Charged particles are accelerated through a potential difference \(V_{0}\) before falling onto a lens consisting of an aperture of height \(y_{0}\) and thickness \(w\), as shown in the figure.

There is a uniform electric field \(\mathbf{E}_{1}\) and \(\mathbf{E}_{2}\) on the left and right of the aperture respectively, as depicted. The figure also shows the trajectory of a charged particle of charge \(q\) emerging a distance \(x_{1}\) from the aperture on the left and focusing a distance \(x_{2}\) on the right. Assume \(V_{0} \gg E_{1} x_{1}\) and \(E_{2} x_{2}\), and \(x_{1}\) and \(x_{2} \gg y_{0}\).

(a) Using \(abla \cdot \mathbf{E}=0\), show that inside the aperture, we have

Transcribed Image Text:

X1 E X2 Yo E2