In Thomson’s experiment for determining e/m the potential difference between the cathode and the anode is same as that between the deflecting plates . If the p.d is doubled, by what factor should the magnetic field by increased to ensure that the electron beam remains undeflected.

Q: In Thomson’s experiment for determining e/m the potential difference between the cathode and the anode (in accelerating column ) is same as that between the deflecting plates (in the region of crossed fields) . If the p.d is doubled, by what factor should the magnetic field by increased to ensure that the electron beam remains undeflected

Sol: $\large \frac{1}{2}m v^2 = q V$

$\large \frac{1}{2}m (\frac{E}{B})^2 = q V$

$\large \frac{1}{2}m (\frac{V}{d B})^2 = q V$

$\large \frac{q}{m} = \frac{V}{2 d^2 B^2}$

$\large B^2 \propto V \Rightarrow B \propto \sqrt{V}$

$\large \frac{B’}{B} = \sqrt{\frac{2 V}{V}}$

B’ = √2 B