A separately excited $$DC$$ shunt motor is driving a fan load whose torque is proportional to the square of the speed. When $$100$$ $$V$$ are applied to the motor, the current taken by the motor is $$8$$ $$A,$$ with the speed being $$500$$ $$rpm.$$ At what applied voltage does the speed reach $$750$$ $$rpm$$ and then what is the current drawn by the armature? Assume the armature circuit resistance to be $$1\,\,\,\Omega .$$ Neglect brush drop and mechanical losses.

Two single phase transformers $$A$$ and $$B$$ have the following parameters.
Transformer $$A$$: $$400V/200V,$$ $$10$$ $$kVA,$$ percentage resistance and percentage reactance are $$3\% $$ and $$4\% $$ respectively.
Transformer B: $$5$$ $$kVA,$$ $$400V/200V,$$ percentage resistance and percentage reactance are $$4\% $$ and $$3\% $$ respectively.
These two transformers are connected in parallel and they share a common load of $$12$$ $$kW$$ at a power factor of $$0.8$$ lag. Determine the active and reactive power delivered by transformer $$A.$$

An electromagnetic field is radiated from

A charge $$+Q$$ is uniformly distributed throughout the volume of a dielectric sphere of radius $$R$$ and dielectric constant $$'{\varepsilon _R}'.$$ Based on Gauss law, determine the expressions for the electric field $$E$$ as a function of distance $$'r'$$ from the center of the sphere, within the ranges $$0 < r < R$$ and $$R \le r.$$ Indicate expression $$(s)$$ for the critical point $$(s)$$ on the sketch.