In the figure, transformer $${T_1}$$ has two secondaries, all three windings having the same number of turns and with polarities having the same number of turns and with polarities as indicated. One secondary is shorted by a $$10\,\Omega $$ resistor $$R,$$ and the other by a $$15\mu F$$ capacitor. The switch $$SW$$ is opened $$(t=0)$$ when the capacitor is charged to $$5$$ $$V$$ with the left plate as positive. At $$t=0+$$ the voltage $${V_P}$$ and current $${I_R}$$ are

A three phase balanced star connected voltage source with frequency $$\omega \,\,rad/s$$ is connected to a star connected balanced load which is purely inductive. The instantaneous line currents and phase to neutral voltages are denoted by $$\left( {{i_a},{i_b},{i_c}} \right)$$ and $$\left( {{V_{an}},\,\,{V_{bn}},\,\,{V_{cn}}} \right)$$ respectively and their $$rms$$ values are denoted by $$V$$ and $$1.$$ If $$$R = \left[ {{V_{an}}\,\,{V_{bn}}\,\,{V_{cn}}} \right]\left[ {\matrix{ 0 & {{1 \over {\sqrt 3 }}} & { - {1 \over {\sqrt 3 }}} \cr { - {1 \over {\sqrt 3 }}} & 0 & {{1 \over {\sqrt 3 }}} \cr {{1 \over {\sqrt 3 }}} & { - {1 \over {\sqrt 3 }}} & 0 \cr } } \right]\left[ {\matrix{ {{i_a}} \cr {{i_b}} \cr {{i_c}} \cr } } \right],$$$
then the magnitude of $$R$$ is

In the circuit shown in Fig. switch $$S{w_1}$$ is initially CLOSED and $$S{w_2}$$ is OPEN. The inductor $$L$$ carries a current of $$10$$ $$A$$ and the capacitor is charged to $$10$$ $$V$$ with polarities as indicated. $$S{w_2}$$ in initially CLOSED at $$t = {0^ - }$$ and $$S{w_1}$$ is OPENED at $$t=0.$$ The current through $$C$$ and the voltage across $$L$$ at $$t = {0^ + }$$ is

A $$3$$ $$V$$ $$dc$$ supply with an internal resistance of $$2$$ $$\Omega $$ supplies a passive non-linear resistance characterized by the relation $${V_{NL}} = {{\rm I}^2}{}_{NL}$$. The power dissipated in the non-linear resistance is