Capacitors
A parallel-plate capacitor stores equal and opposite plate charge and electric-field energy. Its capacitance is set by geometry and dielectric material, while charge, voltage, field and energy respond differently depending on what is held fixed.
Capacitors: quick answer
A parallel-plate capacitor stores equal and opposite plate charge and electric-field energy. Its capacitance is set by geometry and dielectric material, while charge, voltage, field and energy respond differently depending on what is held fixed. Capacitor questions are controlled-variable questions. Capacitance itself is set by geometry and dielectric material, but after a change you must know whether the capacitor is isolated (free plate charge fixed) or connected to a voltage source (potential difference fixed). The same geometry can produce different changes in Q, V, E and U depending on that constraint.
Only parallel-plate capacitor analysis is required here; ignore edge effects unless explicitly stated. Before predicting a dielectric or geometry change, state whether the capacitor is isolated (Q fixed) or connected to a source that keeps ΔV fixed.
What you should be able to explain and do
Use these as closed-note retrieval prompts before looking at the worked examples.
Relate charge, voltage and capacitance without treating C as dependent on instantaneous Q or V.
Predict how plate area, separation and dielectric constant change parallel-plate capacitance.
Connect uniform plate field to charge density and potential difference.
Compare isolated and source-connected capacitor changes, including stored energy.
Key terms you need for this lesson
Definitions stay beside the model so a reader never has to leave the lesson to understand a prerequisite term.
farad
farad is one coulomb per volt, the SI unit of capacitance.
plate area
plate area is the overlapping conducting area A that increases parallel-plate capacitance when other quantities are fixed.
plate separation
plate separation is distance d between plates; larger d lowers ideal parallel-plate capacitance.
dielectric
dielectric is an insulating material placed between plates that polarizes in the applied field.
isolated capacitor
isolated capacitor is a capacitor disconnected from a charge-transfer path, so its free plate charge remains fixed in the ideal model.
source-connected capacitor
source-connected capacitor is a capacitor kept connected to an ideal voltage source that maintains a fixed potential difference.
Know what each relationship means and when it applies
The equation, variables, vector/scalar nature, model condition and physical anchor belong together.
| Relationship / model | Variables & units | When to use it | Physical anchor |
|---|---|---|---|
C = Q/|ΔV| | C capacitance (F); Q magnitude of charge on either plate (C); |ΔV| plate potential difference (V) | definition of capacitance | For a fixed capacitor, doubling Q doubles |ΔV|; C itself does not change. |
C = κε₀A/d; E = |ΔV|/d = Q/(κε₀A) | C capacitance (F); κ dielectric constant; ε₀ vacuum permittivity; A plate area (m²); d plate separation (m); E uniform central field magnitude (N/C) | ideal parallel plates with d much smaller than plate dimensions and edge effects neglected | Increasing plate area raises C; increasing separation lowers C; inserting a dielectric raises C and reduces the field for fixed free plate charge. |
U_C = ½Q|ΔV| = Q²/(2C) = ½C(ΔV)² | U_C energy stored in capacitor/electric field (J); Q charge magnitude (C); C capacitance (F); ΔV voltage (V) | ideal capacitor energy; choose the form matching what is known and what remains fixed | At fixed C, doubling voltage quadruples stored energy; at fixed Q, increasing C lowers stored energy. |
What a strong diagram or graph must show
Translate the same physical model across words, diagrams, graphs and equations.
Parallel-plate diagram
equal-magnitude opposite free charges and approximately uniform central field.
C-versus-A graph
linear through the ideal-model origin at fixed d and dielectric.
C-versus-d graph
inverse relationship at fixed A and dielectric.
Constraint table
decide fixed charge Q versus fixed potential difference ΔV before predicting dielectric/geometry changes; for a charged particle between ideal plates, the approximately uniform field gives constant electric force and therefore constant acceleration.
Change the model and inspect what must respond
Make a prediction first. Then move one control and explain the output using the governing relationship and representation.
Parallel-plate capacitor explorer
Change geometry, dielectric and constraint. The tool separates what determines C from what the external connection holds fixed.
Two examples with the model choice shown
The goal is to expose the reasoning pathway, not only the final number.
Geometry change
Situation: An ideal vacuum parallel-plate capacitor has area A and separation d. The area doubles while separation stays fixed.
Reasoning: C=ε₀A/d, so capacitance is directly proportional to area.
The new capacitance is 2C. What happens to Q, V and U next depends on whether the capacitor is isolated or still source-connected.
Dielectric insertion at fixed voltage
Situation: A capacitor remains connected to an ideal battery while a dielectric with κ>1 fills the gap.
Reasoning: The battery keeps ΔV fixed. C increases by κ, so Q=CΔV increases by κ; U=½C(ΔV)² also increases by κ.
At fixed source voltage, capacitance, stored charge and capacitor energy increase; energy is supplied through the external source interaction.
Recognize the physics before reaching for an equation
Use the cue to choose the model, connect it to nearby ideas, then reject any method whose assumptions do not match the situation.
What should trigger Topic 10.6?
Use the capacitor model when equal-and-opposite plate charge, plate geometry, dielectric material, voltage, or stored field energy are involved.
How this idea fits the course
Topic 10.5 supplies voltage and field language; Unit 11 later combines capacitance with resistor networks and RC transients.
Scope check before solving
Only parallel-plate capacitor analysis is required here; ignore edge effects unless explicitly stated. Before predicting a dielectric or geometry change, state whether the capacitor is isolated (Q fixed) or connected to a source that keeps ΔV fixed.
Turn this topic into measurable evidence
A strong investigation identifies variables, controls competing causes, repeats trials and chooses a graph from the predicted functional dependence.
Investigation target
Vary plate area, separation, or dielectric in a capacitor setup/simulation and test predicted capacitance trends.
Reasoning errors to eliminate
Repair the physical model before doing more arithmetic.
Six things you should be able to do before moving on
Use these as closed-note transfer prompts.
- I can define the core quantities and state whether each is vector or scalar.
- I can choose the governing relationship only after identifying source, probe and system boundary.
- I can explain every variable, unit, sign and reference convention.
- I can translate the situation into a field/equipotential/energy/geometry representation.
- I can predict direction or factor change before calculating.
- I can design or critique an experiment and justify a claim with data or physical principles.
Practice is kept on its dedicated pages
Finish the lesson first. Then open the separate topic bank when you want exam-style practice without replacing the teaching content above.
42 multiple-choice questions
Use the exact-topic MCQ bank to test model selection, representations, calculations and boundaries after learning the concept.
Open 42 Topic MCQs4 multipart free-response tasks
Use the exact-topic FRQ set for mathematical, representation, experimental and qualitative/quantitative reasoning.
Open 4 Topic FRQsTopic 10.6 FAQs
What determines capacitance?
For ideal parallel plates, C=κε₀A/d: geometry and dielectric material determine C.
Does adding charge increase capacitance?
No for an unchanged ideal capacitor. Adding charge increases the potential difference proportionally, leaving C=Q/ΔV unchanged.
What does a dielectric do?
It polarizes and produces an opposing internal field, increasing the capacitance relative to the same vacuum geometry.
What stays constant for an isolated capacitor?
The free charge magnitude Q on the plates remains fixed because there is no conducting path for charge transfer.
What stays constant when connected to an ideal battery?
The potential difference ΔV remains fixed; the source may move charge to or from the plates.
Is the field between capacitor plates exactly uniform?
It is modeled as uniform away from the edges when plate separation is much smaller than plate dimensions; edge effects are ignored unless stated.