Answer to Question #87826 in Electricity and Magnetism for Brayden Oblak

Question

THe plates of a thompson q/m apparatus are 5 cm long and separated by 1 cm. The end of the plates are 25 cm from tube screen. The kinetic energy of the electrons is 2000 eV.
a) If a potential of 20 V is applied across the deflection plates, by how much will the beam deflect

Answer 1

Electron mass: m = 511 keV = 511000 eV

Speed of the electron:

$$E=mv^2/2 \Rightarrow v = \sqrt{\frac{2E}{m}}$$

Amount of time, when electron was in apparatus:$t =a/v,\, a =5\text{cm} = 0.05 \text{m}$

Electric force, applied to electron:

$$F = e U / d,\, U = 20\, \text{V}, \, d = 1\, text{cm} = 0.01\, \text{m}$$

where e - elementary charge.

From second Newtons law follows that change of momentum equals to

$$\Delta p = m \Delta v = F \cdot t = \frac{eUa}{vd} \Rightarrow \Delta v = \frac{eUa}{vdm}$$

Change of speed is comperatively small:

$$\phi = \Delta v / v = \frac{eUa}{v^2 d m} =\frac{eUa}{2 E d} = \frac{20\cdot 0.05}{4000 \cdot 0.01} = 0.025$$

$\Delta v$ perpendicular to the initial direction of speed. Therefore, direction of velocity would be changed on angle $\phi$

Beam would be deflected by distance:

$$l = \phi \cdot L = \frac{eUa}{2 E d} L = 0.025 \cdot 25 = 0.625 \, \text{cm} = 6.25 \, \text{mm}$$