# 0.8 Motion of a charged particle in electric and magnetic fields  (Page 4/4)

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$eE=evB$ $⇒v=\frac{E}{B}$

Note that maximum magnetic force applies as velocity and magnetic field vectors are perpendicular to each other. Substituting expression of v in the kinetic energy equation obtained earlier, we have :

$\frac{m{E}^{2}}{{B}^{2}}=2eV$ $⇒\alpha =\frac{e}{m}=\frac{{E}^{2}}{2V{B}^{2}}$

All the quantities on the right hand side of the equation are measurable, allowing us to measure the specific charge of electron. As a matter of fact, the determination of specific charge of particles composing cathode ray by J.J.Thomson is considered to be the discovery of electron. It can also be easily inferred that he could determine the nature of charge of an electron by studying direction of deviation (upward or downward) when only either of the fields operate. In the derivation above, we measure potential difference applied to accelerate particle between cathode and anode. We should, however, realize that we can determine specific charge measuring some other quantities as well. We can measure the deflection of electron beam when either of two fields operates and use the data to determine specific charge of an electron.

Problem : The d.c. voltage applied to accelerate particle between cathode and anode and the d.c. voltage applied to the plates to produce electric field perpendicular to electrons beam are equal in the Thomson’s experimental set up. If each of the two d.c. voltages as applied are doubled, then by what factor should the magnetic field be changed to keep the electron beam un-deflected.

Solution : Let ${V}_{1}$ , ${E}_{1}$ and ${B}_{1}$ be the potential difference, electric field and magnetic field for un-deflected condition. Then, the specific charge is given by :

$\alpha =\frac{e}{m}=\frac{{E}_{1}^{2}}{2{V}_{1}{B}_{1}^{2}}$

Here, the electric field can be expressed in terms of potential difference provided we know the separation between plates. Let the separation be d.

${E}_{1}=\frac{{V}_{1}}{d}$

Putting in the equation above, we have :

$⇒\alpha =\frac{{V}_{1}^{2}}{2{d}^{2}{V}_{1}{B}_{1}^{2}}=\frac{{V}_{1}}{2{d}^{2}{B}_{1}^{2}}$

Let ${B}_{2}$ be the new magnetic field when two potential differences as applied are doubled. Here,

${V}_{2}=2{V}_{1}$

Putting new values in the expression for specific charge (note that specific charge of electron is a constant),

$\alpha =\frac{2{V}_{1}}{2{d}^{2}{B}_{2}^{2}}$

Combining two equations,

$\frac{2{V}_{1}}{2{d}^{2}{B}_{2}^{2}}=\frac{{V}_{1}}{2{d}^{2}{B}_{1}^{2}}$ $⇒2{V}_{1}{d}^{2}{B}_{2}^{2}=4{V}_{1}{d}^{2}{B}_{1}^{2}$ $⇒\frac{{B}_{2}^{2}}{{B}_{1}^{2}}=2$ $⇒\frac{{B}_{2}}{{B}_{1}}=\sqrt{2}$

## Measurement of deflection by magnetic field

Once the magnetic and electric forces are balanced, electric field is switched off and electron beam is allowed to be deviated due to magnetic field. The magnetic force acts always perpendicular to the direction of motion. The particle, therefore, moves along a circular path inside the region of magnetic field. When electron moves out of the magnetic field, it moves along the straight line and hits the fluorescent screen. If R be the radius of curvature, then :

$\frac{m{v}^{2}}{R}=evB$ $⇒mv=eRB$

Substituting $v=E/B$ as obtained earlier

$\alpha =\frac{e}{m}=\frac{E}{R{B}^{2}}$

We measure R using geometry. We see that the angles enclosed between pairs of two perpendicular lines are equal. Hence,

$\phi =\frac{DG}{R}=\frac{OI}{FO}$ $⇒R=\frac{FOXDG}{OI}$

We approximate DG to be equal to the width of magnetic region.

## Measurement of deflection by electric field

In this case, once the magnetic and electric forces are balanced, electric field is switched off and electron beam is allowed to be deviated due to electric field. The electron beam moving into the region of electric field experiences an upward force. The force in upward (y-direction) imparts acceleration in y-direction. The particle, however, moves with same velocity in x-direction. As a result, path of motion is parabolic. Let the length of plate be L and y be the deflection inside the plate. Then, time to travel through the plate is :

$t=\frac{L}{v}$

and acceleration of the particle in y-direction is :

${a}_{y}=\frac{{F}_{E}}{m}=\frac{eE}{m}$

The vertical displacement is :

$y=\frac{1}{2}{a}_{y}{t}^{2}$

Substituting for time and acceleration, we have :

$y=\frac{1}{2}X\frac{eE}{m}X\frac{{L}^{2}}{{v}^{2}}=\frac{eE{L}^{2}}{2m{v}^{2}}$

Substituting v=E/B as obtained earlier,

$y=\frac{eE{L}^{2}}{2m}X{\left(\frac{B}{E}\right)}^{2}=\frac{\alpha E{L}^{2}}{2}X{\left(\frac{B}{E}\right)}^{2}$ $⇒\alpha =\frac{e}{m}=\frac{2yE}{{B}^{2}{L}^{2}}$

It is clear that measuring GH and HI, we can determine angle φ and then y as required.

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
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What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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