A wire of length $L$ and 3 identical cells of negligible internal resistance are connected in series. Due to the current, the temperature of the wire is raised by $\Delta \text{T}$ in an interval of time $t$. Now find the value of $N$ if, the temperature of the wire is raised by the same amount $\Delta \text{T}$  for the same time $t$ if the number of $N$ similar cells are now connected in series with a wire of the same material and area of cross-section but of length $2L$.

 

A uniform metallic wire of length $L$ is mounted in two configurations. In configuration $1$ (triangle), it is an equilateral triangle and a voltage, $V$ is applied to corners $A$ and $B$. In configuration $2$ (circle), it is bent in the form of a circle, and the potential $V$ is applied at diametrically opposite points $P$ and $Q$. The ratio of the power dissipated is configuration $1$ to configuration $2$ is.

Consider the following two circuits

$\left[A\right] :$ $20$ bulbs are connected in series to a power supply line

$\left[B\right] :$ $20$ bulbs identical to $\left[A\right]$ are connected in a parallel circuit to an identical power supply line

Identify which of the following is not true

A light bulb of resistance $R=16 \mathrm{\Omega }$ is attached in series with an infinite resistor network with identical resistances as shown below. A $10 \mathrm{V}$ battery derives a current in the circuit. What should be the value of $r$ such that the bulb dissipates about $1 \mathrm{W}$ of power?