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 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

The effective resistances of the each of the $3$ parts $AC,CD$ and $DB$ is $\frac{1}{8}\Omega$
So effective resistance across $AB$, $R_{eff}=\frac{3}{8}\Omega$

Current in circuit,
$i=\frac{\epsilon }{R_{eff}+r}=\frac{8}{(\frac{3}{8}+\frac{1}{8})}$
$i=\frac{8\times 8}{4}=16A$

$P=i^2{R}_{eff}$
$=(16{)}^2\times \frac{3}{8}=16\times 16\times \frac{3}{8}=16\times 6$
$P=96W$

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