Assuming ideal elements in the circuit shown below, the voltage $${V_{ab}}$$ will be

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

In Fig., $${R_a},\,\,{R_b},$$ and $${R_c}$$ are $$20\,\Omega ,\,\,10\,\Omega $$ and $$10\,\Omega $$ respectively. The resistances $${R_1},\,\,{R_2}$$ and $${R_3}$$ in $$\Omega $$ of an equivalent star - connection are

In Fig., the value of resistance $$R$$ in $$\Omega $$ is