Most of us have seen dramatizations in which medical personnel use a defibrillator to pass an electric current through a patient’s heart to get it to beat normally. (Review [link].) Often realistic in detail, the person applying the shock directs another person to “make it 400 joules this time.” The energy delivered by the defibrillator is stored in a capacitor and can be adjusted to fit the situation. SI units of joules are often employed. Less dramatic is the use of capacitors in microelectronics, such as certain handheld calculators, to supply energy when batteries are charged. (See [link].) Capacitors are also used to supply energy for flash lamps on cameras.

Energy stored in a capacitor is electrical potential energy, and it is thus related to the charge $Q$ and voltage $V$ on the capacitor. We must be careful when applying the equation for electrical potential energy $\text{\Delta}\text{PE}=q\text{\Delta}V\phantom{\rule{0.25em}{0ex}}$ to a capacitor. Remember that $\text{\Delta}\text{PE}$ is the potential energy of a charge *$q$* going through a voltage $\text{\Delta}V$. But the capacitor starts with zero voltage and gradually comes up to its full voltage as it is charged. The first charge placed on a capacitor experiences a change in voltage $\text{\Delta}V=0$, since the capacitor has zero voltage when uncharged. The final charge placed on a capacitor experiences $\text{\Delta}V=V$, since the capacitor now has its full voltage $V$ on it. The average voltage on the capacitor during the charging process is $V/2$, and so the average voltage experienced by the full charge *$q$* is $V/2$. Thus the energy stored in a capacitor, ${E}_{\text{cap}}$, is

where $Q$ is the charge on a capacitor with a voltage $V$ applied. (Note that the energy is not $\text{QV}$, but $\text{QV}/2$.) Charge and voltage are related to the capacitance $C$ of a capacitor by $Q=\text{CV}$, and so the expression for ${E}_{\text{cap}}$ can be algebraically manipulated into three equivalent expressions:

where $Q$ is the charge and $V$ the voltage on a capacitor $C$. The energy is in joules for a charge in coulombs, voltage in volts, and capacitance in farads.

In a defibrillator, the delivery of a large charge in a short burst to a set of paddles across a person’s chest can be a lifesaver. The person’s heart attack might have arisen from the onset of fast, irregular beating of the heart—cardiac or ventricular fibrillation. The application of a large shock of electrical energy can terminate the arrhythmia and allow the body’s pacemaker to resume normal patterns. Today it is common for ambulances to carry a defibrillator, which also uses an electrocardiogram to analyze the patient’s heartbeat pattern. Automated external defibrillators (AED) are found in many public places ([link]). These are designed to be used by lay persons. The device automatically diagnoses the patient’s heart condition and then applies the shock with appropriate energy and waveform. CPR is recommended in many cases before use of an AED.

A heart defibrillator delivers $4.00\times {\text{10}}^{\text{2}}\phantom{\rule{0.25em}{0ex}}\text{J}$ of energy by discharging a capacitor initially at $1.00\times {\text{10}}^{\text{4}}\phantom{\rule{0.25em}{0ex}}\text{V}$. What is its capacitance?

**Strategy**

We are given ${E}_{\text{cap}}$ and $V$, and we are asked to find the capacitance $C$. Of the three expressions in the equation for ${E}_{\text{cap}}$, the most convenient relationship is

**Solution**

Solving this expression for $C$ and entering the given values yields

**Discussion**

This is a fairly large, but manageable, capacitance at $1.00\times {\text{10}}^{\text{4}}\phantom{\rule{0.25em}{0ex}}\text{V}$.

# Section Summary

- Capacitors are used in a variety of devices, including defibrillators, microelectronics such as calculators, and flash lamps, to supply energy.
- The energy stored in a capacitor can be expressed in three ways:
${E}_{\text{cap}}=\frac{\text{QV}}{2}=\frac{{\text{CV}}^{2}}{2}=\frac{{Q}^{2}}{2C},$where $Q$ is the charge, $V$ is the voltage, and $C$ is the capacitance of the capacitor. The energy is in joules when the charge is in coulombs, voltage is in volts, and capacitance is in farads.

# Conceptual Questions

How does the energy contained in a charged capacitor change when a dielectric is inserted, assuming the capacitor is isolated and its charge is constant? Does this imply that work was done?

What happens to the energy stored in a capacitor connected to a battery when a dielectric is inserted? Was work done in the process?

# Problems & Exercises

(a) What is the energy stored in the $\text{10.0 \mu F}$ capacitor of a heart defibrillator charged to $9.00\times {\text{10}}^{\text{3}}\phantom{\rule{0.25em}{0ex}}\text{V}$? (b) Find the amount of stored charge.

(a) $\text{405 J}$

(b) $\text{90.0 mC}$

In open heart surgery, a much smaller amount of energy will defibrillate the heart. (a) What voltage is applied to the $\text{8.00 \mu F}$ capacitor of a heart defibrillator that stores 40.0 J of energy? (b) Find the amount of stored charge.

(a) 3.16 kV

(b) 25.3 mC

A $1\text{65 \xb5F}$ capacitor is used in conjunction with a motor. How much energy is stored in it when 119 V is applied?

Suppose you have a 9.00 V battery, a $\text{2.00 \mu F}$ capacitor, and a $\text{7.40 \mu F}$ capacitor. (a) Find the charge and energy stored if the capacitors are connected to the battery in series. (b) Do the same for a parallel connection.

(a) $1.42\times {\text{10}}^{\text{\u22125}}\phantom{\rule{0.25em}{0ex}}\text{C}$, $6.38\times {\text{10}}^{\text{\u22125}}\phantom{\rule{0.25em}{0ex}}\text{J}$

(b) $8.46\times {\text{10}}^{\text{\u22125}}\phantom{\rule{0.25em}{0ex}}\text{C}$, $3.81\times {\text{10}}^{\text{\u22124}}\phantom{\rule{0.25em}{0ex}}\text{J}$

A nervous physicist worries that the two metal shelves of his wood frame bookcase might obtain a high voltage if charged by static electricity, perhaps produced by friction. (a) What is the capacitance of the empty shelves if they have area $1.00\times {\text{10}}^{\text{2}}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{\text{2}}$ and are 0.200 m apart? (b) What is the voltage between them if opposite charges of magnitude 2.00 nC are placed on them? (c) To show that this voltage poses a small hazard, calculate the energy stored.

(a) $4\text{.}\text{43}\times {\text{10}}^{\u2013\text{12}}\phantom{\rule{0.25em}{0ex}}\text{F}$

(b) $\text{452}\phantom{\rule{0.25em}{0ex}}\text{V}$

(c) $4\text{.}\text{52}\times {\text{10}}^{\u20137}\phantom{\rule{0.25em}{0ex}}\text{J}$

Show that for a given dielectric material the maximum energy a parallel plate capacitor can store is directly proportional to the volume of dielectric ($\text{Volume =}\phantom{\rule{0.25em}{0ex}}A\xb7d$). Note that the applied voltage is limited by the dielectric strength.

**Construct Your Own Problem**

Consider a heart defibrillator similar to that discussed in [link]. Construct a problem in which you examine the charge stored in the capacitor of a defibrillator as a function of stored energy. Among the things to be considered are the applied voltage and whether it should vary with energy to be delivered, the range of energies involved, and the capacitance of the defibrillator. You may also wish to consider the much smaller energy needed for defibrillation during open-heart surgery as a variation on this problem.

**Unreasonable Results**

(a) On a particular day, it takes $9.60\times {\text{10}}^{\text{3}}\phantom{\rule{0.25em}{0ex}}\text{J}$ of electric energy to start a truck’s engine. Calculate the capacitance of a capacitor that could store that amount of energy at 12.0 V. (b) What is unreasonable about this result? (c) Which assumptions are responsible?

(a) $\text{133}\phantom{\rule{0.25em}{0ex}}\text{F}$

(b) Such a capacitor would be too large to carry with a truck. The size of the capacitor would be enormous.

(c) It is unreasonable to assume that a capacitor can store the amount of energy needed.

- College Physics
- Preface
- Introduction: The Nature of Science and Physics
- Kinematics
- Introduction to One-Dimensional Kinematics
- Displacement
- Vectors, Scalars, and Coordinate Systems
- Time, Velocity, and Speed
- Acceleration
- Motion Equations for Constant Acceleration in One Dimension
- Problem-Solving Basics for One-Dimensional Kinematics
- Falling Objects
- Graphical Analysis of One-Dimensional Motion

- Two-Dimensional Kinematics
- Dynamics: Force and Newton's Laws of Motion
- Introduction to Dynamics: Newton’s Laws of Motion
- Development of Force Concept
- Newton’s First Law of Motion: Inertia
- Newton’s Second Law of Motion: Concept of a System
- Newton’s Third Law of Motion: Symmetry in Forces
- Normal, Tension, and Other Examples of Forces
- Problem-Solving Strategies
- Further Applications of Newton’s Laws of Motion
- Extended Topic: The Four Basic Forces—An Introduction

- Further Applications of Newton's Laws: Friction, Drag, and Elasticity
- Uniform Circular Motion and Gravitation
- Work, Energy, and Energy Resources
- Linear Momentum and Collisions
- Statics and Torque
- Rotational Motion and Angular Momentum
- Introduction to Rotational Motion and Angular Momentum
- Angular Acceleration
- Kinematics of Rotational Motion
- Dynamics of Rotational Motion: Rotational Inertia
- Rotational Kinetic Energy: Work and Energy Revisited
- Angular Momentum and Its Conservation
- Collisions of Extended Bodies in Two Dimensions
- Gyroscopic Effects: Vector Aspects of Angular Momentum

- Fluid Statics
- Fluid Dynamics and Its Biological and Medical Applications
- Introduction to Fluid Dynamics and Its Biological and Medical Applications
- Flow Rate and Its Relation to Velocity
- Bernoulli’s Equation
- The Most General Applications of Bernoulli’s Equation
- Viscosity and Laminar Flow; Poiseuille’s Law
- The Onset of Turbulence
- Motion of an Object in a Viscous Fluid
- Molecular Transport Phenomena: Diffusion, Osmosis, and Related Processes

- Temperature, Kinetic Theory, and the Gas Laws
- Heat and Heat Transfer Methods
- Thermodynamics
- Introduction to Thermodynamics
- The First Law of Thermodynamics
- The First Law of Thermodynamics and Some Simple Processes
- Introduction to the Second Law of Thermodynamics: Heat Engines and Their Efficiency
- Carnot’s Perfect Heat Engine: The Second Law of Thermodynamics Restated
- Applications of Thermodynamics: Heat Pumps and Refrigerators
- Entropy and the Second Law of Thermodynamics: Disorder and the Unavailability of Energy
- Statistical Interpretation of Entropy and the Second Law of Thermodynamics: The Underlying Explanation

- Oscillatory Motion and Waves
- Introduction to Oscillatory Motion and Waves
- Hooke’s Law: Stress and Strain Revisited
- Period and Frequency in Oscillations
- Simple Harmonic Motion: A Special Periodic Motion
- The Simple Pendulum
- Energy and the Simple Harmonic Oscillator
- Uniform Circular Motion and Simple Harmonic Motion
- Damped Harmonic Motion
- Forced Oscillations and Resonance
- Waves
- Superposition and Interference
- Energy in Waves: Intensity

- Physics of Hearing
- Electric Charge and Electric Field
- Introduction to Electric Charge and Electric Field
- Static Electricity and Charge: Conservation of Charge
- Conductors and Insulators
- Coulomb’s Law
- Electric Field: Concept of a Field Revisited
- Electric Field Lines: Multiple Charges
- Electric Forces in Biology
- Conductors and Electric Fields in Static Equilibrium
- Applications of Electrostatics

- Electric Potential and Electric Field
- Electric Current, Resistance, and Ohm's Law
- Circuits, Bioelectricity, and DC Instruments
- Magnetism
- Introduction to Magnetism
- Magnets
- Ferromagnets and Electromagnets
- Magnetic Fields and Magnetic Field Lines
- Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field
- Force on a Moving Charge in a Magnetic Field: Examples and Applications
- The Hall Effect
- Magnetic Force on a Current-Carrying Conductor
- Torque on a Current Loop: Motors and Meters
- Magnetic Fields Produced by Currents: Ampere’s Law
- Magnetic Force between Two Parallel Conductors
- More Applications of Magnetism

- Electromagnetic Induction, AC Circuits, and Electrical Technologies
- Introduction to Electromagnetic Induction, AC Circuits and Electrical Technologies
- Induced Emf and Magnetic Flux
- Faraday’s Law of Induction: Lenz’s Law
- Motional Emf
- Eddy Currents and Magnetic Damping
- Electric Generators
- Back Emf
- Transformers
- Electrical Safety: Systems and Devices
- Inductance
- RL Circuits
- Reactance, Inductive and Capacitive
- RLC Series AC Circuits

- Electromagnetic Waves
- Geometric Optics
- Vision and Optical Instruments
- Wave Optics
- Introduction to Wave Optics
- The Wave Aspect of Light: Interference
- Huygens's Principle: Diffraction
- Young’s Double Slit Experiment
- Multiple Slit Diffraction
- Single Slit Diffraction
- Limits of Resolution: The Rayleigh Criterion
- Thin Film Interference
- Polarization
- *Extended Topic* Microscopy Enhanced by the Wave Characteristics of Light

- Special Relativity
- Introduction to Quantum Physics
- Atomic Physics
- Introduction to Atomic Physics
- Discovery of the Atom
- Discovery of the Parts of the Atom: Electrons and Nuclei
- Bohr’s Theory of the Hydrogen Atom
- X Rays: Atomic Origins and Applications
- Applications of Atomic Excitations and De-Excitations
- The Wave Nature of Matter Causes Quantization
- Patterns in Spectra Reveal More Quantization
- Quantum Numbers and Rules
- The Pauli Exclusion Principle

- Radioactivity and Nuclear Physics
- Medical Applications of Nuclear Physics
- Particle Physics
- Frontiers of Physics
- Atomic Masses
- Selected Radioactive Isotopes
- Useful Information
- Glossary of Key Symbols and Notation