Calculating Current in a Parallel Cell Circuit

The problem asks us to find the current flowing through an external resistance when it is connected to a parallel combination of two cells with different electromotive forces (emfs) and internal resistances. To solve this, we can first find the equivalent emf and equivalent internal resistance of the parallel combination of cells. Then, we can use Ohm's law for the entire circuit to find the total current.

Given Information:

• Emf of the first cell, $E_1 = 2\text{ V}$
• Internal resistance of the first cell, $r_1 = 1\text{ }\Omega$
• Emf of the second cell, $E_2 = 1\text{ V}$
• Internal resistance of the second cell, $r_2 = 2\text{ }\Omega$
• External resistance, $R = \frac{4}{3}\text{ }\Omega$

Method: Equivalent Source Approach

When cells are connected in parallel, the equivalent internal resistance ($r_{eq}$) and equivalent emf ($E_{eq}$) can be calculated using specific formulas.

Calculating Equivalent Internal Resistance ($r_{eq}$)

For two cells with internal resistances $r_1$ and $r_2$ connected in parallel, the reciprocal of the equivalent internal resistance is the sum of the reciprocals of individual internal resistances:

$\frac{1}{r_{eq}} = \frac{1}{r_1} + \frac{1}{r_2}$

Alternatively, we can use the formula:

$r_{eq} = \frac{r_1 r_2}{r_1 + r_2}$

Let's calculate $r_{eq}$ using the given values:

$r_{eq} = \frac{(1\text{ }\Omega) \times (2\text{ }\Omega)}{1\text{ }\Omega + 2\text{ }\Omega} = \frac{2\text{ }\Omega^2}{3\text{ }\Omega} = \frac{2}{3}\text{ }\Omega$

So, the equivalent internal resistance of the parallel combination is $\frac{2}{3}\text{ }\Omega$.

Calculating Equivalent EMF ($E_{eq}$)

For two cells with emfs $E_1$, $E_2$ and internal resistances $r_1$, $r_2$ connected in parallel, the equivalent emf is given by the formula:

$E_{eq} = \frac{\frac{E_1}{r_1} + \frac{E_2}{r_2}}{\frac{1}{r_1} + \frac{1}{r_2}}$

We can simplify the denominator $\left(\frac{1}{r_1} + \frac{1}{r_2}\right)$ using the $r_{eq}$ calculation, as $\frac{1}{r_1} + \frac{1}{r_2} = \frac{1}{r_{eq}}$.

Substituting the given values into the formula for $E_{eq}$:

$E_{eq} = \frac{\frac{2\text{ V}}{1\text{ }\Omega} + \frac{1\text{ V}}{2\text{ }\Omega}}{\frac{1}{1\text{ }\Omega} + \frac{1}{2\text{ }\Omega}} = \frac{2 + 0.5}{1 + 0.5} = \frac{2.5}{1.5}$

To work with fractions:

$E_{eq} = \frac{\frac{5}{2}}{\frac{3}{2}} = \frac{5}{2} \times \frac{2}{3} = \frac{5}{3}\text{ V}$

So, the equivalent emf of the parallel combination is $\frac{5}{3}\text{ V}$.

Calculating Total Current ($I$)

Now we have a single equivalent source with emf $E_{eq} = \frac{5}{3}\text{ V}$ and internal resistance $r_{eq} = \frac{2}{3}\text{ }\Omega$ connected to an external resistance $R = \frac{4}{3}\text{ }\Omega$. The total current ($I$) flowing through the external resistance is given by Ohm's Law for the entire circuit:

$I = \frac{E_{eq}}{R + r_{eq}}$

Substituting the calculated values:

$I = \frac{\frac{5}{3}\text{ V}}{\frac{4}{3}\text{ }\Omega + \frac{2}{3}\text{ }\Omega} = \frac{\frac{5}{3}\text{ V}}{\frac{4+2}{3}\text{ }\Omega} = \frac{\frac{5}{3}\text{ V}}{\frac{6}{3}\text{ }\Omega} = \frac{\frac{5}{3}\text{ V}}{2\text{ }\Omega}$

$I = \frac{5}{3} \times \frac{1}{2}\text{ A} = \frac{5}{6}\text{ A}$

Conclusion

The current through the external resistance is $\frac{5}{6}\text{ A}$.

Revision Table: Electric Circuits and Cells

Additional Information on Cell Combinations

Combining cells can increase the total emf or decrease the effective internal resistance, depending on the connection.

• Cells in Series: When cells are connected in series such that the positive terminal of one is connected to the negative terminal of the next, their emfs add up (if connected in the same polarity) and their internal resistances also add up.
  • $E_{eq} = E_1 + E_2 + ...$
  • $r_{eq} = r_1 + r_2 + ...$
  This arrangement is useful when a higher voltage is needed.
• Cells in Parallel: When cells are connected in parallel, their positive terminals are connected together, and their negative terminals are connected together. This arrangement is typically used to provide a higher current capacity or when cells have different emfs. The equivalent emf is calculated as shown above, and the equivalent internal resistance is the reciprocal of the sum of reciprocals. This reduces the effective internal resistance of the source.
• Kirchhoff's Laws: For more complex circuits with multiple sources that cannot be easily reduced to equivalent sources, Kirchhoff's laws (Junction Rule and Loop Rule) can be used to set up equations and solve for unknown currents and voltages.

Understanding how to combine cells in series and parallel is fundamental to analyzing more complex electrical circuits containing multiple power sources.