Source: PHY 212 Electromagnetism Practice Exam, University of Illinois at Urbana-Champaign
Difficulty: Intermediate | Prerequisites: Electric field, electric potential, conductor behaviour at equilibrium
Tags: capacitor, capacitance, parallel plate capacitor, dielectric, dielectric constant, kappa, series capacitors, parallel capacitors, energy stored, energy density, charge on plates, voltage, battery connected, isolated capacitor
Capacitors are one of the most practically important components in electronics, and they are also a favourite exam topic because they let examiners test whether you can keep track of which quantities stay fixed and which change when conditions shift. This set of notes covers the parallel-plate capacitor, what happens when you insert a dielectric (with and without a battery), how capacitors combine in series and parallel, and the concept of energy density in an electric field. You should already be comfortable with electric fields between parallel plates and with the relationship V = Ed before starting here.
Capacitance depends on geometry and the dielectric material, not on the charge or voltage applied. Inserting a dielectric always increases capacitance. Whether charge, voltage, or field changes when you insert it depends entirely on whether the capacitor is connected to a battery (voltage fixed) or isolated (charge fixed). Energy density tells you how much energy is stored per unit volume in the field itself.
Capacitance (C)
The ratio of the charge stored on one plate to the voltage across the plates: C = Q / V. It depends only on the geometry of the capacitor and the dielectric material between the plates, not on Q or V themselves. In simple terms, capacitance measures how much charge a capacitor can hold per volt applied.
Parallel-plate capacitor
Two parallel conducting plates of area A separated by distance d. Without a dielectric, C = epsilon_0 A / d. Think of it as the simplest capacitor geometry and the one exams default to.
Dielectric
An insulating material placed between capacitor plates. It reduces the electric field inside and increases the capacitance by a factor of kappa (the dielectric constant). In simple terms, it is a non-conducting filling that lets the capacitor store more charge for the same voltage.
Dielectric constant (kappa, sometimes written as K)
A dimensionless number (always greater than 1 for real materials) that multiplies the capacitance: C_new = kappa C_0. Vacuum has kappa = 1.
Energy density (u)
The energy stored per unit volume in an electric field: u = (1/2) epsilon_0 E^2. In simple terms, the electric field itself carries energy, spread throughout the space it occupies.
Capacitors in series
Capacitors connected end-to-end so that the same charge flows through each. The charge on each capacitor is the same; the voltages are different (unless the capacitances happen to be equal).
Capacitors in parallel
Capacitors connected across the same two nodes so that the voltage across each is the same. The charges are different (unless the capacitances happen to be equal).
C = epsilon_0 A / d (no dielectric)
C = kappa epsilon_0 A / d (with dielectric filling the gap)
The electric field between the plates (no dielectric): E = V / d = sigma / epsilon_0
Capacitance is a property of the device. Doubling the charge doubles the voltage, but C = Q/V stays the same.
The battery maintains a constant voltage V across the plates. V does not change.
Inserting the dielectric increases C by a factor of kappa.
Since Q = CV and V is constant, Q increases by a factor of kappa. The battery pushes extra charge onto the plates.
The electric field E = V/d stays the same (V and d are both unchanged).
Energy stored U = (1/2)CV^2 increases by a factor of kappa (C went up, V stayed the same).
Summary for battery connected:
V: unchanged
C: increases by kappa
Q: increases by kappa
E: unchanged
U: increases by kappa
The capacitor is disconnected from any battery. No charge can flow on or off the plates. Q does not change.
Inserting the dielectric increases C by a factor of kappa.
Since V = Q/C and Q is constant, V decreases by a factor of kappa.
The electric field E = V/d also decreases by a factor of kappa.
Energy stored U = Q^2 / (2C) decreases by a factor of kappa (C went up, Q stayed the same).
Summary for isolated:
Q: unchanged
C: increases by kappa
V: decreases by kappa
E: decreases by kappa
U: decreases by kappa
Increasing the plate area A (C is proportional to A)
Decreasing the plate separation d (C is inversely proportional to d)
Inserting a dielectric material (C multiplied by kappa)
Increasing the voltage of the connected battery does not change capacitance. C depends on geometry and material, not on the applied voltage.
The charge on each capacitor is the same: Q_1 = Q_2 = Q_total.
The voltages add: V_total = V_1 + V_2.
The equivalent capacitance is found from 1/C_eq = 1/C_1 + 1/C_2. The equivalent is always smaller than the smallest individual capacitance.
The voltage across each capacitor is the same: V_1 = V_2 = V_total.
The charges add: Q_total = Q_1 + Q_2.
The equivalent capacitance is C_eq = C_1 + C_2. The equivalent is always larger than the largest individual capacitance.
If you double the charge on the plates, the voltage also doubles (V = Q/C), but the capacitance does not change. Capacitance is determined by the geometry and dielectric, not by how much charge is on the plates.
Energy stored in a capacitor: U = (1/2)QV = (1/2)CV^2 = Q^2 / (2C). These are all equivalent; choose the form that uses the quantities you know.
Energy density in an electric field: u = (1/2) epsilon_0 E^2 (in vacuum) or u = (1/2) kappa epsilon_0 E^2 (in a dielectric).
The energy density tells you that energy is stored in the field itself, distributed throughout the volume where the field exists. It is not localised on the plates.
Parallel-plate capacitance: C = kappa epsilon_0 A / d
Charge-voltage relation: Q = CV
Energy stored: U = (1/2) CV^2 = (1/2) QV = Q^2 / (2C)
Energy density: u = (1/2) epsilon_0 E^2
Series combination: 1/C_eq = 1/C_1 + 1/C_2 + ...
Parallel combination: C_eq = C_1 + C_2 + ...
Field between plates: E = V / d
Capacitors are everywhere: the flash unit in a camera stores energy in a capacitor and releases it rapidly, touchscreens detect changes in capacitance when your finger approaches the screen, and defibrillators use large capacitors to deliver a controlled burst of energy to the heart. The dielectric concept is why capacitors in circuits use ceramic, plastic film, or electrolyte fillings rather than air: higher kappa means more capacitance in the same volume.
Students often think that doubling the charge on a capacitor doubles its capacitance. It does not. Capacitance is fixed by geometry and the dielectric. Doubling Q doubles V, leaving C unchanged.
Students often confuse the two dielectric scenarios. When the battery stays connected, V is constant and Q changes. When the capacitor is isolated, Q is constant and V changes. Mixing these up is one of the most common exam errors.
Students often think increasing the battery voltage increases the capacitance. It does not. A higher voltage stores more charge (Q = CV), but C itself is unchanged.
Students often think that in series, the voltage across each capacitor must be the same. That is the rule for parallel capacitors. In series, the charge is the same.
⚠️ The "battery connected vs. isolated" dielectric question is almost guaranteed on any electrostatics exam. Know both cases cold.
⚠️ Series vs. parallel: remember "series means same charge, parallel means same voltage."
⚠️ "What happens to capacitance if you double the charge?" is a standard trap. The answer is nothing.
⚠️ Energy density u = (1/2) epsilon_0 E^2 is a common short-answer definition question. Know its physical meaning: energy is stored in the field itself, per unit volume.
True or false: Inserting a dielectric into a capacitor always increases its capacitance. (True.)
Fill in the blank: When a dielectric is inserted while a battery remains connected, the quantity that stays constant is the __________. (voltage)
True or false: For capacitors in series, the voltage across each must be the same. (False. The charge is the same; the voltages differ.)
Fill in the blank: Energy density in an electric field is u = (1/2) __________ E^2. (epsilon_0)
True or false: Doubling the charge on a capacitor doubles its capacitance. (False. Capacitance depends on geometry and dielectric, not on charge.)
Q: A parallel-plate capacitor is connected to a battery with voltage V. A dielectric with kappa > 1 is inserted while the battery stays connected. Which quantity increases: voltage, field, or charge stored?
A: The charge stored on the plates increases. The battery holds voltage constant, and C increases by kappa, so Q = CV increases by kappa. The field E = V/d is unchanged.
Q: The dielectric is instead inserted while the capacitor is isolated (no battery). Which quantity remains constant?
A: The charge Q remains constant. With no battery, no charge can flow on or off the plates.
Q: Two capacitors are connected in series. Which property must be the same for both?
A: The charge on each capacitor. In a series connection, the same charge flows onto each capacitor.
Q: Which of the following increase the capacitance of a parallel-plate capacitor: (A) increasing plate area, (B) increasing plate separation, (C) inserting a dielectric, (D) increasing battery voltage?
A: (A) and (C). Capacitance C = kappa epsilon_0 A / d, so increasing A or inserting a dielectric (increasing kappa) raises C. Increasing d decreases C. Battery voltage does not affect C.
Q: If you double the charge on the plates of a capacitor, what happens to its capacitance?
A: Nothing. Capacitance is determined by the geometry (plate area, separation) and the dielectric material. Doubling Q doubles V, but C = Q/V remains the same.
Q: Define "energy density" as it relates to an electric field.
A: Energy density u is the energy stored per unit volume in the electric field. It equals (1/2) epsilon_0 E^2 in vacuum. It tells you that the field itself is the carrier of the stored energy, distributed throughout the region of space where the field is nonzero.
Capacitors connect to RC circuits in later chapters: the time constant for charging and discharging depends on C. The energy-density concept reappears with magnetic fields (u = B^2 / 2 mu_0) and is essential for understanding electromagnetic waves, where energy is carried jointly by E and B fields. The dielectric concept also ties into polarisation and bound charges, which some courses cover in more depth.
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