Electrons pass through the parallel electrodes shown in FIGURE EX37.9 with a speed of $5\times {10}^{6}m/s$. What magnetic field strength and direction will allow the electrons to pass through without being deflected? Assume that the magnetic field is confined to the region between the electrodes.

We have been given that Voltage$V=200V$, Length$L=8.0mm$, Width$w=4.0cm$, Speed$v=5\times {10}^{6}m/s$towards left.

Electric field$E=\frac{V}{L}=\frac{200}{8\times {10}^{-3}}=2.5\times {10}^{4}N/C$

Since the upper plate is at a higher potential so Electric field is in downward direction

But electron being negatively charged will have a force in upward direction of magnitude qE

where e is charge on electron.

Magnetic force on electron is given by $qv\times B$where B is magnetic field

Since electron is going left and we need the resulting magnetic force to be in downward direction (opposite to electric field) so that net force is zero.

Using the right hand rule we can see that magnetic field is directed into the plane of paper.

Comparing magnitudes

$$\begin{gathered} qE=qv\times B \\ E=v\times B \\ B=\frac{E}{v}=\frac{2.5\times {10}^4}{5\times {10}^6}=5.0\times {10}^{-3}T \end{gathered}$$