What Happens When You Connect Cells in Series and Parallel?

Cells in series and parallel combinations are fundamental concepts in understanding electric circuits. These configurations determine the total electromotive force (EMF), current capacity, and internal resistance of a battery system. A clear comprehension of these arrangements is essential for solving JEE Main and Advanced physics problems on current electricity.

Definition and Structure of Electric Cells

An electric cell is a device that converts chemical energy into electrical energy, generating a potential difference between its two terminals. Each cell consists of an anode and cathode, immersed in an electrolyte, allowing the movement of charges required for current flow.

The anode is the terminal through which current enters the circuit, while the cathode is the terminal where current exits to the external circuit. Cells are the basic units for establishing potential difference in electric circuits, and their arrangement impacts the resultant output.

Cells Connected in Series

When cells are connected in series, the positive terminal of one cell is linked to the negative terminal of the next. The same current passes through each cell, and the total EMF is the sum of individual EMFs. This combination is used when a higher voltage output is required.

If $n$ cells with EMFs $E_1, E_2, ..., E_n$ and internal resistances $r_1, r_2, ..., r_n$ are connected in series, the total EMF ($E_{\text{eq}}$) and total internal resistance ($r_{\text{eq}}$) are:

$E_{\text{eq}} = E_1 + E_2 + \ldots + E_n$

$r_{\text{eq}} = r_1 + r_2 + \ldots + r_n$

For $n$ identical cells, each with EMF $E$ and resistance $r$, connected to an external resistance $R$, the current ($I$) is given by:

$I = \dfrac{nE}{R + nr}$

Cells Connected in Parallel

In a parallel combination, the positive terminals of all cells are connected together, and so are the negative terminals. The voltage across each cell remains the same, but the total current is divided among them. This configuration is used to increase the current supplying capacity without changing the voltage.

For $n$ cells with identical EMF $E$ and internal resistance $r$, the equivalent EMF ($E_{\text{eq}}$) and internal resistance ($r_{\text{eq}}$) for parallel arrangement are:

$E_{\text{eq}} = E$

$r_{\text{eq}} = \dfrac{r}{n}$

If the cells are not identical, the general expressions become:

$\dfrac{1}{r_{\text{eq}}} = \dfrac{1}{r_1} + \dfrac{1}{r_2} + \ldots + \dfrac{1}{r_n}$

$E_{\text{eq}} = \dfrac{E_1/r_1 + E_2/r_2 + \ldots + E_n/r_n}{1/r_1 + 1/r_2 + \ldots + 1/r_n}$

Comparison of Series and Parallel Connection of Cells

In series connections, the total EMF increases with the number of cells, while the internal resistance also increases. In parallel connections, the voltage remains the same as a single cell, but the internal resistance decreases, allowing more current to be supplied to the external circuit.

For further understanding of the difference between these two types, refer to Difference Between Series And Parallel Circuits.

Derivation of Current in Series and Parallel Combination

In a series combination of $n$ identical cells (EMF $E$, internal resistance $r$) with external resistance $R$, the current is $I = \dfrac{nE}{R + nr}$. The total EMF adds, making it suitable when a higher voltage is necessary.

In parallel, for $n$ identical cells with the same resistance $r$ and external resistance $R$, current is $I = \dfrac{E}{R/n + r}$. This makes the parallel arrangement preferable for higher current output at the same voltage.

Applications and Practical Selection of Cell Combinations

Series combinations are used in devices requiring higher voltages, such as flashlights and electronic instruments. Parallel combinations are employed when increased current supply is needed, such as in backup power systems.

The choice between series and parallel arrangement depends on the application requirement for voltage and current. For analysis of current electricity in circuits, see Current Electricity.

Key Points on Resistance and Potential Difference in Series and Parallel Circuits

In a series circuit, the same current flows through each component, but the total resistance is the sum of individual resistances. The total voltage is divided among the components based on resistance.

In a parallel circuit, the voltage across all branches remains the same. The total resistance decreases as more branches are added, leading to increased total current.

The behavior of resistors in different configurations is further elaborated in Combination Of Capacitors, which provides useful analogies for understanding cell combinations.

Solved Example: Series and Parallel Connection of Cells

Consider three identical cells each with EMF $E=1.5\,\text{V}$ and internal resistance $r=1~\Omega$, connected to an external resistor $R=2~\Omega$.

• Series connection: Total EMF = $3 \times 1.5$ V = $4.5$ V
• Series connection: Total internal resistance = $3 \times 1~\Omega$ = $3~\Omega$
• Series current, $I = \dfrac{4.5}{2 + 3} = 0.9$ A
• Parallel connection: Total EMF = $1.5$ V
• Parallel connection: Total internal resistance = $1/3~\Omega$
• Parallel current, $I = \dfrac{1.5}{2 + 1/3} = 0.643$ A

In series combination, the current is higher due to increased voltage, but internal resistance also increases. In parallel arrangement, the current is lower but with reduced internal resistance and same voltage as a single cell.

Summary of Formulas for Cells in Series and Parallel

• Series: $E_{\text{eq}} = nE,\quad r_{\text{eq}} = nr$
• Parallel: $E_{\text{eq}} = E,\quad r_{\text{eq}} = r/n$
• Series, current: $I = \dfrac{nE}{R + nr}$
• Parallel, current: $I = \dfrac{E}{R/n + r}$

These formulas are fundamental for analyzing various electric circuits in both academic problems and practical applications. Detailed derivations can be found in advanced resources related to the topic at Cells In Series And Parallel.

Additional Practice and Related Concepts

Further exploration of circuit configurations and logical arrangements in physics can be enhanced by referring to concepts such as Basic Logic Gates and the functioning of devices like Transformer.