1.8.12 · D5Electromagnetism
Question bank — Series and parallel capacitors — derivations
Look at the two wiring pictures before you start: they show what is physically shared in each case (the single idea every trap below turns on).
This page belongs to the parent derivations note — the formulas are derived there; here we only stress-test understanding.
True or false — justify
True or false: In a series combination the equivalent capacitance can be larger than one of the members.
False. Adding reciprocals makes bigger than any single , so drops below the smallest member — never above any of them.
True or false: In parallel, the biggest capacitor stores the most charge.
True. All share the same , and , so — the larger capacitance grabs proportionally more charge.
True or false: In series, the biggest capacitor stores the most charge.
False. In series every capacitor carries the identical charge (the middle plates are isolated islands that stay neutral). Size changes the voltage it takes, not its charge.
True or false: Adding one more capacitor in parallel always increases the total capacitance.
True. Parallel means ; each extra term is positive, so the sum can only grow (you added more plate area to hold charge).
True or false: Adding one more capacitor in series always decreases the total capacitance.
True. Each extra positive raises , pushing down (you stacked another gap, making it "thicker" and harder to fill).
True or false: Two identical capacitors in series give exactly half the capacitance of one.
True. , so — a clean, memorable special case.
True or false: In series, the voltage always splits equally between two capacitors.
False. It splits equally only when the capacitances are equal. Otherwise , so the smaller capacitor hogs the larger voltage.
True or false: Capacitors follow the same series/parallel rules as resistors.
False — they are the opposite. In resistor networks the series resistances add and the parallel ones take reciprocals; for capacitors the roles swap — add in parallel, reciprocals in series.
True or false: The equivalent capacitor stores the same total charge the network draws from the battery.
True — that is the very definition of equivalent, . It draws the same charge at the same terminal voltage.
Spot the error
"In series the battery pushes charge onto and a different charge onto ." — find the flaw.
The middle conductor (right plate of joined to left plate of ) is an isolated island. By charge conservation it started neutral and no charge can enter or leave it, so on one plate forces exactly on the next — hence , not different.
"For capacitors in series I just add them: ." — find the flaw.
That's the parallel rule wrongly transplanted from resistors. Series capacitors add reciprocals: .
"In parallel each capacitor gets a share of the total voltage that adds up to ." — find the flaw.
In parallel every capacitor feels the full (they sit across the same two nodes / equipotential wires). It's the charges that add up to the total, not the voltages.
"Series equivalent of a and a is ." — find the flaw.
The product-over-sum shortcut is , i.e. product on top. The student inverted it. Correct value is .
"Because the same charge flows in series, each capacitor stores the same energy." — find the flaw.
By the energy formula the stored energy is ; with common it scales as , so the smaller capacitor stores more energy. Same charge ≠ same energy.
"Inserting a dielectric into one capacitor in a series chain changes only that capacitor and nothing else." — find the flaw.
A dielectric raises that capacitor's , which lowers its term, so rises, the total charge rises, and every capacitor's voltage shifts. The whole chain re-balances.
Why questions
Why does "same voltage" follow automatically for parallel capacitors?
Both plates of each capacitor connect to the same two wires (nodes A and B). A wire is an equipotential, so the potential difference across every capacitor is forced to equal the same value — the battery voltage.
Why is the total charge in parallel the sum of the branch charges, ?
At node A the wire from the battery splits into the branches. By charge conservation the charge arriving each second must equal the charge fanning out into the branches, so the total delivered charge is exactly the branch charges added together: .
Why is it the reciprocals that add in series and not the capacitances?
Because in series the shared quantity is and voltages add: . Dividing by leaves . The reciprocal appears because sits in the denominator of .
Why does the smaller capacitor take the larger voltage in series?
With common to all, means . A small makes a big — the narrow "bucket" needs more pressure for the same charge.
Why can we legally add the voltage drops around a series loop to get the battery voltage?
Because of Kirchhoff's voltage law: potential is single-valued, so going once around the loop the drops must sum to the applied EMF, .
Why is the series equivalent always smaller than the smallest member?
Every extra reciprocal term only increases , so exceeds every individual , which means is below every individual — including the smallest.
Why does stacking capacitors in series behave like making a thicker dielectric gap?
Capacitance falls as the plate separation grows. Series capacitors put their gaps end-to-end, so the charge must cross more total "insulating distance", lowering the combined ability to store charge.
Edge cases
What is the equivalent when one capacitor in a parallel group has (an ideal open gap)?
It contributes to the sum , so it's as if it weren't there — the others determine everything. A zero-capacitance branch simply stores no charge.
What is the equivalent when one capacitor in a series chain has ?
Its term dominates the sum, so and . One "impossible to fill" element chokes the whole chain to nearly zero capacitance.
What happens to a series combination if one member becomes infinitely large (, effectively a plain wire)?
Its reciprocal vanishes from the sum, so that capacitor "disappears" and is set by the remaining members alone. It acts like a short across a filled bucket.
In a parallel bank, does adding a tiny capacitor noticeably change ?
Barely — it adds only its small value to the sum. Parallel is dominated by the largest member; small ones are near-negligible.
In a series chain, does adding a huge capacitor noticeably change ?
Barely — its reciprocal is tiny, so it drops out of the sum. Series is dominated by the smallest member; huge ones are near-negligible.
If two capacitors are equal, do series and parallel give the same answer?
No. For equal : parallel gives , series gives — a factor of four apart. Equality of the members does not collapse the two rules together.
What is for a single capacitor "in series" or "in parallel" with nothing?
Just itself. A sum or reciprocal-sum with one term returns that term — both formulas gracefully reduce to the trivial case.
Active recall
Recall Two-line self-test
- State one reason series charges must be equal — in physics, not by formula.
- Name the single member that dominates a parallel bank, and the one that dominates a series chain.
Answers ::: (1) The middle conductor between two series capacitors is isolated and stays net-neutral, forcing equal induced charge. (2) Parallel is ruled by the largest member; series by the smallest.