3.5 Energy stored in capacitors

Capacitor Fundamentals

Capacitors store electrical energy in the electric field between two conductive plates separated by an insulating material (called a dielectric). They're one of the most common circuit components you'll encounter, and understanding how they store energy is the core focus of this section.

Definition and Structure

A capacitor is a passive component with two parallel conductive plates (electrodes) separated by a dielectric material. The dielectric can be air, ceramic, plastic, or an electrolytic solution.

Two physical features control how much charge a capacitor can store:

• Larger plate area → more capacitance
• Smaller separation distance → more capacitance

Types of Capacitors

• Ceramic capacitors use ceramic dielectrics for high stability and low losses
• Electrolytic capacitors use a conductive electrolyte to achieve high capacitance in a small package
• Film capacitors use thin plastic films for improved temperature stability
• Variable capacitors allow you to adjust capacitance mechanically (used in radio tuning circuits)

Electric Field in Capacitors

When a voltage is applied across a capacitor, an electric field forms between the plates. This field is what actually holds the stored energy.

Field Distribution

Between the plates, the electric field is uniform: field lines run perpendicular from the positive plate to the negative plate. Near the edges of the plates, the field bends slightly (called edge effects), but for most problems you can treat the field as perfectly uniform throughout the dielectric region.

Field Strength Calculation

$E = \frac{V}{d}$

• $V$ = voltage across the plates
• $d$ = separation distance between the plates

Field strength is directly proportional to voltage and inversely proportional to plate separation. Double the voltage, double the field. Double the distance, halve the field.

Capacitance

Capacitance measures how much charge a capacitor can store per volt of applied potential difference.

Defining Equation

$C = \frac{Q}{V}$

• $C$ = capacitance (in farads)
• $Q$ = stored charge (in coulombs)
• $V$ = applied voltage (in volts)

Factors Affecting Capacitance

The capacitance of a parallel-plate capacitor depends on geometry and the dielectric material:

$C = \frac{\kappa \varepsilon_0 A}{d}$

• $A$ = plate area
• $d$ = plate separation
• $\kappa$ = dielectric constant of the insulating material
• $\varepsilon_0$ = permittivity of free space ($8.85 \times 10^{-12}$ F/m)

Units

Capacitance is measured in farads (F). One farad equals one coulomb per volt. A farad is enormous for most practical purposes, so you'll usually see microfarads ($\mu$F = $10^{-6}$ F), nanofarads (nF = $10^{-9}$ F), or picofarads (pF = $10^{-12}$ F).

Energy Storage Mechanism

Charge Separation

When you apply a voltage, electrons move from one plate to the other. One plate builds up negative charge while the other becomes positively charged. The dielectric between them prevents the charges from recombining. The amount of separated charge is proportional to both the applied voltage and the capacitance: $Q = CV$.

Potential Difference and Stored Energy

The voltage difference between the plates creates an electric field, and energy is stored in that field. The greater the potential difference, the more energy is stored. You can always recover the voltage from the charge using $V = \frac{Q}{C}$.

Energy Calculation

This is the central topic of section 3.5. The energy stored in a capacitor has three equivalent forms, and you should be comfortable using whichever one fits the information you're given.

Deriving the Energy Formula

The tricky part: as charge builds up on the plates, the voltage across the capacitor increases. You can't just use $W = QV$ because $V$ isn't constant during charging. Instead, you integrate the work done to move each small bit of charge $dq$ against the growing voltage $v = q/C$:

$U = \int_0^Q \frac{q}{C}\, dq = \frac{Q^2}{2C}$

This gives three equivalent expressions:

$U = \frac{1}{2}CV^2 = \frac{1}{2}\frac{Q^2}{C} = \frac{1}{2}QV$

Use $\frac{1}{2}CV^2$ when you know capacitance and voltage. Use $\frac{1}{2}\frac{Q^2}{C}$ when you know charge and capacitance. Use $\frac{1}{2}QV$ when you know charge and voltage.

Energy vs. Charge Relationship

• Energy scales with the square of the charge: $U = \frac{1}{2}\frac{Q^2}{C}$. Doubling the charge quadruples the energy.
• For a fixed charge, energy is inversely proportional to capacitance.
• For a fixed voltage, energy is directly proportional to capacitance.

Capacitor Charging Process

Charging Curve

When a capacitor charges through a resistor, the voltage doesn't jump instantly to its final value. Instead, it rises exponentially:

$V(t) = V_0\left(1 - e^{-t/(RC)}\right)$

• $V_0$ = supply voltage
• $R$ = circuit resistance
• The voltage approaches $V_0$ asymptotically (it never quite reaches it in theory)

Time Constant

The time constant $\tau = RC$ (measured in seconds) sets the pace of charging and discharging.

• After $1\tau$, the capacitor reaches 63.2% of full charge
• After $5\tau$, it's at 99.3%, which is typically treated as "fully charged"

A large resistance or large capacitance means slower charging. This is the basis for timing circuits and filters.

Energy Density

Energy density tells you how much energy is packed into a given volume of the electric field.

$u = \frac{1}{2}\varepsilon E^2$

• $\varepsilon$ = permittivity of the dielectric ($\varepsilon = \kappa \varepsilon_0$)
• $E$ = electric field strength
• Units: joules per cubic meter (J/m³)

Comparison with Other Storage Devices

Capacitors have high power density (they can deliver energy very quickly) but low energy density compared to batteries. Supercapacitors narrow that gap. In practice, capacitors are best for rapid charge/discharge tasks like power smoothing, while batteries are better for sustained energy delivery.

Applications of Stored Energy

Power Supplies

Capacitors smooth the output of rectifier circuits by filling in voltage dips, provide short-term backup during brief power outages, and supply high-current bursts in switch-mode power supplies.

Flash Photography

A camera flash stores energy in a capacitor, then dumps it all at once through a flash tube. This produces a high-intensity burst of light lasting only milliseconds, and the capacitor recharges quickly for the next shot.

Defibrillators

A defibrillator charges a large capacitor (often to several thousand volts) and then delivers a controlled pulse of energy through the patient's chest to restore normal heart rhythm. Typical defibrillator capacitors store around 200–360 J per shock.

Capacitor Networks

Series vs. Parallel Connections

Series: Capacitors share the same charge $Q$, but voltages divide among them.

$\frac{1}{C_{\text{total}}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots$

Total capacitance is always less than the smallest individual capacitor.

Parallel: Capacitors share the same voltage, but charge distributes among them.

$C_{\text{total}} = C_1 + C_2 + \cdots$

Total capacitance is the sum of all individual capacitances.

Energy Distribution in Networks

• In parallel networks, total stored energy is the sum of each capacitor's energy (each at the same voltage).
• In series networks, each capacitor holds the same charge $Q$, but the capacitor with the smallest capacitance stores the most energy (since $U = Q^2/(2C)$).
• Voltage divides in series inversely with capacitance: the smaller capacitor gets the larger voltage.
• Charge divides in parallel proportionally with capacitance: the larger capacitor gets more charge.

Dielectrics and Energy Storage

Dielectric Materials

A dielectric is an insulating material placed between the plates. Each dielectric is characterized by its dielectric constant $\kappa$ (also called relative permittivity). Some typical values: air $\approx 1.0$, paper $\approx 3.7$, glass $\approx 4{-}10$, water $\approx 80$.

Effect on Capacitance and Energy

Inserting a dielectric multiplies the capacitance by $\kappa$:

$C = \kappa C_0$

where $C_0$ is the capacitance without the dielectric. Higher capacitance means more energy storage at the same voltage. However, every dielectric has a dielectric strength, the maximum electric field it can withstand before it breaks down and conducts. This limits the maximum voltage you can safely apply.

Energy Loss and Efficiency

Real capacitors aren't perfect. Two main loss mechanisms reduce efficiency:

Dielectric Loss

As the electric field alternates, molecules in the dielectric polarize back and forth, generating heat through molecular friction. This loss is quantified by the dissipation factor (or loss tangent, $\tan \delta$). Dielectric loss increases with both frequency and temperature, which matters in AC circuits.

Leakage Current

No dielectric is a perfect insulator. A small current slowly flows through it, gradually discharging the capacitor over time. The magnitude depends on the capacitor type and quality. Electrolytic capacitors tend to have higher leakage than ceramic or film types.

Safety Considerations

High Voltage Hazards

Capacitors can hold dangerous voltages long after a circuit is powered off. A charged capacitor is essentially a stored energy source that can deliver a painful or lethal shock, cause arc flash, or damage components if shorted accidentally.

Proper Discharge Procedures

1. Disconnect the circuit from its power source.
2. Connect an appropriate discharge resistor across the capacitor terminals to safely drain the stored energy.
3. Wait for the voltage to drop (at least $5\tau$).
4. Verify complete discharge with a voltmeter before touching anything.
5. In permanent installations, bleed resistors can be wired in parallel with capacitors so they discharge automatically when power is removed.

Always follow manufacturer guidelines for the specific capacitor type and voltage rating you're working with.