Physics

Physics · electromagnetism

Problem

(b) An incoming electron plane wave with wave vector

k_{z}

is deflected by the "biprism" due to the

x

-component of the force exerted on the electron. Determine

k_{x}

the

x

-component of the wave vector due to the "biprism" in terms of the electron charge,

e, v_{\mathrm{z}}, V_{a}, k_{\mathrm{z}}, a

and

b

, where

e

and

v_{z}

are the charge and the

z

-component of the velocity of the electrons

\left(k_{x} \ll k_{z}\right)

. Note that

\vec{k}=\frac{2 \pi \vec{p}}{h}

where

h

is the Planck constant. Given constants: The two-slit electron interference experiment was first performed by Möllenstedt et al, MerliMissiroli and Pozzi in 1974 and Tonomura et al in 1989. In the two-slit electron interference experiment, a monochromatic electron point source emits particles at

S

that first passes through an electron "biprism" before impinging on an observational plane;

S_{1}

and

S_{2}

are virtual sources at distance

d

. In the diagram, the filament is pointing into the page. Note that it is a very thin filament (not drawn to scale in the diagram). <image_1> The electron "biprism" consists of a grounded cylindrical wire mesh with a fine filament

F

at the center. The distance between the source and the "biprism" is

\ell

, and the distance between the distance between the "biprism" and the screen is

L

. Context question: (a) Taking the center of the circular cross section of the filament as the origin

O

, find the electric potential at any point

(x, z)

very near the filament in terms of

V_{a}, a

and

b

where

V_{a}

is the electric potential of the surface of the filament,

a

is the radius of the filament and

b

is the distance between the center of the filament and the cylindrical wire mesh. (Ignore mirror charges.) Context answer:

V(r)=V_{a} \frac{\ln (b / r)}{\ln (b / a)} \text { where } r=\sqrt{x^{2}+z^{2}}

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