Calculate Total Current Between Terminals in Resistor Network

To find the current between terminals $A$ and $B$, we can analyze the given resistor network branch by branch.

1. Analysis of the Source Branches (Top Three Branches)

Each of the upper three parallel branches consists of three $5\text{ V}$ DC sources and three $3\,\Omega$ resistors connected in series.

For any one of the top three branches:

• Total EMF of one branch ($E_b$): $E_b = 5\text{ V} + 5\text{ V} + 5\text{ V} = 15\text{ V}$

• Total resistance of one branch ($r_b$): $r_b = 3\,\Omega + 3\,\Omega + 3\,\Omega = 9\,\Omega$

Since these three identical branches are connected in parallel, their equivalent internal EMF ($E_{\text{eq}}$) and equivalent internal resistance ($r_{\text{eq}}$) are: $E_{\text{eq}} = 15\text{ V}$ $r_{\text{eq}} = \frac{r_b}{3} = \frac{9\,\Omega}{3} = 3\,\Omega$

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2. Analysis of the Load Branch (Bottom Branch)

The bottom branch connected between terminals $A$ and $B$ contains three $3\,\Omega$ resistors in series without any voltage source.

• Total load resistance ($R_L$): $R_L = 3\,\Omega + 3\,\Omega + 3\,\Omega = 9\,\Omega$

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3. Calculation of Current

The total circuit can be modeled as an equivalent voltage source of $E_{\text{eq}} = 15\text{ V}$ with internal resistance $r_{\text{eq}} = 3\,\Omega$ supplying current to the load resistor $R_L = 9\,\Omega$ between terminals $A$ and $B$.

Using Ohm's law, the current $I$ flowing between terminals $A$ and $B$ is: $I = \frac{E_{\text{eq}}}{r_{\text{eq}} + R_L}$

Substitute the values into the equation: $I = \frac{15\text{ V}}{3\,\Omega + 9\,\Omega} = \frac{15}{12}\text{ A} = 1.25\text{ A}$

Correct Option: B ($1.25$)