Question

A rectangular region of dimensions $w\times l(W\ll l)$ has a constant magnetic field into the plane of the paper as shown. On one side the region is bounded by a screen. On the other side positive ions of mass m and charge q are accelerated from rest and towards the screen by a parallel plate capacitor at constant potential difference $V<0$, and come out through a small hole in the upper plate. Which one of the following statements is correct regarding the charge on the ions that hit the screen?

• A. Ions with $q>\frac{2|v|m}{B^2{w}^2}$ will hit the screen
• B. Ions with $q<\frac{2|v|m}{B^2{w}^2}$ will hit the screen
• C. All ions will hit the screen
• D. Only ions with $q=\frac{2|v|m}{B^2{w}^2}$ will hit the screen

Answer 1

The correct option is B Ions with $q<\frac{2|v|m}{B^2{w}^2}$ will hit the screen

Using conservation of energy,

$\frac{1}{2}m{v}^2=qV$

Ions will take a circular path inside the field. Ions with $r<w$ will not hit the screen.

$r=\frac{mv}{qB}<w$

$\frac{m\sqrt{2qV}}{qB\sqrt{m}}<w$

$⟹q>\frac{2Vm}{B^2{w}^2}$ will miss the screen.

Similarly, $q<\frac{2Vm}{B^2{w}^2}$ will hit the screen.