Capacitors
A capacitor stores separated equal-and-opposite charge on two conductors. Capacitance C=Q/|ΔV| is a geometric/material property, not the amount of charge currently stored. For an ideal parallel-plate capacitor C=κε₀A/d; the field is nearly uniform away from edges, energy can be written as U=Q²/(2C)=½C(ΔV)²=½QΔV, and the current course also expects quantitative concentric-spherical and coaxial-cylindrical capacitor models.
Topic sequence · 3 of 4
Learning objectives
Objective 1
Relate stored charge, potential difference, capacitance, electric field, and stored energy for capacitors.
Objective 2
Derive and use capacitance for parallel-plate, concentric-spherical, and coaxial-cylindrical geometries within the current course boundary.
Objective 3
Predict how geometry changes C and distinguish fixed-charge from fixed-potential reasoning before making field or energy claims.
- Topic 9.2 electric potential and line integrals.
- Topic 8.6 Gauss’s law for planar, spherical, and cylindrical symmetry.
- Topic 10.1 conductor boundary conditions and equipotential surfaces.
Definitions first
Representations you must read and create
Representation 1
Q-versus-ΔV graph: straight line through the origin with slope C for an ideal linear capacitor.
Representation 2
Parallel-plate field diagram showing nearly uniform E between plates and fringing only near edges.
Representation 3
Energy-versus-Q graph at fixed C: U=Q²/(2C), a parabola whose slope dU/dQ equals the instantaneous potential difference.
Representation 4
Cross-sections of concentric spherical and coaxial cylindrical capacitors with Gaussian surfaces used to derive E(r), then ΔV and C.
Core relationships and calculus
Variables: C F; Q C; ΔV V
Use: definition of capacitance
Variables: A m²; d m
Use: ideal parallel-plate capacitor; κ=1 for vacuum
Variables: E N/C
Use: uniform field between ideal oppositely charged parallel plates, away from edges
Variables: V volts
Use: uniform-field parallel-plate model
Variables: U J
Use: energy stored by an ideal capacitor
Variables: a,b m
Use: concentric spherical capacitor with inner radius a and outer radius b
Variables: L,a,b m
Use: long coaxial cylindrical capacitor, neglecting end effects
Capacitance derivations follow one repeated calculus pathway: use Gauss’s law to find E from the charge and symmetry, then evaluate |ΔV|=|−∫E⃗·dℓ⃗| between conductors, and finally form C=Q/|ΔV|. For a spherical capacitor E=kQ/r² integrates to a 1/a−1/b potential difference; for a coaxial capacitor E=λ/(2πεr) integrates to ln(b/a).
Parallel plates: Gauss’s law plus superposition gives E=σ/(κε₀)=Q/(κε₀A). Because the field is uniform, |ΔV|=Ed=Qd/(κε₀A), hence C=Q/|ΔV|=κε₀A/d. Stored energy follows from incremental charging: dU=V dq=(q/C)dq. Integrating q from 0 to Q gives U=∫₀^Q(q/C)dq=Q²/(2C), which is equivalent to ½Q|ΔV| and ½C(ΔV)².
For quantitative capacitor geometry, the current course expects parallel-plate, concentric spherical, and coaxial cylindrical capacitors. Edge/end effects are neglected in the ideal models. Topic 10.4 treats dielectric insertion and polarization in depth; here κ may appear as the material factor in capacitance, but do not mix fixed-Q and fixed-V conclusions without stating the constraint.
Physical examples
Example 1
Doubling plate area at fixed separation doubles parallel-plate capacitance.
Example 2
Doubling plate separation at fixed area halves capacitance, but whether E or stored energy rises/falls depends on whether Q or ΔV is held fixed.
Example 3
A charged particle between ideal plates experiences approximately constant electric force qE and therefore constant acceleration while it remains in the uniform-field region.
Example 4
A coaxial cable is a cylindrical capacitor: its capacitance depends logarithmically on the ratio of outer to inner radius.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
Functional dependence: identify whether C, Q, ΔV, E, or U is being held fixed before applying proportional reasoning.
Reasoning pattern 2
Derivation pathway: Gauss → field E(r) → potential difference integral → C=Q/ΔV.
Reasoning pattern 3
Energy representation: the area under a V-versus-Q charging graph is stored energy; for a linear capacitor it is ½QΔV.
Reasoning pattern 4
Particle motion: within an ideal uniform plate field, qE is constant, so motion perpendicular to the initial velocity parallels constant-acceleration projectile mathematics.
Deeper calculus and limiting-case check
At fixed capacitance, dU/dQ=Q/C=V, so the slope of an energy-versus-charge curve is the instantaneous potential difference required to add more charge. Combining U=½εE²(volume) for a uniform parallel-plate field with volume Ad gives U=½εE²Ad=½C(ΔV)², linking the lumped capacitor model to field energy.
Model-selection workflow: decide what the problem is before choosing an equation
For Capacitors, the fastest reliable route is not equation hunting. Start by identifying the conductor/capacitor geometry, encode it with the charge-field-potential representation, choose the electrostatic boundary or capacitance equation, and then test the equilibrium and ideal-geometry assumption. This sequence prevents a mathematically correct manipulation from being attached to the wrong physical model.
Define the physical object or system
Name what is being modeled, what is outside the system, the coordinate or orientation convention, and the interval or geometry over which the statement applies. Treat capacitor and capacitance C as physical quantities first and symbols second.
Choose a representation
Build Q-versus-ΔV graph: straight line through the origin with slope C for an ideal linear capacitor.. If the prompt supplies Parallel-plate field diagram showing nearly uniform E between plates and fringing only near edges., translate it into the same sign, axis, orientation, or reference convention before combining information.
Write the governing relationship
A useful starting relationship is C = Q/|ΔV|. State what the variables mean and why this relation—not merely a familiar formula—matches the physical situation.
Check conditions and limits
For quantitative capacitor geometry, the current course expects parallel-plate, concentric spherical, and coaxial cylindrical capacitors. Edge/end effects are neglected in the ideal models. Topic 10.4 treats dielectric insertion and polarization in depth; here κ may appear as the material factor in capacitance, but do not mix fixed-Q and fixed-V conclusions without stating the constraint. After solving, test units, sign or direction, zero/large-parameter limits, and whether the answer remains consistent with the original representation.
Representation → equation → calculus → physics chain
| Stage | What to write or inspect | What proves the step is physically valid |
|---|---|---|
| Representation | Q-versus-ΔV graph: straight line through the origin with slope C for an ideal linear capacitor. | Every arrow, sign, axis, node, surface, path, or reference has an explicit meaning. The diagram must contain enough information to determine the relevant components or constraints. |
| Primary model | C = Q/|ΔV| | definition of capacitance. Preserve the variables and units long enough to see dependence before inserting numerical values. |
| Second relationship | C_pp = κε₀A/d | ideal parallel-plate capacitor; κ=1 for vacuum. Combine relationships only after confirming they describe the same system, interval, frame, or geometry. |
| Calculus operation | Capacitance derivations follow one repeated calculus pathway: use Gauss’s law to find E from the charge and symmetry, then evaluate |ΔV|=|−∫E⃗·dℓ⃗| between conductors, and finally form C=Q/|ΔV|. For a spherical capacitor E=kQ/r² integrates to a 1/a−1/b potential difference; for a coaxial capacitor E=λ/(2πεr) integrates to ln(b/a). | State what the derivative, integral, differential equation, or limiting process represents physically; do not treat calculus notation as an algebraic decoration. |
| Physical check | Return to the original words and representation. | The final sign, direction, units, endpoint behavior, and functional dependence must all agree. A correct-looking expression that violates one of these checks needs repair. |
Advanced synthesis: how to handle an unfamiliar version of this topic
A vacuum parallel-plate capacitor has A=0.020 m², d=1.0 mm, and Q=3.0 nC. C=ε₀A/d≈1.77×10⁻¹⁰ F. Then |ΔV|=Q/C≈16.9 V, E=|ΔV|/d≈1.69×10⁴ N/C, and stored energy U=Q²/(2C)≈2.54×10⁻⁸ J. The same answers are internally consistent with E=Q/(ε₀A). Always state whether Q or ΔV is fixed before asking how a geometry change affects these quantities. To deepen this example, change one assumption or parameter, predict the effect before recomputing, and then use the symbolic dependence to verify the prediction.
FRQ-quality response architecture for Topic 10.3
Mathematical routines
Start from the governing law or definition, keep the derivation symbolic, identify the condition that permits any simplification, and finish with units plus a limiting-case check. For this topic, the relationship C = Q/|ΔV| should appear only when its physical conditions have been established.
Translation between representations
Move deliberately between Q-versus-ΔV graph: straight line through the origin with slope C for an ideal linear capacitor. and Parallel-plate field diagram showing nearly uniform E between plates and fringing only near edges.. Explain what a slope, area, direction, curvature, intercept, node value, field orientation, or other feature means instead of merely redrawing it.
Experimental design and analysis
Turn the topic into a measurable relationship. State the independent and dependent variables, important controls, repeated/ranged trials, and the graph or transformed graph whose slope/intercept would test the model or extract a physical parameter.
Qualitative/quantitative translation
Make the comparison or direction claim first, justify it through the model, derive the symbolic dependence independently, and then reconcile the derivation with the original qualitative claim.
Do not let an equation stand in for an explanation. A complete response names what changes, identifies the governing physical relationship, follows the causal chain, and states the requested outcome explicitly.
Experiment and data-analysis upgrade
Build or simulate a parallel-plate capacitor. Vary plate separation d and overlapping area A while measuring C, then test linear relationships C versus A and C versus 1/d. A second analysis can charge the capacitor to several voltages and test whether Q=CΔV is linear. Record whether the source remains connected so fixed-V and fixed-Q interpretations are not mixed.
Operationalize the variables
Say exactly what instrument or observable supplies each measured quantity. Distinguish a directly measured variable from a quantity calculated later from those measurements.
Predict the graph before collecting data
Use the governing relationship to decide what should be plotted. If the raw relationship is nonlinear, choose a transformation that produces a line whose slope or intercept has a defined physical meaning.
Control and repeat for a reason
Hold variables fixed when they would change the predicted relationship. Repeat trials to estimate scatter; vary the independent quantity over a useful range to test functional dependence. These solve different experimental problems.
Evaluate the model, not only the number
Compare the best-fit result with the theoretical prediction, inspect residual or endpoint behavior when appropriate, and identify whether a discrepancy is random scatter, instrument resolution, or a systematic violation of an ideal-model assumption.
Error-repair matrix
| Failure pattern | Repair move | Self-check |
|---|---|---|
| Treating capacitance as the amount of charge stored. C is a property of geometry/material for the linear capacitor model. | Return to the charge-field-potential representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Saying increasing d always decreases E. At fixed Q, ideal parallel-plate E=Q/(εA) is unchanged; at fixed voltage, E=V/d decreases. | Return to the electrostatic boundary or capacitance equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Using U=Q²/(2C) and U=½CV² to predict opposite trends without noticing that one calculation holds Q fixed and the other holds V fixed. | Return to the charge-field-potential representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Using parallel-plate C=εA/d for spherical or coaxial capacitor geometry. | Return to the electrostatic boundary or capacitance equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Forgetting that each plate carries equal charge magnitude Q with opposite sign; Q in C=Q/ΔV is the magnitude on either conductor, not the algebraic total of both plates. | Return to the charge-field-potential representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
Spiral review: make this topic transferable
A topic is exam-ready only when it can be recognized inside a mixed problem. Use the connections below as transfer tests rather than as optional reading.
Transfer 1
Topic 8.6 supplies the Gauss-law field derivations for planar, spherical, and cylindrical capacitor geometries. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 2
Topic 9.2 supplies potential difference as a field line integral. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 3
Topic 9.3 supplies conservation-of-energy reasoning for charged particles accelerated between plates. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Formula-condition retrieval
From memory, write C = Q/|ΔV| and one condition that must be true before using it. Then state one situation where a more general derivative, integral, conservation law, or field/force model is required.
Topic 10.3 mastery ladder
Interactive parallel-plate capacitor model
Enter plate area, separation, dielectric constant, and stored charge magnitude. The tool returns C, |ΔV|, ideal internal field, and stored energy.
Experimental connection
Build or simulate a parallel-plate capacitor. Vary plate separation d and overlapping area A while measuring C, then test linear relationships C versus A and C versus 1/d. A second analysis can charge the capacitor to several voltages and test whether Q=CΔV is linear. Record whether the source remains connected so fixed-V and fixed-Q interpretations are not mixed.
Connections across Physics C
Connection 1
Topic 8.6 supplies the Gauss-law field derivations for planar, spherical, and cylindrical capacitor geometries.
Connection 2
Topic 9.2 supplies potential difference as a field line integral.
Connection 3
Topic 9.3 supplies conservation-of-energy reasoning for charged particles accelerated between plates.
Connection 4
Topic 10.4 will show how polarization in a dielectric changes capacitance and the field/potential relationships under different constraints.
Related lessons and resources
Use these links when the current model depends on a prerequisite or becomes the foundation for the next model.
Common mistakes
Avoid this
Treating capacitance as the amount of charge stored. C is a property of geometry/material for the linear capacitor model.
Avoid this
Saying increasing d always decreases E. At fixed Q, ideal parallel-plate E=Q/(εA) is unchanged; at fixed voltage, E=V/d decreases.
Avoid this
Using U=Q²/(2C) and U=½CV² to predict opposite trends without noticing that one calculation holds Q fixed and the other holds V fixed.
Avoid this
Using parallel-plate C=εA/d for spherical or coaxial capacitor geometry.
Avoid this
Forgetting that each plate carries equal charge magnitude Q with opposite sign; Q in C=Q/ΔV is the magnitude on either conductor, not the algebraic total of both plates.
Mastery check
- Define the main quantity without relying on an equation.
- Choose the correct representation and sign convention.
- Explain why the calculus relationship matches the physics.
- Design or interpret an experiment that could test the model.
Frequently asked questions
Does capacitance increase when more charge is placed on a capacitor?
For an ideal linear capacitor, no. More Q produces proportionally larger ΔV, so C=Q/ΔV remains set by geometry and material.
Why is the field between large parallel plates nearly uniform?
Each plate approximates an infinite charged sheet away from edges; superposition makes the fields add between the plates and largely cancel outside.
What is the difference between fixed charge and fixed voltage?
An isolated charged capacitor approximately keeps Q fixed, while a capacitor connected to an ideal battery keeps ΔV fixed by exchanging charge with the source.
Why is capacitor energy one-half QV rather than QV?
The voltage rises from zero to its final value as the capacitor is charged, so the average voltage during linear charging is one-half the final voltage.
Are only parallel-plate capacitors required?
No. The current Physics C E&M framework also expects quantitative concentric-spherical and coaxial-cylindrical capacitor models.