A teacher in his physics laboratory allotted an experiment to...

If a large resistance $R$ is connected with a battery, then current will be approximately $\dfrac{ε}{R}$, and $t$ will remain almost constant.

In ${1}^{st}$ case, only $\left(G\right)$ is connected $⇒ ig = i$

In ${2}^{nd}$ case,

${i}_{g}{R}_{0} = \left(I - {i}_{g}\right)S ⇒ {\left({i}_{g}\right)}_{2} = \left(\dfrac{S}{{R}_{g} + S}\right)i$

If ${\left({i}_{g}\right)}_{2} = \dfrac{1}{3}\left({i}_{g}\right), ⇒ \left(\dfrac{S}{{R}_{g} + S}\right)i = \dfrac{1}{3}i$

${R}_{g} = 2S$