A 3-phase, 415 V, 50 Hz induction motor draws 5 times the rated current at rated voltage at starting. It is required to bring down the starting current from the supply to 2 times of the rated current using a 3-phase autotransformer. If the magnetizing impedance of the induction motor and no load current of the autotransformer is neglected, then the transformation ratio of the autotransformer is given by ________. (round off to two decimal places).

A long conducting cylinder having a radius b is placed along the z-axis. The current density is $$\overrightarrow J = {J_a}{r^3}\widehat z$$ for the region r < b where r is the distance in the radial direction. The magnetic field intensity ($$\overrightarrow H $$) for the region inside the conductor (i.e., for r < b) is

If the magnetic field intensity ($$\overrightarrow H $$) in a conducting region is given by the expression, $$\overrightarrow H = {x^2}\widehat i + {x^2}{y^2}\widehat j + {x^2}{y^2}{z^2}\widehat k$$ A/m. The magnitude of the current density, in A/m², at x = 1 m, y = 2 m and z = 1 m is

As shown in the figure below, to concentric conducting spherical shells, centred at r = 0 and having radii r = c and r = d are maintained at potentials such that the potential V(r) at r = c is V₁ and V(r) at r = d is V₂. Assume that V(r) depends only on r, where r is the radial distance. The expression for V(r) in the region between r = c and r = d is