In order to determine the $e.m.f.$ of a storage battery,it was connected in series with a standard cell in a certain circuit,and a current $I_1$ was obtained. When the battery is connected to the same circuit opposite to the standard cell,a current $I_2$ flows in the external circuit from the positive pole of the storage battery. What is the $e.m.f.$ $\varepsilon_1$ of the storage battery? The $e.m.f.$ of the standard cell is $\varepsilon_2$.

$A$ brass disc and a carbon disc of same radius are assembled alternatively to make a cylindrical conductor. The resistance of the cylinder is independent of the temperature. The ratio of thickness of the brass disc to that of the carbon disc is [$\alpha$ is temperature coefficient of resistance and neglect linear expansion].

Shown in the figure is a semicircular metallic strip that has thickness $t$ and resistivity $\rho$. Its inner radius is $R_1$ and outer radius is $R_2$. If a voltage $V_0$ is applied between its two ends,a current $I$ flows in it. In addition,it is observed that a transverse voltage $\Delta V$ develops between its inner and outer surfaces due to purely kinetic effects of moving electrons (ignore any role of the magnetic field due to the current). Then (figure is schematic and not drawn to scale)-
$(A)$ $I = \frac{V_0 t}{\pi \rho} \ln \left(\frac{R_2}{R_1}\right)$
$(B)$ the outer surface is at a higher voltage than the inner surface
$(C)$ the outer surface is at a lower voltage than the inner surface
$(D)$ $\Delta V \propto I^2$

The two ends of a conductor are connected to a cell having an $e.m.f. \ E$ and some internal resistance. Starting from the midpoint $P$ of the conductor,moving in the direction of the current,and returning to point $P$,which of the following graphs best represents the potential $V$ versus the distance $x$ covered along the path?

Two wires '$A$' and '$B$' of the same material have their lengths in the ratio $1 : 2$ and radii in the ratio $2 : 1$. The two wires are connected in parallel across a battery. The ratio of the heat produced in '$A$' to the heat produced in '$B$' for the same time is

There are two electric bulbs of $40\, W$ and $100\, W$. Which one will be brighter when first connected in series and then in parallel?