To light, a $$50 \mathrm{~W}, 100 \mathrm{~V}$$ lamp is connected, in series with a capacitor of capacitance $$\frac{50}{\pi \sqrt{x}} \mu F$$, with $$200 \mathrm{~V}, 50 \mathrm{~Hz} \,\mathrm{AC}$$ source. The value of $$x$$ will be ___________.

In a coil of resistance $$8 \,\Omega$$, the magnetic flux due to an external magnetic field varies with time as $$\phi=\frac{2}{3}\left(9-t^{2}\right)$$. The value of total heat produced in the coil, till the flux becomes zero, will be _____________ $$J$$.

The effective current I in the given circuit at very high frequencies will be ___________ A.

Magnetic flux (in weber) in a closed circuit of resistance 20 $$\Omega$$ varies with time t(s) at $$\phi$$ = 8t² $$-$$ 9t + 5. The magnitude of the induced current at t = 0.25 s will be ____________ mA.