bodies should get separated from each other, is positive

2. Examples (Different Cases for COR) :

We have 2 bodies (body 1 and body 2) undergoing collision. We will be trying to write the Coefficient of Restitution for each case.

Case 1 :

Writing Velocity of approach (Vapp) first :

  • Both u1 and u2 want the approach of the bodies to happen. So both will be positive. Hence Vapp is ‘u1+u2’

Writing Velocity of separation (Vsep) :

  • v1 and v2, both want separation to happen. So, +v1 and +v2. Hence Vsep will be ‘v1 + v2’
Case 2 :

Writing Velocity of approach (Vapp) first :

  • u1 wants approach but u2 doesn’t want that. So, +u1 but -u2. Hence Vapp is ‘u1-u2’

Writing Velocity of separation (Vsep) first :

  • v1 wants to separate but v2 doesn’t want that. So, +v1 but -v2. Hence Vsep is ‘v1-v2’
Case 3 :

Writing Velocity of approach (Vapp) first :

  • Both u1 and u2 want the approach of the bodies to happen. So both will be positive. Hence Vapp is ‘u1+u2’

Writing Velocity of separation (Vsep) first :

  • v2 wants to separate but v1 doesn’t want that. So, -v1 but +v2. Hence Vsep is ‘-v1+v2’
Case 4 :

Writing Velocity of approach (Vapp) first :

  • u1 wants approach but u2 doesn’t want that. So, +u1 but -u2. Hence Vapp is ‘u1-u2’

Writing Velocity of separation (Vsep) first :

  • v1 and v2, both want separation to happen. So, +v1 and +v2. Hence Vsep will be ‘v1 + v2’

FAQ section :



What is Coefficient of Restitution ?

Coefficient of Restitution or COR is the Ratio of the Rebound speed (velocity of separation) to Collision Speed (velocity of approach). It helps in determining the intensity/type of collision that can take place



How is elastic and inelastic collision determined by COR ?

If value of Coefficient of Restitution (COR) is 1, it implies that the collision is elastic in nature i.e. no kinetic energy will be lost during collision. If the COR is <1, it implies partially elastic while if COR is 0, then collision is called ‘perfectly inelastic’

What is Total Internal Reflection ? – Geometrical Optics

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Introduction :

Total Internal Reflection also known as TIR is one of the useful phenomenon which has applications in lot of areas. One such example is of Optical Fiber Cables. In this Article, we will be discussing about TIR : What is it ? Under what conditions does it happen ? What is the difference between ‘normal reflection’ and ‘total internal reflection’ ? 

Topics Covered :

  • What is meant by reflection ?
  • How does TIR happen ?
  • Difference between normal reflection and TIR

1. What is meant by reflection ?

In simple words, Reflection is nothing but bouncing back of light into the ‘same medium’ once it gets hit onto a polished hard surface (usually mirror).

2. How does Total Internal Reflection happen ?

When a light travels from one medium to another, it bends. But does it bend toward the normal or away from the normal depends on the fact that from which medium to which medium it is going.

  • Rarer to Denser – bends towards the normal
  • Denser to Rarer – bends away from the normal


Now, this specific case of Total Internal Reflection (TIR) happens when the light travels from denser medium to rarer medium.

From the figure, as we keep increasing the angle of incidence, the angle of refraction also increases until the critical angle is reached.

Critical Angle :
For each medium interface, we have a special angle defined, known as critical angle. In simple words, it is the angle of incidence at which the refracted ray grazes along the medium-separating interface (the angle of refraction becomes 90 deg).


TIR Condition :
When the angle of incidence goes beyond this critical angle, the ray gets ‘reflected back’ into the ‘same medium’. This phenomena is called ‘Total Internal Reflection’

3. Difference between Normal Reflection and TIR :

Normal Reflection :
  • The intensity of the incident ray is much greater than the intensity of reflected ray. This is because during normal reflection, a part of the light gets absorbed by the material of which it hits and some of it gets transmitted further. 

Total Internal Reflection :
  • In this case of the incident ray intensity is retained 100 % by the reflected ray. This is the major differnce between Normal Reflection and Total Internal reflection

A small Question (JEE Advance PYQ) :

Question : 

A light ray travelling in glass medium is incident on glass-air interface at an angle of incidence. The reflected (R) and transmitted (T) intensities, both as function of theta, are plotted. The correct sketch is : 

Answer : (C) option

  • At angle of incidence = 0 deg : Most of the light (not 100%) is transmitted.
  • At angle of incidence > critical angle : 100 % of light is reflected and hence 0% transmission of light

FAQ section :



What is Total Internal Reflection (TIR) ?

TIR is the phenomenon in which the incident light travelling from denser medium to rarer medium gets reflected back into the denser medium retaining its full intensity (100%).



What is a grazing ray ?

We call an refracted ray as a grazing ray when it passes along the interface separating two mediums



What is meant by critical angle in TIR ?

For each medium interface, we have a special angle defined, known as critical angle. In simple words, it is the angle of incidence at which the refracted ray grazes along the medium-separating interface (the angle of refraction becomes 90 deg).



Who can read this article ?

Students from class 10, class 11, class 12 preparing for boards or competitive exams like JEE, NEET, etc. or someone just having interest in Physics can refer to this article



Under which chapter, does this topic come ?

This topic of Total Internal Reflection or TIR comes under the ‘Ray Optics’ Chapter, also known as ‘Geometrical Optics’ 



What are the main concepts needed to study this ?

Reflection and Refraction (applying Snell’s Law) are the two important concepts needed

Charge Induction in Metals and Non-metals

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Topic under the Chapter : Electrostatics

1. Short Introduction

To understand this topic, we need to know about ‘free electrons’. These free electrons behave very similar to the gas atoms in a container. Both of them are under continuous random motion throughout the space given to them.

Application – Electroscope

Free electrons are nothing but some loosely bonded valence electrons which come out of the atom very easily ; just as the loosely stitched button comes out of the shirt very easily : )

Note that :

  • A neutral body has equal number of positive charges and negative charges. Presence of free electrons doesn’t disturb the neutrality of the body as free electrons are also negative charges but the only difference being that ; they are free to move inside the body

2. Induction in Metal and Non-metals

When we talk of bodies, we classify them as –

  • Metals – The ones in which there are a lot of free electrons
  • Non-metals – electrons are bounded to the Atom tightly (Atom loves them !)

2.1 Charge Induction in Metals

Now suppose you have a metal conductor placed in a region. And you bring a positive charge ‘+q’  in that region ‘externally’.

Important Note : 

  • excess of electrons implies negative charge
  • deficiency of electrons implies positive charge


These charges are ‘induced’ on the conductor due to the external charge. This phenomenon of separation of charges in a body by some external factor is called ‘Charge Induction’

2.2 Charge Induction in Non-Metals

As discussed, the basic difference between non-metal and metals is the absence of free electrons in case of non-metals.

Setup – Let’s consider the same condition. An non-metallic body has been placed in a region. Now, we bring a positive charge in vicinity of this body.

Important Note :

  • Atom is made up of a positively charged nucleus and an negatively charged electron cloud surrounding it. In a neutral, undisturbed atom, the negative center and the positive center, both, coincide.
(Cloud represents negatively charged electron cloud)


This alignment in dipoles in a non-conducting body due to an external charge is called induction in non-conducting bodies or ‘Polarization’. The separation between positive and negative charges is very very small. So, usually, we ignore it in our problem solving, etc.

Special Note :
This is why a Charged body always attracts a Neutral body. We can also conclude that :
“Attractive force on neutral conducting body will be more due to any external charge”

FAQ section :

What is charge induction ?

In presence of an external electric field set up by an charge placed outside a body; charge separation happens within the body. This phenomenon is called charge induction.

What is Polarization in Electrostatics ?

The formation of dipoles within the atoms of non-metal due to the external electric field set up by the charge placed outside the body is called Polarization

Conversion of Galvanometer to Ammeter & Voltmeter

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Topics Covered :

  • What is Galvanometer ? & it’s Types
  • Conversion to Ammeter
  • Conversion to Voltmeter
  • Some Examples

1. What is Galvanometer ?

Galvanometer is a deflection type meter which is used to measure the current value. The needle present in the Galvanometer gets deflected when a current passes through it and the amount of deflection produced is proportional to the current passing through the device.

There are two types of Galvanometers : 

  • Uni-Directional
  • Bi-Directional


  • Uni-Directional : In this case, the markings on the dial start from 0 till the maximum range. It has a red terminal which indicates that it has to be given high potential connection and the other one is black terminal for lower potential connection.
  • Bi-Directional : The zero of the dial is in the center and we have the maximum on either side. So, from the direction of deflection, we get to know the direction of current in the conductor and the amount of deflection gives us the magnitude. 

So, the basic difference between the two is that, Unidirectional can give information only about magnitude while Bi-directional can tell direction as well as magnitude.

The inner setup of Galvanometer has something known as Coil Resistance ‘G‘ and at maximum deflection, the safe current which flows through the galvanometer is ‘ig‘ . The symbol for Galvanometer is :

2. Conversion to Ammeter

The purpose of an Ammeter is also to measure current but the range for current measurement is much higher. 

  • How to Convert ? – Just add a resistor with very small resistance (Shunt ‘S’) in parallel to Galvanometer

What happens because of this ?
  • Now suppose, ‘I‘ (I > ig) is the current flowing in conductor. Since S and G are connected in parallel, ‘I‘ will be divided in ‘ig‘ and ‘I-ig‘. 
  • The shunt resistance S, being very small in magnitude will attract a lot of current (since current always prefers least resistance path). The shunt resistance S, is the reason why we are able to supply a larger current than ig.
  • This helps us to measure a larger current, resulting in increase in the range of the galvanometer
How to calculate this ‘I‘ ?

3. Conversion to Voltmeter

The Voltmeter is used specifically to measure the potential difference across the given terminals.

  • How to Convert ? – Add a very high ‘Load’ resistance R in series to Galvanometer

What happens because of this ?

4. Examples

(Question from Arihant – Electricity and Magnetism)

Solution :


(Question from Arihant – Electricity and Magnetism)

Solution :

Conclusion :

So, we have learnt about the Galvanometer and how can we use it as an Ammeter and a Voltmeter.

  • This topic is important from not only practical point of view but also theory exam point of view
  • And other than marks, it’s always good to know about our electrical instruments !

All the Best !

Physics Experiment – Calculating e/m Ratio

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https://www.youtube.com/watch?v=rxL8DkdYCTw

*Note : (Prefer Desktop for better Experience)

Calculation of charge-to-mass ratio is of great importance when it comes to the subject of Modern Physics. Understanding the procedure behind this experiment is equivalent to revising the following topics as well :

  • Moving charge in Magnetic Field
  • Behaviour of Charge in Electric Field
  • Projectile Motion
Sections :
  • Importance of calculating e/m ratio
  • Setup of the experiment
  • Procedure (*Imp) 
  • Final Data & Conclusion

1. Importance of e/m Ratio

Ok, so you have a value called e/m ratio ! But why is it important to caluculate this value? Is there any applications of it ?

  • In simple words, the charge-to-mass ratio of a charge helps us to predict the behaviour of the particle under electric and magnetic fields. This ability to predict the particles behaviour enables us to have an idea of adjusting the setup in order to have so and so outcomes.

We can see it’s application in : 

Electron Microscopes :

  • Electron Microscopes are known for their ability to magnify the images to a very high resolution & this is done with the help of a beam of electrons
  • Knowing the e/m ratio enables the scientists to control the movement of electrons and a result, they produce the required resolution of the image

Particle Accelerometers :

  • These are used in order to accelerate the charged particles.
  • By knowing the charge-to-mass ratios, we can actually control the trajectories of the particles.

2. Setup of the Experiment

The Setup mainly consists of the following things :

  • Filament F
  • Battery V
  • Pump
  • 2 parallel plates, across which another battery has been connected 
  • Current carrying coil wire (not shown in setup)
  • Screen S

                                                                                              Fig. Setup for the Experiement

Purpose :
  • Filament F : The filament is inclusive of that battery (not V) shown in figure.

  • Voltage V : 

       –   The plate attached to the positive terminal is used as anode to attract the electron cloud. This is done to make the electrons                         accelerate.

       –   Each electron has different energies when they come out from atoms. And when they are accelerated due to potential                                   difference of V, then they all end having different set of velocities. 

  • Pump : To create vacuum inside the tube
  •  The parallel plates kept facing each other + battery setup,  is used to create a uniform electric field E in the region between the two plates. Direction will be from positive plate to negative plate
  • Current carrying coil wire : This is done in order to setup a steady magnetic field B (going into the plane)
  • Screen S : Whenever an electron strikes the screen S; it creates a spot on the screen which helps us to detect and hence analyze the trajectory/path taken by the electron.

Note that : The E and B vectors are perpendicular to each other

3. Procedure

Step-1 :

As discussed, the anode attracts the electron cloud which makes them to accelerate towards the screen S. But well before they reach the screen, the electrons are made to pass through a region R where, for now, only Electric field is applied (B is turned off).

The current setup for Step-1 looks like : 

As the electrons pass through ‘Region R‘ , they undergo deflection ‘y’ due to the electric field and follow a trajectory as shown (green) . We zoom into the Region R to get a better understanding of what’s happening

Zoomed picture of Region R :

We need the expression for deflection ‘y’.

Important : Note that the deflection is going to be measured from axis

Some of the Projectile comes into picture now !


  • This ‘y’ is measured during the experiment
Step-2 :

Now, our aim is to find the velocity ‘v’ of the electron. Recall that the ‘y’ is the deflection –> BUT Deflection from which path ? The answer is ‘the axis’ . We need to find the deflection caused in electron’s trajectory due to the electric field E because otherwise in the absence of E, it would just follow the straight path along axis.

  • For getting the speed (v) of the electrons which go undeflected, we introduce B now in addition to E in order to make zero deflection. This is equivalent to saying that there was none of the fields present in region R

We have to balance the forces (to get zero deflection). Remember it’s a negative charge.


We then adjust the values of E and B until the magnitudes of forces are same. This allows us to build a ‘velocity selector‘

Velocity Selector :

As discussed in Section 2 of this article, all the electrons come with different set of velocities. But, for continuing our experiment, we need only the electrons of specific velocity to be focused on. So, how to exactly distinguish those electrons ?

We can clearly see the relation of E and B with velocity. This means that controlling the values of E and B allows us to select the electrons which have their velocities as E/B. The electrons possessing this specific velocity will go through the region undeflected and hence we can separate them. 

Step-3 :

Calculating the e/m ratio with the expressions and equations we got till now :

Substituting v = E/B in the expression of y obtained in step-1, we get  :

4. Final Data

Conclusion

  •  This completes our e/m calculation experiment performed by Sir J.J. Thomson.  
  • The article or the whole experiment procedure itself has a lot of concepts involved in it which makes it even more important to understand, both as an Experiment as well as an good multi-concept level problem

Keep Learning !