(a) What is the energy of a photon that has wavelength 0.10 mm ? (b) Through approximately what potential difference must electrons be accelerated so that they will exhibit wave nature in passing through a pinhole 0.10 mm in diameter? What is the speed of these electrons? (c) If protons rather than electrons were used, through what potential difference would protons have to be accelerated so they would exhibit wave nature in passing through this pinhole? What would be the speed of these protons?

(a) The energy of a photon of wavelength /\ is given by:

$0, for a photon with wavelength A = 0.10 X 10—6m , its energy is:

therefore the energy of the photon is

(b) An electron exhibits wave nature in passing through a pinhole when it

has a wavelength approximately equal to the diameter of that pinhole, which

is .\ : 0.10 mu.

“’9 know that de Broglie wavelength of a. particle of momentum p is given

is A : 0.10 mu.

“’9 know that de Broglie wavelength of :1. particle of momentum p is given

by:

So, for an electron of wavelength ,\ = 0.10 x 10“"i m, it momentum is:

For nonrelativistic: :‘II‘JDI‘OW‘JIt the kinetic energy and I‘ncm'lerlllll‘n are related

by the following formula:

Substituting for our values for p and mp, we got the kinetic energy of the

electron:

When an electron is accelerated through a potential difference 1-", it gains a

When an electron is accelerated through a potential difference V, i1. gains a

kinetic energy of K CV;

So, the potential (li?‘crcncc through which the electron must be accelerated

is:

Therefore the potential Difference is

The relativistic momentum is given by:

, the speed of the electrons is:

(c) For a proton With the same wavelength, its momentum is the same as

the electron,

But the mass of £1 proton is "’1' : 1.67 x 10“27 kg, so its kinetic energy is:

\

Since the proton has the same charge us the electron, the potential difference

through which the proton must be accelerated is:

Using equation (1), the speed of the protons is:

\

Therefore the potential difference of the proton is