what is the energy stored in the capacitor of a heart defibrillator charged to
151 19.7 Energy Stored in Capacitors
Summary
- List some uses of capacitors.
- Limited in equation grade the energy stored in a capacitor.
- Explain the role of a defibrillator.
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 information technology to beat out normally. (Review Effigy 1.) Often realistic in detail, the person applying the shock directs another person to "brand it 400 joules this time." The free energy delivered by the defibrillator is stored in a capacitor and tin be adapted to fit the situation. SI units of joules are often employed. Less dramatic is the apply of capacitors in microelectronics, such as certain handheld calculators, to supply free energy when batteries are charged. (See Figure one.) Capacitors are also used to supply free energy for flash lamps on cameras.
Energy stored in a capacitor is electric potential energy, and it is thus related to the charge $latex \boldsymbol{Q} $ and voltage $latex \boldsymbol{V}$ on the capacitor. We must exist careful when applying the equation for electric potential energy $latex \boldsymbol{\Delta \textbf{PE} = q \Delta 5} $ to a capacitor. Retrieve that $latex \boldsymbol{\Delta \textbf{PE}} $ is the potential energy of a charge $latex \boldsymbol{q} $ going through a voltage $latex \boldsymbol{\Delta V} $. Only the capacitor starts with zero voltage and gradually comes up to its full voltage as it is charged. The beginning charge placed on a capacitor experiences a modify in voltage $latex \boldsymbol{\Delta V = 0}$, since the capacitor has zero voltage when uncharged. The terminal charge placed on a capacitor experiences $latex \boldsymbol{\Delta 5 = V} $, since the capacitor now has its total voltage $latex \boldsymbol{V} $ on it. The average voltage on the capacitor during the charging process is $latex \boldsymbol{V/ii} $, and and so the average voltage experienced by the total charge $latex \boldsymbol{q} $ is $latex \boldsymbol{V/2} $. Thus the energy stored in a capacitor, $latex \boldsymbol{E_{\textbf{cap}}} $, is
$latex \boldsymbol{E_{\textbf{cap}} =}$ 
where $latex \boldsymbol{Q} $ is the charge on a capacitor with a voltage $latex \boldsymbol{5} $ applied. (Note that the free energy is not $latex \boldsymbol{QV} $, but $latex \boldsymbol{QV/2} $.) Charge and voltage are related to the capacitance $latex \boldsymbol{C} $ of a capacitor by $latex \boldsymbol{Q = CV} $, and then the expression for $latex \boldsymbol{E_{\textbf{cap}}}$ tin exist algebraically manipulated into three equivalent expressions:
$latex \boldsymbol{E_{\textbf{cap}} =}$ 
$latex \boldsymbol{=} $ 
$latex \boldsymbol{=} $ 
where $latex \boldsymbol{Q} $ is the charge and $latex \boldsymbol{V} $ the voltage on a capacitor $latex \boldsymbol{C} $. The energy is in joules for a charge in coulombs, voltage in volts, and capacitance in farads.
Energy Stored in Capacitors
The energy stored in a capacitor can be expressed in three means:
$latex \boldsymbol{E_{\textbf{cap}} =}$ $latex \boldsymbol{=} $ $latex \boldsymbol{=} $
where $latex \boldsymbol{Q} $ is the charge, $latex \boldsymbol{V} $ is the voltage, and $latex \boldsymbol{C} $ is the capacitance of the capacitor. The free energy is in joules for a accuse 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 beyond a person'due south chest can be a lifesaver. The person's heart attack might have arisen from the onset of fast, irregular chirapsia of the centre—cardiac or ventricular fibrillation. The application of a large stupor of electrical energy can terminate the arrhythmia and allow the body'south pacemaker to resume normal patterns. Today information technology is common for ambulances to comport a defibrillator, which also uses an electrocardiogram to analyze the patient's heartbeat pattern. Automatic external defibrillators (AED) are found in many public places (Figure 2). These are designed to be used by lay persons. The device automatically diagnoses the patient'due south middle condition and and so applies the daze with advisable energy and waveform. CPR is recommended in many cases before use of an AED.
Example 1: Capacitance in a Heart Defibrillator
A eye defibrillator delivers $latex \boldsymbol{iv.00 \times 10^2 \;\textbf{J}} $ of energy by discharging a capacitor initially at $latex \boldsymbol{1.00 \times 10^4 \;\textbf{V}} $. What is its capacitance?
Strategy
Nosotros are given $latex \boldsymbol{E_{\textbf{cap}}} $ and $latex \boldsymbol{5} $, and nosotros are asked to find the capacitance $latex \boldsymbol{C} $. Of the iii expressions in the equation for E$latex \boldsymbol{E_{\textbf{cap}}} $, the almost convenient relationship is
$latex \boldsymbol{E_{\textbf{cap}} =} $ 
Solution
Solving this expression for $latex \boldsymbol{C} $ and entering the given values yields
$latex \begin{array}{r @{{}={}} fifty} \boldsymbol{C} & \boldsymbol{\frac{2E_{\textbf{cap}}}{V^2} = \frac{2(4.00 \times 10^2 \;\textbf{J})}{(1.00 \times 10^iv \;\textbf{V})^two} = 8.00 \times 10^{-half-dozen} \;\textbf{F}} \\[1em] & \boldsymbol{eight.00 \;\mu \textbf{F}}. \cease{array} $
Discussion
This is a fairly large, but manageable, capacitance at $latex \boldsymbol{1.00 \times 10^iv \;\textbf{V}} $.
Section Summary
- Capacitors are used in a variety of devices, including defibrillators, microminiaturization such as calculators, and wink lamps, to supply energy.
- The energy stored in a capacitor can exist expressed in 3 ways:
$latex \boldsymbol{E_{\textbf{cap}} =}$
$latex \boldsymbol{=} $ $latex \boldsymbol{=} $ where $latex \boldsymbol{Q} $ is the charge, $latex \boldsymbol{V} $ is the voltage, and $latex \boldsymbol{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
one: How does the free 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?
2: What happens to the energy stored in a capacitor connected to a bombardment when a dielectric is inserted? Was piece of work done in the process?
Issues & Exercises
1: (a) What is the energy stored in the $latex \boldsymbol{ten.0 \;\mu \textbf{F}} $ capacitor of a centre defibrillator charged to $latex \boldsymbol{9.00 \times 10^3 \;\textbf{V}} $? (b) Detect the amount of stored accuse.
two: In open heart surgery, a much smaller amount of energy will defibrillate the heart. (a) What voltage is applied to the $latex \boldsymbol{8.00 \; \mu \textbf{F}} $ capacitor of a heart defibrillator that stores 40.0 J of energy? (b) Notice the amount of stored charge.
3: A $latex \boldsymbol{165 \;\mu\textbf{F}} $ capacitor is used in conjunction with a motor. How much energy is stored in it when 119 V is applied?
four: Suppose yous have a 9.00 V battery, a $latex \boldsymbol{two.00 \;\mu \textbf{F}} $ capacitor, and a $latex \boldsymbol{seven.xl \;\mu \textbf{F}} $ capacitor. (a) Notice the charge and free energy stored if the capacitors are continued to the battery in series. (b) Practise the same for a parallel connection.
five: 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 expanse $latex \boldsymbol{1.00 \times 10^ii \;\textbf{one thousand}^2} $ and are 0.200 one thousand autonomously? (b) What is the voltage betwixt them if opposite charges of magnitude 2.00 nC are placed on them? (c) To show that this voltage poses a small take a chance, calculate the energy stored.
6: Show that for a given dielectric material the maximum energy a parallel plate capacitor can store is directly proportional to the book of dielectric ($latex \boldsymbol{\textbf{Book} = A \cdot d} $). Note that the applied voltage is express by the dielectric strength.
seven: Construct Your Own Problem
Consider a middle defibrillator similar to that discussed in Example ane. Construct a problem in which yous examine the charge stored in the capacitor of a defibrillator equally a function of stored free energy. Among the things to exist considered are the practical voltage and whether it should vary with energy to exist delivered, the range of energies involved, and the capacitance of the defibrillator. Y'all may likewise wish to consider the much smaller energy needed for defibrillation during open-center surgery as a variation on this problem.
8: Unreasonable Results
(a) On a particular day, information technology takes $latex \boldsymbol{9.60 \times 10^3 \;\textbf{J}} $ of electric energy to start a truck's engine. Summate the capacitance of a capacitor that could shop that amount of energy at 12.0 V. (b) What is unreasonable well-nigh this consequence? (c) Which assumptions are responsible?
Glossary
- defibrillator
- a machine used to provide an electrical shock to a heart assault victim's eye in order to restore the heart's normal rhythmic pattern
Solutions
Problems & Exercises
1: (a) 405 J
(b) 90.0 mC
2: (a) 3.16 kV
(b) 25.3 mC
4: (a) $latex \boldsymbol{1.42 \times 10^{-5} \;\textbf{C}} $, $latex \boldsymbol{6.38 \times 10^{-5} \;\textbf{J}} $
(b) $latex \boldsymbol{8.46 \times ten^{-v} \;\textbf{C}} $, $latex \boldsymbol{3.81 \times x^{-4} \;\textbf{J}} $
5: (a) $latex \boldsymbol{iv.43 \times 10^{-12} \;\textbf{F}} $
(b) 452 Five
(c) $latex \boldsymbol{4.52 \times 10^{-7} \;\textbf{J}} $
eight: (a) 133 F
(b) Such a capacitor would be besides large to carry with a truck. The size of the capacitor would exist enormous.
(c) It is unreasonable to assume that a capacitor can store the amount of energy needed.
Source: https://pressbooks.uiowa.edu/clonedbook/chapter/energy-stored-in-capacitors/
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