Is current density a vector or a scalar quantity? Deduce the relation between current density and the potential difference across a current-carrying conductor of length $l$, area of cross-section $A$, and number density of free electrons $n$. How does the current density, in a conductor vary with increase in potential gradient?

Twenty electric bulbs are connected in series with the mains of a $220 \mathrm{V}$ supply. After one bulb is fused, the remaining $19$ bulbs are again connected in series across the same mains. What will be the effect on illumination?

In an electric kettle, water boils in $20$ minutes after the kettle is switched on. With the same supply voltage if the water is to boil in $10$ minutes, should the length of the heating element be decreased or increased?

Fig. shows a piece of pure semiconductor, $S$ in series with a variable resistor $R$, and a source of constant voltage $V$. Would you increase or decrease the value of $R$ to keep the reading of ammeter ($A$) constant, when semiconductor $S$ is heated? Give reason.