Question

An electron enters with a velocity v = v₀i into a cubical region (faces parallel to coordinate planes) in which there are uniform electric and magnetic fields. The orbit of the electron is found to spiral down inside the cube in plane parallel to the x-y plane. Suggest a configuration of fields E and B that can lead to it.

Answer 1

Due to magnetic force charge particle revolves in uniform circular motion in x-y plane and due to electric field charge particle increases the speed along the x-direction, which in turn increases the radius of circular path and hence, particle traversed on spiral path.

Let us consider a magnetic field B = B₀ present in the region and an electron enters with a velocity into cubical region (faces parallel to coordinate planes). The force on the electron, using magnetic Lorentz force, is given by

The velocity of electron is v = v₀i, i.e., along X-axis as magnetic field is perpendicular to velocity so it is in Y-direction.

The moving electron enters into cubical region. The force on electron due to Lorentz force

F = q(E + v × B)

By putting the values

F_m = `-e[v_ohati xx bhatk] = -ev_0Bhatj` 

Which revolves around the electron in X-Y plane.

The force due to electric field `F_m = evecKhatk` accelerates electron along z-axis and force due to magnetic field keeps it in circular motion, which in turn increases the radius of circular path. So the motion becomes helical path.

Which revolves the electron in x-y plane.

The electric force F = eE₀j accelerates e along the z-axis which in turn increases the radius of circular path and hence particle traversed on spiral path.