Cells in Series & Parallel 🔋
When you hook up batteries (cells) together, you create equivalent sources that behave like one “super-cell.” Knowing how their emf (ε) and internal resistance (r) add up lets you predict the current and voltage in any circuit.
1. Single-Cell Refresher ⚡
The current drawn from one cell with an external resistor R is
\( I = \dfrac{\varepsilon}{R + r} \)
Set R to zero (a short-circuit) and you get the maximum current the cell can safely supply:
\( I_{\text{max}} = \dfrac{\varepsilon}{r} \)
Real cells can’t handle this huge current for long, so always keep some external resistance in the loop! 💡
2. Cells in Series ➕➕
Imagine placing the positive plate of one cell against the negative plate of the next (like stacking coins). The same current I flows through each cell.
2.1 Two cells
The voltage between terminals A and C equals
\( V_{AC} = \varepsilon_1 + \varepsilon_2 – I\,(r_1 + r_2) \)
We can replace both cells by one equivalent cell:
\( \boxed{\varepsilon_{\text{eq}} = \varepsilon_1 + \varepsilon_2},\qquad \boxed{r_{\text{eq}} = r_1 + r_2} \)
🔄 Change the connection (negative tied to negative) and the second emf reverses sign:
\( \varepsilon_{\text{eq}} = \varepsilon_1 – \varepsilon_2 \;\;(\varepsilon_1 > \varepsilon_2) \)
2.2 n cells in series
- Emf adds: \( \varepsilon_{\text{eq}} = \displaystyle\sum_{i=1}^{n} \varepsilon_i \)
- Resistance adds: \( r_{\text{eq}} = \displaystyle\sum_{i=1}^{n} r_i \)
If current leaves a cell’s negative plate, treat that ε as negative in the sum. 🚦
3. Cells in Parallel 🔀
Parallel means all the positives meet at one junction, all the negatives at another. Currents split inside the batteries but reunite in the external circuit.
3.1 Two cells
The total current leaving the junction is
\( I = I_1 + I_2 \)
After some algebra you can write the common terminal voltage as
\( V = \frac{\varepsilon_1 r_2 + \varepsilon_2 r_1}{\,r_1 + r_2\,} \;-\; I\,\frac{r_1 r_2}{\,r_1 + r_2\,} \)
So the single equivalent cell must obey
\( \boxed{\varepsilon_{\text{eq}} = \dfrac{\varepsilon_1 r_2 + \varepsilon_2 r_1}{r_1 + r_2}},\qquad \boxed{r_{\text{eq}} = \dfrac{r_1 r_2}{r_1 + r_2}} \)
3.2 Two-cell shortcut
Many prefer these handy forms:
- \( \dfrac{1}{r_{\text{eq}}} = \dfrac{1}{r_1} + \dfrac{1}{r_2} \)
- \( \dfrac{\varepsilon_{\text{eq}}}{r_{\text{eq}}} = \dfrac{\varepsilon_1}{r_1} + \dfrac{\varepsilon_2}{r_2} \)
3.3 n cells in parallel
- \( \displaystyle \frac{1}{r_{\text{eq}}} = \sum_{i=1}^{n} \frac{1}{r_i} \)
- \( \displaystyle \frac{\varepsilon_{\text{eq}}}{r_{\text{eq}}} = \sum_{i=1}^{n} \frac{\varepsilon_i}{r_i} \)
Reverse the polarity of any branch and its emf enters the sum with a minus sign—but the resistance part stays positive.
4. High-Yield Ideas for NEET 🎯
- Series boost voltage: Adding cells in series raises ε linearly, while r also piles up—great for high-voltage, low-current needs.
- Parallel cuts resistance: Parallel reduces the net r and gives a larger current reserve without changing ε much.
- Sign convention matters: Flipping one cell reverses its ε in the series sum; you can even use one cell to “cancel” part of another’s emf for fine control.
- Short-circuit caution: \( I_{\text{max}} = \varepsilon/r \) sets a hard limit—exceed it and you risk damaging the cell.
- Quick formulas: Remember “S-add, P-reciprocate” (Series: add ε and r; Parallel: add 1/r and ε/r). A 5-second check can save precious exam time!
✨ Happy studying! Keep practicing, and may your circuits always close (safely) ✨
