Cells in Series and Parallel — Revision Notes

⚡ 30-Second Revision

Series Connection: — $E_{eq} = \sum E_i$, $r_{eq} = \sum r_i$. Current $I = \frac{E_{eq}}{R + r_{eq}}$.
Parallel Connection (Identical Cells): — $E_{eq} = E$, $r_{eq} = r/n$. Current $I = \frac{E}{R + r/n}$.
Mixed Grouping ($m$ rows, $n$ cells/row): — $E_{eq} = nE$, $r_{eq} = nr/m$. Current $I = \frac{nE}{R + nr/m}$.
Maximum Current (Mixed Grouping): — $R = nr/m$.
Terminal Voltage: — $V = E - Ir$ (discharging).
Internal Resistance ($r$): — Always adds in series, decreases in parallel.

2-Minute Revision

Cells in series and parallel are fundamental circuit configurations. In a series connection, cells are linked end-to-end, typically positive to negative. This arrangement sums up the individual EMFs to give a higher total voltage ($E_{eq} = nE$ for $n$ identical cells) and also sums up their internal resistances ($r_{eq} = nr$).

Series is chosen when a higher voltage is required. The total current is $I = \frac{nE}{R + nr}$. In a parallel connection, all positive terminals are joined, and all negative terminals are joined.

For identical cells, the equivalent EMF remains the same as a single cell ($E_{eq} = E$), but the equivalent internal resistance significantly decreases ($r_{eq} = r/n$). Parallel connections are preferred for increasing current capacity and reducing overall internal resistance.

Mixed grouping combines both, with $m$ parallel rows each containing $n$ series cells, yielding $E_{eq} = nE$ and $r_{eq} = nr/m$. A key condition for maximum current from a mixed grouping is when the external resistance $R$ equals the equivalent internal resistance ($R = nr/m$).

Always remember that internal resistance causes a voltage drop, making terminal voltage ($V = E - Ir$) less than EMF when discharging.

5-Minute Revision

Revisiting cells in series and parallel is crucial for NEET. Start with the basics: every real cell has an EMF ($E$) and an internal resistance ($r$). The terminal voltage ($V$) is $E - Ir$ when discharging.

Series Connection: When $n$ identical cells are connected in series (positive to negative), their EMFs add up: $E_{eq} = nE$. Their internal resistances also add up: $r_{eq} = nr$. The total current through an external resistor $R$ is $I = \frac{nE}{R + nr}$. If cells are connected with opposing polarities, their EMFs subtract (e.g., $E_{eq} = E_1 - E_2$), but internal resistances still add. Series connections are for higher voltage requirements.

Parallel Connection: When $n$ identical cells are connected in parallel (all positives together, all negatives together), the equivalent EMF remains the same as a single cell: $E_{eq} = E$. The equivalent internal resistance decreases: $r_{eq} = \frac{r}{n}$.

The total current is $I = \frac{E}{R + r/n}$. Parallel connections are for higher current capacity and lower effective internal resistance. For non-identical cells in parallel, internal currents flow, and the equivalent EMF is a weighted average: $E_{eq} = r_{eq} \sum \frac{E_i}{r_i}$, where $\frac{1}{r_{eq}} = \sum \frac{1}{r_i}$.

Mixed Grouping: This involves $m$ parallel rows, each with $n$ identical cells in series. The equivalent EMF is $E_{eq} = nE$ (EMF of one series row). The equivalent internal resistance is $r_{eq} = \frac{nr}{m}$ (internal resistance of one series row divided by number of parallel rows).

The current is $I = \frac{nE}{R + nr/m}$. A common NEET question involves finding the condition for maximum current, which occurs when the external resistance $R$ equals the equivalent internal resistance: $R = \frac{nr}{m}$.

Example: Two cells ($E=1.5,\text{V}, r=0.1,Omega$) in series, connected to $R=2.8,Omega$. $E_{eq} = 2 \times 1.5 = 3,\text{V}$. $r_{eq} = 2 \times 0.1 = 0.2,Omega$. $I = \frac{3}{2.8 + 0.2} = \frac{3}{3} = 1,\text{A}$.

Example: Two cells ($E=1.5,\text{V}, r=0.1,Omega$) in parallel, connected to $R=1.45,Omega$. $E_{eq} = 1.5,\text{V}$. $r_{eq} = \frac{0.1}{2} = 0.05,Omega$. $I = \frac{1.5}{1.45 + 0.05} = \frac{1.5}{1.5} = 1,\text{A}$.

Remember to always consider internal resistance and the specific configuration when solving problems.

Prelims Revision Notes

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Single Cell: — EMF ($E$) is open-circuit voltage. Internal resistance ($r$) causes terminal voltage $V = E - Ir$ (discharging) or $V = E + Ir$ (charging). Power dissipated internally is $I^2r$.
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Cells in Series:

* Aiding: $n$ identical cells ($E, r$) in series: $E_{eq} = nE$, $r_{eq} = nr$. Current $I = \frac{nE}{R + nr}$. * Opposing: If $m$ cells out of $n$ are reversed: $E_{eq} = (n-2m)E$, $r_{eq} = nr$. Internal resistances always add. * Advantage: Higher voltage. * Disadvantage: If one cell fails, circuit breaks. Current limited by $E/r$ if $R \ll nr$.

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Cells in Parallel:

* **Identical Cells ($E, r$):** $n$ cells in parallel: $E_{eq} = E$, $r_{eq} = r/n$. Current $I = \frac{E}{R + r/n}$. * **Non-identical Cells ($E_i, r_i$):** $\frac{1}{r_{eq}} = \sum \frac{1}{r_i}$, $E_{eq} = r_{eq} \sum \frac{E_i}{r_i}$. Internal currents flow if EMFs differ. * Advantage: Higher current capacity, lower effective internal resistance, redundancy (if one fails, others work). * Disadvantage: Voltage not increased. Inefficient with non-identical cells.

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Mixed Grouping: — $m$ parallel rows, each with $n$ identical cells ($E, r$) in series.

* $E_{eq} = nE$. * $r_{eq} = \frac{nr}{m}$. * Current $I = \frac{nE}{R + nr/m}$. * Condition for Maximum Current/Power: $R = r_{eq} = \frac{nr}{m}$.

1
Key Formulas:

* $I = \frac{E_{eq}}{R + r_{eq}}$ * Power delivered to external load: $P = I^2R = \left(\frac{E_{eq}}{R + r_{eq}}\right)^2 R$ * Power dissipated internally: $P_{internal} = I^2 r_{eq}$

Vyyuha Quick Recall

S.V.A.P.C.I.R.

Series: Voltage Adds, Polarity matters for EMF, Current is same, Internal Resistance adds.

Parallel: Current Increases, Resistance decreases, Voltage is same (for identical cells), Internal Resistance divides.