An infinite number of charges, each equal to $$'q'$$ are placed along the $$x=1,x=2, x=4, x=8, x=16$$ and so on, Find the potential and electric field at point $$x=0,$$ due to these system of charges.

An electron moves in the $$X$$-$$Y$$ plane with a speed of $${10^6}\,\,m/s.$$ Its velocity vector makes an angle of $${60^ \circ }$$ with $$X$$ axis. A magnetic field of magnitude $${10^{ - 2}}T$$ exists along the $$Y$$ axis. Compute the magnetic force exerted on the electron and its direction.

The laws of electromagnetic induction (Faraday's and Lenz's law) are summarized in the following equation

$$A = \left[ {\matrix{ 2 & 0 & 0 & { - 1} \cr 0 & 1 & 0 & 0 \cr 0 & 0 & 3 & 0 \cr { - 1} & 0 & 0 & 4 \cr } } \right].$$ The sum of the eigen values of the matrix $$A$$ is