Internal resistance, potential difference and emf of a cell, combinations of cells in series and parallel

Internal Resistance of a Cell

Every cell, whether it's a simple voltaic cell or a more complex battery, has an internal resistance. This resistance arises from the materials within the cell itself, such as the electrolyte and the electrodes. When current flows through the cell, it encounters this opposition, leading to a voltage drop within the cell. This internal resistance is a crucial factor in determining the actual voltage delivered by the cell to an external circuit.

The internal resistance ($r$) of a cell depends on several factors:

• The nature of the electrolyte: A more resistive electrolyte will lead to a higher internal resistance.
• The distance between the electrodes: Greater distance means more electrolyte for the current to traverse, thus increasing resistance.
• The surface area of the electrodes: A larger surface area can sometimes decrease internal resistance by providing more pathways for current.
• The concentration of the electrolyte: Changes in concentration can affect the electrolyte's conductivity and thus its resistance.

We can model a real cell as an ideal electromotive force (emf) source in series with a resistor representing its internal resistance.

Electromotive Force (emf) and Potential Difference (PD)

It's essential to distinguish between electromotive force (emf) and potential difference (PD). While both are measured in volts, they represent different concepts related to a cell.

Electromotive Force (emf)

The electromotive force (emf), denoted by the Greek letter epsilon ($\epsilon$), is the total energy supplied by the cell per unit charge that passes through it. It represents the work done by the non-electrical forces (like chemical reactions) inside the cell in moving a unit positive charge from the lower potential terminal to the higher potential terminal.

When there is no current flowing through the cell (i.e., the external circuit is open, or the switch is open), the potential difference across the terminals of the cell is equal to its emf. This is because there is no voltage drop across the internal resistance.

Mathematically, emf is defined as:

$$ \epsilon = \frac{W}{q} $$

where $W$ is the work done by the cell in moving charge $q$.

Potential Difference (PD)

The potential difference (PD) across the terminals of a cell is the actual voltage available to drive current through the external circuit. When a current ($I$) flows through the cell, there is a voltage drop across its internal resistance ($r$). This voltage drop is given by $Ir$.

Therefore, the potential difference ($V$) across the terminals of the cell when current is flowing is the emf minus the voltage drop across the internal resistance:

$$ V = \epsilon - Ir $$

This equation is fundamental. It shows that the terminal voltage ($V$) is always less than the emf ($\epsilon$) when the cell is discharging (supplying current). If the cell is being charged (current flowing into the positive terminal), the equation becomes $V = \epsilon + Ir$.

When the external circuit is open, $I = 0$, so $V = \epsilon$.

Ohm's Law for a Complete Circuit

We can combine Ohm's law with the concept of internal resistance to analyze a complete circuit. Consider a cell with emf $\epsilon$ and internal resistance $r$ connected to an external resistor $R$. The total resistance in the circuit is the sum of the external resistance and the internal resistance: $R_{total} = R + r$.

According to Ohm's law applied to the entire circuit, the current ($I$) flowing is given by:

$$ I = \frac{\text{Total EMF}}{\text{Total Resistance}} = \frac{\epsilon}{R + r} $$

From this, we can also express the terminal potential difference $V$ in terms of the external resistance $R$:

Since $V = IR$ (Ohm's law for the external resistor), substituting the expression for $I$:

$$ V = \left(\frac{\epsilon}{R + r}\right) R = \frac{\epsilon R}{R + r} $$

This formula shows how the terminal voltage depends on the external resistance. As $R$ becomes very large (approaching infinity, i.e., an open circuit), $V$ approaches $\epsilon$. As $R$ becomes very small (approaching zero, i.e., a short circuit), $V$ approaches zero, and the current becomes $I = \epsilon / r$, which can be very large and potentially damaging.

Combinations of Cells

Often, a single cell may not provide sufficient voltage or current for a particular application. In such cases, multiple cells are combined. There are two primary ways to combine cells: in series and in parallel.

Cells in Series

When cells are connected in series, the positive terminal of one cell is connected to the negative terminal of the next cell, and so on. This arrangement is used to increase the total emf of the combination.

Consider $n$ identical cells, each with emf $\epsilon$ and internal resistance $r$, connected in series.

• Total emf: The emfs add up. The total emf of the combination is $n\epsilon$.
• Total internal resistance: The internal resistances also add up. The total internal resistance is $nr$.

If this combination is connected to an external resistor $R$, the current flowing in the circuit is:

$$ I_{\text{series}} = \frac{\text{Total emf}}{\text{Total resistance}} = \frac{n\epsilon}{R + nr} $$

Example: If you have three 1.5V cells, each with an internal resistance of 0.5 ohms, and you connect them in series to a 3-ohm resistor:

• Total emf = $3 \times 1.5V = 4.5V$
• Total internal resistance = $3 \times 0.5\Omega = 1.5\Omega$
• Current = $\frac{4.5V}{3\Omega + 1.5\Omega} = \frac{4.5V}{4.5\Omega} = 1A$

This arrangement is effective when the external resistance $R$ is much larger than the total internal resistance $nr$. If $R \gg nr$, then $I_{\text{series}} \approx \frac{n\epsilon}{R}$, which is $n$ times the current from a single cell.

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 primarily used to increase the total current capacity or to maintain a stable voltage when a large current is drawn. For parallel combination to be effective, the cells should be identical (same emf and same internal resistance).

Consider $n$ identical cells, each with emf $\epsilon$ and internal resistance $r$, connected in parallel.

• Total emf: The emf of the combination is the same as that of a single cell, $\epsilon$. This is because connecting identical cells in parallel does not increase the potential difference.
• Total internal resistance: The internal resistances are connected in parallel. The equivalent internal resistance is $\frac{r}{n}$.

If this combination is connected to an external resistor $R$, the current flowing in the circuit is:

$$ I_{\text{parallel}} = \frac{\text{Total emf}}{\text{Total resistance}} = \frac{\epsilon}{R + \frac{r}{n}} $$

Example: Using the same three 1.5V cells (each with 0.5 ohms internal resistance) connected in parallel to a 3-ohm resistor:

• Total emf = $1.5V$
• Total internal resistance = $\frac{0.5\Omega}{3} \approx 0.167\Omega$
• Current = $\frac{1.5V}{3\Omega + 0.167\Omega} = \frac{1.5V}{3.167\Omega} \approx 0.474A$

This arrangement is effective when the external resistance $R$ is much smaller than the equivalent internal resistance $\frac{r}{n}$. If $R \ll \frac{r}{n}$, then $I_{\text{parallel}} \approx \frac{\epsilon}{R/n} = \frac{n\epsilon}{R}$. However, the more common scenario is when $R$ is comparable or larger than $r/n$. The key benefit here is that the total current drawn from the combination ($I_{\text{parallel}}$) is distributed among the $n$ cells, so each cell supplies only $I_{\text{parallel}}/n$ current, reducing the load on each individual cell and prolonging its life.

Mixed Combination of Cells

Sometimes, cells are connected in a combination of series and parallel to achieve specific voltage and current requirements. This involves arranging $m$ rows, with each row containing $n$ identical cells connected in series. Then, these $m$ rows are connected in parallel.

In such a mixed combination:

• Each series row has an emf of $n\epsilon$ and an internal resistance of $nr$.
• There are $m$ such rows connected in parallel.
• The total emf of the combination is $n\epsilon$ (same as one row).
• The total internal resistance of the combination is $\frac{nr}{m}$ (since $m$ rows are in parallel).

If this mixed combination is connected to an external resistor $R$, the current flowing in the circuit is:

$$ I_{\text{mixed}} = \frac{\text{Total emf}}{\text{Total resistance}} = \frac{n\epsilon}{R + \frac{nr}{m}} $$

For maximum current to be delivered to the external resistance $R$, the condition $R = \frac{nr}{m}$ must be met. In this case, the external resistance is equal to the total internal resistance of the combination.

Substituting $R = \frac{nr}{m}$ into the formula for $I_{\text{mixed}}$:

$$ I_{\text{max}} = \frac{n\epsilon}{\frac{nr}{m} + \frac{nr}{m}} = \frac{n\epsilon}{2 \frac{nr}{m}} = \frac{m n \epsilon}{2 nr} = \frac{m\epsilon}{2r} $$

This condition ensures that the power delivered to the external circuit is maximized (maximum power transfer theorem).

Non-identical Cells

If cells are not identical, connecting them in series is straightforward: the total emf is the algebraic sum of individual emfs, and the total internal resistance is the sum of individual internal resistances.

However, connecting non-identical cells in parallel is generally not recommended. It can lead to unequal current distribution, and in some cases, a cell with a lower emf might get charged by a cell with a higher emf, which is inefficient and can damage the cells. If they must be connected in parallel, the calculation of equivalent emf and resistance becomes more complex, involving weighted averages and specific formulas that are usually beyond the scope of introductory physics for competitive exams. For exam purposes, assume cells are identical unless stated otherwise for parallel or mixed combinations.

Summary Table of Combinations

Combination	Total emf	Total Internal Resistance	Current (for external R)
Single Cell	$\epsilon$	$r$	$I = \frac{\epsilon}{R+r}$
$n$ Identical Cells in Series	$n\epsilon$	$nr$	$I_{\text{series}} = \frac{n\epsilon}{R+nr}$
$n$ Identical Cells in Parallel	$\epsilon$	$\frac{r}{n}$	$I_{\text{parallel}} = \frac{\epsilon}{R+\frac{r}{n}}$
$m$ Rows (each $n$ series cells) in Parallel	$n\epsilon$	$\frac{nr}{m}$	$I_{\text{mixed}} = \frac{n\epsilon}{R+\frac{nr}{m}}$