From the negative plate of a parallel plate capacitor charged to a potential difference V , an electron is emitted with negligible speed. The separation between the plates is d , and a magnetic field B exists in the space as shown in the figure. Which of the following conditions ensures that the electron will fail to strike the upper plate?

Step-by-step solution

Let the lower plate be at y = 0 and the upper plate be at y = d . The electric field between the plates is E = V d , directed downwards. The magnetic field is B = B k . The equations of motion for the electron are given by the Lorentz force: m_e d v dt = -e( E + v B ) Considering the x-component of the force: m_e dv_x dt = -e B v_y = -e B dy dt Integrating with respect to time and using the initial condition v_x = 0 at y = 0 : m_e v_x = -e B y v_x = - e B m_e y By the work-energy theorem, the kinetic energy of the