Question

40 wires each of resistance $1\Omega$ are arranged as shown in figure. A battery of emf $8V$ and internal resistance $0.125\Omega$ is connected between $A$ and $B$. Rate of heat loss from cage is $12\times \beta$ W where $\beta$ is an integer. Then the value of $\beta$ is . (Answer upto two digits after the decimal point)

Answer 1

Since all the points around each octagon have same potential, those 16 marked resistors are ineffective.

Leaving us with 24 resistors, which are connected as show in figure below

So effective resistance ($R_{eff}$) across AB is =$\frac{3}{8}$

$I$ in circuit is
$i=\frac{8}{(\frac{3}{8}+\frac{1}{8})}=\frac{8\times 8}{4}=16A$

$P=i^2{R}_{eff}$

$=(16{)}^2\times \frac{3}{8}=16\times 16\times \frac{3}{8}=16\times 6=96W$

So, the value of $\beta$ is 8.