The radius of the circular conducting loop shown in the figure is $R.$ The magnetic field is decreasing at a constant rate $\alpha.$ The resistance per unit length of the loop is $r.$ Find the current in the wire $AB,$ where $AB$ is one of the diameters.

In the branch $AB$ of a circuit,as shown in the figure,a current $I = (t + 2) \ A$ is flowing,where $t$ is the time in seconds. At $t = 0$,the value of $(V_A - V_B)$ will be: (in $V$)

The circular wire in the figure below encircles a solenoid in which the magnetic flux is increasing at a constant rate out of the plane of the page. The clockwise emf around the circular loop is $\varepsilon_{0}$. By definition,a voltmeter measures the voltage difference between two points given by $V_{b}-V_{a}=-\int_{a}^{b} E \cdot ds$. We assume that $a$ and $b$ are infinitesimally close to each other. The values of $V_{b}-V_{a}$ along path $1$ and $V_{a}-V_{b}$ along path $2$,respectively,are

$A$ charged particle of charge $q$ and mass $m$ is placed at a distance $2R$ from the centre of a vertical cylindrical region of radius $R$ where the magnetic field varies as $\vec{B} = (4t^2 - 2t + 6) \hat{k}$, where $t$ is time. Then which of the following statement$(s)$ is/are true?

$A$ conducting loop is placed in a time-varying magnetic field $B = \frac{\alpha}{t^2}$,where $\alpha$ is a positive constant. The magnetic field is directed into the plane of the loop. Determine the nature of the charge on plate $A$ of the capacitor $C$ connected in the loop.

$A$ magnetic field at a distance $r$ from the $z$-axis is given by $\vec{B} = B_0 r t \hat{k}$,where $B_0$ is a constant and $t$ is time. The magnitude of the induced electric field at a distance $r$ from the $z$-axis is: