Equivalent EMF Calculation for Battery Combinations

This problem involves calculating the equivalent EMF of a complex battery arrangement, starting with a parallel battery combination and then connecting a third battery in a series connection. Understanding how to combine EMFs and internal resistances is crucial for solving such electrical circuit problems.

Parallel Battery Combination Analysis

First, let's analyze the two batteries, E₁ and E₂, connected in parallel. Their positive terminals are connected to point A and negative terminals to point B. This setup implies they are aiding each other.

Given values for the parallel batteries:

• Battery E₁: EMF ($E_1$) = 6 V, Internal Resistance ($r_1$) = 0.5 Ω
• Battery E₂: EMF ($E_2$) = 12 V, Internal Resistance ($r_2$) = 1.0 Ω

For batteries connected in parallel, the formula for the equivalent EMF ($E_{parallel}$) and equivalent internal resistance ($r_{parallel}$) are:

Equivalent Internal Resistance formula:

\[ \frac{1}{r_{parallel}} = \frac{1}{r_1} + \frac{1}{r_2} \]

Equivalent EMF formula:

\[ E_{parallel} = \frac{\frac{E_1}{r_1} + \frac{E_2}{r_2}}{\frac{1}{r_1} + \frac{1}{r_2}} \]

Calculating Parallel Equivalent Resistance

Let's calculate the equivalent internal resistance ($r_{parallel}$) of the parallel battery combination:

\[ \frac{1}{r_{parallel}} = \frac{1}{0.5 \text{ } \Omega} + \frac{1}{1.0 \text{ } \Omega} \]

\[ \frac{1}{r_{parallel}} = 2 \text{ } \Omega^{-1} + 1 \text{ } \Omega^{-1} = 3 \text{ } \Omega^{-1} \]

\[ r_{parallel} = \frac{1}{3} \text{ } \Omega \]

Calculating Parallel Equivalent EMF

Now, let's calculate the equivalent EMF ($E_{parallel}$) for the parallel battery combination:

\[ E_{parallel} = \frac{\frac{6 \text{ V}}{0.5 \text{ } \Omega} + \frac{12 \text{ V}}{1.0 \text{ } \Omega}}{\frac{1}{0.5 \text{ } \Omega} + \frac{1}{1.0 \text{ } \Omega}} \]

\[ E_{parallel} = \frac{12 \text{ V/}\Omega + 12 \text{ V/}\Omega}{2 \text{ } \Omega^{-1} + 1 \text{ } \Omega^{-1}} \]

\[ E_{parallel} = \frac{24 \text{ V/}\Omega}{3 \text{ } \Omega^{-1}} \]

\[ E_{parallel} = 8 \text{ V} \]

So, the parallel combination of E₁ and E₂ effectively acts as a single battery with an EMF of 8 V and an internal resistance of $\frac{1}{3}$ Ω. For this combination, point A is the positive terminal and point B is the negative terminal.

Series Connection with the Third Battery

Next, a third battery, E₃, is connected in series with this parallel combination. Its positive terminal is connected to point B.

Given values for the third battery:

• Battery E₃: EMF ($E_3$) = 6 V, Internal Resistance ($r_3$) = $\frac{2}{3}$ Ω

Let's consider the polarity for the series connection:

• The parallel combination (E₁ || E₂) has its positive terminal at A and negative terminal at B. We can represent this as A + —[8V] — B −.
• Battery E₃ has its positive terminal connected to B. Let's call its negative terminal point C. So, it can be represented as B + —[6V] — C −.

When batteries are connected in series such that the negative terminal of one is connected to the positive terminal of the next (as is the case here, B acts as the negative for the parallel combination and positive for E₃), their EMFs add up. This is an aiding connection.

The overall configuration is A + —[Parallel Combo] — B + —[E₃] — C −.

The total equivalent EMF ($E_{total}$) for batteries in series (aiding) is the sum of their individual EMFs:

\[ E_{total} = E_{parallel} + E_3 \]

Total Equivalent EMF

Let's calculate the total equivalent EMF for the entire combination:

\[ E_{total} = 8 \text{ V} + 6 \text{ V} \]

\[ E_{total} = 14 \text{ V} \]

For completeness, let's also calculate the total equivalent internal resistance ($r_{total}$) for the series connection. In series, internal resistances simply add up:

\[ r_{total} = r_{parallel} + r_3 \]

\[ r_{total} = \frac{1}{3} \text{ } \Omega + \frac{2}{3} \text{ } \Omega \]

\[ r_{total} = \frac{3}{3} \text{ } \Omega = 1 \text{ } \Omega \]

However, the question specifically asks only for the equivalent EMF of this combination.

The equivalent EMF of the entire combination is 14 V.