CP Electrons go through a single slit 300 nm wide and strike a screen 24.0 cm away. At angles of 20.0 from the centre of the diffraction pattern, no electrons hit the screen, but electrons hit at all points closer to the centre. (a) How fast were these electrons moving when they went through the slit? (b) What will be the next pair of larger angles at which no electrons hit the screen?

From equation 36.2, when a beam of wavelength /\ is diffracted from a single slit of width (1, the condition of a dark fringe is:

sia = ”7% (1)

Where 0m is the Angle of line from centre of slit to mth dark fringe on screen-

From equation 39.], the de Broglie wavelength of a particle of momentum p is given by:

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Givens

The width of the slit is , the distance between the screen and the slit is and the angle from

the centre of the diffraction pattern where no electrons hit the screen is-

(a) The points where no electrons hit the screen is dark fringe, so the given angle 0 is the angle of the ?rst fringe, \.

(m = 1)-

So, we plug our values for 91 and a into equation (1) and solve for A, so We get the Wavelength of the electrons:

m

Then, we plug this value into equation (2), so we get the momentum of the electrons:

The nonrelativistic momentum is given by:

So, we substituted for me and p, so we get the speed of the electrons

Therefore speed is very small compared to the speed of light, so using the nonrelativistic approach is acceptable.

(b) The next fringe is m = 2; So, we plug our values for A, a and m into equation (I) and solve for 62, so we get:

Therefore indicates that there are two second fringe, one on each side of the centre of the pattern-