1.8.11 · D5Electromagnetism
Question bank — Capacitance — parallel plate derivation, cylindrical, spherical
True or false — justify
Capacitance of a given capacitor doubles if you double the charge on it.
False. but doubling doubles the field and hence doubles in exact step, so the ratio is unchanged — is fixed by geometry and dielectric alone.
A capacitor with no charge on it has zero capacitance.
False. Capacitance is evaluated as a ratio that survives the limit: it's a property of the shape, present even when , just as a cup's volume exists when the cup is empty.
The two plates of a real capacitor carry the same charge.
False. They carry equal and opposite charge and ; "the charge on the capacitor" means the magnitude on the positive plate, not the net (which is zero).
Filling the gap with a dielectric of constant always increases the capacitance.
True. The polarized dielectric partially cancels the field, so the same produces a smaller , raising by the factor (see Dielectrics and Polarization).
For a spherical capacitor you can find by plugging the outer radius into .
False. here is the difference of potential between the two shells, ; a single is the potential of one lone charge, not a two-conductor gap.
Cylindrical capacitance scales as , just like the plate scales as .
False. The coaxial field falls as , so integrating gives a logarithm: . Only the plate's uniform field integrates to a plain gap .
An isolated single sphere has no capacitance because there's only one conductor.
False. Its "second plate" is infinity; taking in gives the finite value .
At fixed charge , pushing parallel plates closer together stores more energy.
False. Smaller means larger , and falls; energy actually decreases, which is why the plates attract (moving them apart is what costs work).
Spot the error
"Field between plates is , because that's a single charged sheet."
The factor-2 error. Between the two oppositely-charged plates the two sheet fields add, giving ; the pillbox in Gauss's Law already bakes this in because flux only exits the inner face.
" from − plate to + plate gives the capacitor voltage."
Sign slip. Going against the field (− to +) gives a negative number; you must integrate from the + plate to the − plate (along the field) to get the positive magnitude .
"For the coaxial cable I set the Gaussian cylinder area to ."
Wrong geometry's area. A cylinder's curved side has area ; the belongs to a sphere. Using the wrong area destroys the field.
" works even when is comparable to the plate width."
Ignores fringing. The formula assumes an ideal uniform field (); when is large the fringe fields at the edges add capacitance, so the formula underestimates .
"Doubling the plate area doubles the field for the same charge."
Backwards. For fixed , doubling halves and hence halves ; the field weakens, which is exactly why grows with .
"In the spherical derivation I use ."
Wrong antiderivative. (that's the field); is what the cylindrical field gives.
Why questions
Why does always cancel in every capacitance derivation?
Because the field is linear in charge ( from Gauss's law) and too, so the ratio has no left — proof capacitance is pure geometry.
Why is the parallel-plate field independent of position inside the gap?
The pillbox Gaussian area cancels on both sides of Gauss's law, leaving with no or in it — the fingerprint of an infinite-plane field.
Why does the coaxial capacitor produce a logarithm but the sphere a ?
Their fields fall off differently: integrates to , while integrates to ; the geometry of the Gaussian surface (its area's -power) dictates the field's -power and thus the integral.
Why do we say capacitance measures how "generous" a conductor pair is?
A high- pair swallows a lot of charge before its voltage (its "push-back") rises much — means large gives large per volt, exactly the intuition in Electric Potential.
Why does a thin high- film beat a huge air gap for making big capacitors?
rewards small and large ; a rolled thin film with a dielectric maximizes both, so nanofarads become microfarads in a tiny package.
Why does making the outer coax conductor bigger (larger ) decrease the capacitance?
A bigger increases , which sits in the denominator, so shrinks — a wider gap means more voltage to climb per unit charge.
Edge cases
What happens to the spherical capacitance as (shells almost touching)?
: a vanishing gap makes blow up, matching the plate intuition that tiny separation gives huge capacitance.
What is the capacitance of a spherical capacitor as ?
It tends to — the finite capacitance of an isolated sphere, since the outer "plate" retreats to infinity but the inner sphere still holds charge at finite voltage.
What happens to any capacitor's field and voltage as ?
Both go to zero together, but their ratio stays ; the geometry-defined capacitance is the well-defined limit even though numerator and denominator both vanish.
If two identical capacitors are wired in series, is the combined capacitance bigger or smaller than one alone?
Smaller — series capacitors add reciprocally, giving half the value here, because the same charge must climb two voltage gaps (see Capacitors in Series and Parallel).
For the coaxial cable, what does physically mean for a real transmission line?
Every extra metre of cable adds the same fixed capacitance, so a long line stores/couples proportionally more charge per volt — a key spec in Coaxial Cable Transmission Lines.
What limits how much energy a capacitor can hold before it fails?
The dielectric's breakdown field: raising raises until the insulator ionizes and sparks across, so and the material — not the formula — set the ceiling.
Recall One-line self-test
Cover every answer, then answer: does depend on , on , or on neither? On neither — only geometry and dielectric. ::: Correct if you also said the reason: and both scale with , so the ratio is -free.