$A$ battery of $e.m.f.$ $10\, V$ and internal resistance $0.5\, \Omega$ is connected across a variable resistance $R$. The value of $R$ for which the power delivered in it is maximum is given by ......... $\Omega$.

Current $I$ is flowing through two materials having electrical conductivities $\sigma_1$ and $\sigma_2$ respectively $(\sigma_1 > \sigma_2)$ as shown in the figure. The total amount of charge at the junction of the materials is

In the given circuit,the value of $\left|\frac{I_1+I_3}{I_2}\right|$ is

Four identical electrical lamps are labelled $1.5\,V$,$0.5\,A$,which describes the condition necessary for them to operate at normal brightness. $A$ $12\,V$ battery of negligible internal resistance is connected to the lamps as shown. Then:

$A$ thermocouple is formed by two metals $X$ and $Y$. Metal $X$ comes earlier than $Y$ in the Seebeck series. If the temperature of the hot junction increases beyond the temperature of inversion,then the direction of the current in the thermocouple will be:

Three equal resistances each of value $R$ are joined as shown in the figure. The equivalent resistance between $M$ and $N$ is