The magnetic vector potential in a region is defined by $$\overrightarrow A = {e^{ - y}}\sin \left( x \right){\widehat a_z}.$$ An infinitely long conductor, having a cross section area, $$a=5$$ $$m{m^2}$$ and carrying a $$dc$$ current, $${\rm I} = 5\,A$$ in the $$Y$$ direction, passes through this region as shown in Fig. Determine the expression for $$(a)$$ $$\overrightarrow B $$ and $$(b)$$ force density $$\overrightarrow f $$ exerted on the conductor

A current $${\rm I}$$ in the short conducting element shown in Fig. produces a flux density $${B_1}$$ at point $$1.$$ Determine the magnitude and the direction of the flux density vector at point $$2$$

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.

A infinitely long straight wire carries $$1000$$ $$A$$ of current and in the vicinity, there is a circular conducting loop of $$100$$ $$mm$$ diameter with the center of the loop $$1$$ $$m$$ away from the straight conductor. Both the wire and the loop are coplanar. Determine the magnitude and direction of current in the loop that produces a zero flux density at its center.