Ncert Exercises & Solutions

An electron enters with a velocity $v = v_{0} \hat{i}$ 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 a plane parallel to the x-y plane. Suggest a configuration of fields E and B that can lead to it.

Hint: Find the directions of electric force and magnetic force.

Step 1: Considering the magnetic field

B = B_{0} \hat{k}

, and an electron enters with a velocity

v = v_{0} \hat{i}

into a cubical region (faces parallel to coordinate planes).
The force on electron, using magnetic Lorentz force, is given by;

F_{m} = - e (v_{0} \hat{i} x B_{0} \hat{k}) = {ev}_{0} B_{0} \hat{i}

which revolves around the electron in the x-y plane.

Step 2: The electric force,

F_{e} = - {eE}_{0} \hat{k}

accelerates the electron along the z-axis which in turn increases the radius of the circular path and hence particle traversed on the spiral path.

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