A parallel $L-R$ circuit consists of two ideal inductances $L$ and $2L$ and a resistor $R.$ At an instant the same current $I_0$ is flowing in both the inductor in the same direction. How much heat (in $\mu \mathrm{J}$) will be dissipated in resistance after a long time? $\left(L{I}_0{}^2=24 \mu \mathrm{J}\right)$

In a square loop of side length $\text{'}a\text{'},$ current $I_0$ is flowing as shown. If magnetic flux associated with region $x>0$ in $xy$ plane is $\frac{{\mu }_0{I}_{0}a}{2\pi }\ln N$, then value of $N$ is

A conducting wire of length $L$ fixed at both ends is vibrating with displacement equation given by $y=2A \sin \left(kx\right) \cos \omega t.$ A constant magnetic field $B$ is present in the region as shown. If maximum value of emf induced across the wire is $\frac{4 AB\omega L}{5\pi }$, then value of $\frac{kL}{\pi }$ is

A thin superconducting ring is held above a vertical long solenoid as shown in the figure. Axis of solenoid is same as that of a ring. The cylindrically symmetric magnetic field around the ring in terms of vertical and radial component, is $B_z=B_0\left(1-\alpha z\right)$ and $B_r=B_{0}Br$ where $B_0, \alpha$ and $B$ are positive constant and $z$ and $r$ are vertical and radial components respectively. Initially plane of ring is horizontal with no current flowing through it. Mass of the ring is $m$, radius $r_0$ and self inductance $L.$ If initially, coordinates of ring are $\left(0, 0, 0\right)$ and it is released then

When the ring is at vertical position $z$, then current in the ring is $\frac{k{B}_0\alpha z\pi r_0^2}{L}.$ Find constant $k$ in current expression.

A thin superconducting ring is held above a vertical long solenoid as shown in the figure. Axis of solenoid is same as that of a ring. The cylindrically symmetric magnetic field around the ring in terms of vertical and radial component, is $B_z=B_0\left(1-\alpha z\right)$ and $B_r=B_{0}Br$ where $B_0, \alpha$ and $B$ are positive constant and $z$ and $r$ are vertical and radial components respectively. Initially plane of ring is horizontal with no current flowing through it. Mass of the ring is $m$, radius $r_0$ and self inductance $L.$ If initially, coordinates of ring are $\left(0, 0, 0\right)$ and it is released then

If ring is at equilibrium at vertical coordinate $z=-\frac{mgL}{2B_0^2\alpha B{\pi }^2{r}_0^N}$, then value of $N$ (Power of $r_0$) is

A thin superconducting ring is held above a vertical long solenoid as shown in the figure. Axis of solenoid is same as that of a ring. The cylindrically symmetric magnetic field around the ring in terms of vertical and radial component, is $B_z=B_0\left(1-\alpha z\right)$ and $B_r=B_{0}Br$ where $B_0, \alpha$ and $B$ are positive constant and $z$ and $r$ are vertical and radial components respectively. Initially plane of ring is horizontal with no current flowing through it. Mass of the ring is $m$, radius $r_0$ and self inductance $L.$ If initially, coordinates of ring are $\left(0, 0, 0\right)$ and it is released then

If tension in the ring is maximum when $z=\frac{1}{c\alpha }$, then value of $c$ is