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$$y=\frac{1}{2}{a}_{y}{t}^{2}=\frac{1}{2}\alpha E{t}^{2}$$
The position vector of the particle after time t is :
$$\mathbf{r}=x\mathbf{i}+yj+z\mathbf{k}$$ $$\Rightarrow \mathbf{r}=\frac{{v}_{0}}{\alpha B}\mathrm{sin}\left(\alpha Bt\right)\mathbf{i}+\frac{1}{2}\alpha E{t}^{2}\mathbf{j}+\frac{{v}_{0}}{\alpha B}[1-\mathrm{cos}\left(\alpha Bt\right)]\mathbf{k}$$
Let electric and magnetic fields are aligned along z and x directions and charge is placed at the origin of coordinate system. Initially, there is no magnetic force as charge is at rest. However, there is electric force, which accelerates the charge in z-direction. As the particle acquires velocity in z-direction, the magnetic force comes into play and tries to rotate the particle in xz plane about a center on x-axis.
However, z-component of velocity keeps increasing with time due to electric force in that direction. The magnetic force though draws the charged particle away from z-axis along a curved path. This action of magnetic force is countered by electric force in z-direction. The velocity of charged particle ultimately reduces to zero at x-axis. This cycle repeats itself forming cycloid motion. The cycloid path is generated by a point on the circumference of a rolling wheel. Here, we shall skip the mathematical derivation and limit ourselves to a descriptive analysis only.
The specific charge of an electron is ratio of charge and mass of electron. The specific charge (α) of electron is measured employing crossed fields on a beam of electrons. The beam of electrons emerging from cathode plate passes through a very narrow slit in anode plate. The electrons are accelerated between cathode and anode due to applied electrical potential V. The kinetic energy of the electron emerging from the slit is given by :
$$\frac{1}{2}m{v}^{2}=eV$$ $$\Rightarrow m{v}^{2}=2eV$$
where v is the velocity of electron moving into the region of force fields.
Two parallel plates connected to an electric source produce a uniform electric field E from positive plate to negative plate. The electrical force works in the direction opposite to the direction of field E as charge on electron is negative. In the figure, electric field is directed in downward direction. Hence, electric force acts in upward direction.
On the other hand, the magnetic field is produced by a solenoid in a circular region covering the plate as shown in the figure. Its direction is chosen such that it applies a force in the opposite direction to that applied by the electrical field. For a magnetic field into the plane of drawing as shown by uniformly distributed cross signs, the magnetic field applies a upward magnetic force on a positive charge. However, as the charge on the electron is negative, the magnetic force acts in downward direction.
The beam of electrons hit the center of fluorescent screen, producing light as electrons collide with it when electric and magnetic fields are switched off. The point on the fluorescent screen is noted. Then, the electric field is switched on which moves the electron beam in upward direction following a parabolic path. Finally, magnetic field is turned and its magnitude is adjusted such that electric and magnetic forces acting in opposite directions balance each other and the electron is brought to hit original spot as noted earlier for the fields in switched off condition. In this situation:
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