Wave Particle Duality: Magnifying Power of Electron Microscope

In summary, using an electron microscope, which uses highly accelerated electrons, you can see objects that are smaller (in order of magnitude) than you can using red light. You would need to calculate the Debroglie wavelength of this highly accelerated electron.
  • #1
Dinno
4
0

Homework Statement



Microscopes are inherently limited by the wavelength of the light used. How much smaller (in order of magnitude) can we "see" using an electron microscope whose electrons have been accelerated through a potential difference of 50,000V than using red light(500nm)?

Homework Equations



k=1/2mv2


The Attempt at a Solution



Don't know how to start
 
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  • #2
You would need to calculate the Debroglie wavelength of this highly accelerated electron. I'm going to assume that they taught you the equation for debroglie wavelength.

Apparently wave-partiicle duality describes every massive object as exhibiting wave like behaviour at high speeds.

You know that the kinetic of an electron = eV
1/2mv^2 = eV

But you need to find the momentum (p = mv) of this said electron in order to work out it's wavelength.

Have a go at deriving this momentum of electron formula in terms of it's kinetic energy, if you can't I will help you.

Anywho, if you manage, plug in the numbers for momentum, then plug momentum into your de broglie equation. The wavelengths will obviously be different than that of red light, you may then decide why it is better to use electrons.
 
  • #3
Thank you for your help,
I am not sure if I'm on the right track but I used the kinetic energy formula to find velocity then I used that velocity in the lambda=h/mv formula to find the wavelength, is that right?
 
  • #4
No.

If you simply equated eV = 1/2mv^2, you would end up getting a relativistic value for the speed which is ridiculous.

-----------------------------------------

Here is how to derive the momentum of an electron:

p = mv (1)

eV = 1/2mv^2 (2)

These are 2 separate expressions

What we can do to number (2) is rearrange it for v and substitute into equation number (1)#

So number (2) becomes:

sqrt(2eV/m) = v (where aqrt stands for square root). Substitute this into v for (1):

p = m sqrt(2eV/m)

Square both sides and get rid of one of the m's, and square root again. Final expression for momentum should be:

p = sqrt(2meV)

h (plancks constant) is 6.63E-34

You should end up getting a wavelength of 5.49E-12m, I'm sure you can imagine why this would give a better resolution (when veiwing a microscope) than an optical microscope with wavelengths of 500nm.

----------------------------------

HOWEVER

I'm unsure whether to bring relativity into this. Because I did use KE = eV straight off the bat and the electron was moving at an extremely relativistic velocity. I decided I would calculate it's momentum by using it's rest mass, kinetic energy and total energy, but I ended up with an unuseable value. If you were supposed to apply laws of relativity to this equation then ignore what I said.

Can a more experienced individual please advise, I don't want to give some wrong answers.
 

Related to Wave Particle Duality: Magnifying Power of Electron Microscope

1. What is wave-particle duality?

Wave-particle duality is the concept that states that particles, such as electrons, can exhibit both wave-like and particle-like behavior. This means that they can have properties of both waves, like wavelength and frequency, and particles, like mass and momentum.

2. How does wave-particle duality apply to the magnifying power of an electron microscope?

In an electron microscope, electrons are used to create a magnified image of a sample. These electrons exhibit both wave-like and particle-like behavior, allowing them to pass through the sample and create a detailed image. This is made possible by the wave-like property of electrons, which allows them to diffract and create interference patterns, and the particle-like property, which allows them to interact with the sample and create an image.

3. Why is an electron microscope able to magnify images more than a light microscope?

The magnifying power of an electron microscope is determined by the wavelength of the electrons used. Since electrons have a much smaller wavelength than visible light, they are able to resolve smaller details of a sample, resulting in a higher magnification compared to a light microscope.

4. What are the limitations of the magnifying power of an electron microscope?

The magnifying power of an electron microscope is limited by the size and stability of the electron beam. As the beam becomes smaller, it becomes more difficult to maintain its stability, which can affect the quality of the image. Additionally, the high energy of the electrons can also damage the sample being observed.

5. How does the principle of wave-particle duality impact our understanding of the microscopic world?

The concept of wave-particle duality has revolutionized our understanding of the microscopic world by allowing us to observe and study particles that were previously impossible to see with visible light microscopy. It has also led to the development of various advanced technologies, such as the electron microscope, which have greatly enhanced our understanding of the microscopic world and its role in various scientific fields.

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