to what potential should you charge a 2.0 μf capacitor to store 1.0 j of energy?

Learning Objectives

By the cease of this section, yous volition exist able to:

• List some uses of capacitors.
• Express in equation form the free energy stored in a capacitor.
• Explain the office of a defibrillator.

Near of united states of america take seen dramatizations in which medical personnel employ a defibrillator to pass an electrical current through a patient'due south heart to get it to trounce commonly. (Review Figure ane.) Often realistic in particular, the person applying the shock directs another person to "make information technology 400 joules this fourth dimension." The energy delivered by the defibrillator is stored in a capacitor and can be adapted to fit the state of affairs. 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. (Encounter Effigy 1.) Capacitors are besides used to supply energy for flash lamps on cameras.

Figure 1. Free energy stored in the large capacitor is used to preserve the memory of an electronic calculator when its batteries are charged. (credit: Kucharek, Wikimedia Commons)

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 exist careful when applying the equation for electric potential energy ΔPE = qΔV to a capacitor. Remember that ΔPE is the potential energy of a charge q going through a voltage ΔV. Just the capacitor starts with zero voltage and gradually comes upwardly to its full voltage every bit information technology is charged. The first charge placed on a capacitor experiences a change in voltage Δ5 = 0, since the capacitor has zero voltage when uncharged. The final charge placed on a capacitor experiences Δ5 =V, since the capacitor at present has its total voltage V on it. The average voltage on the capacitor during the charging process is [latex]\frac{5}{two}\\[/latex], then the average voltage experienced by the full charge q is [latex]\frac{V}{2}\\[/latex]. Thus the free energy stored in a capacitor, E _cap, is [latex]E_{\text{cap}}=\frac{QV}{2}\\[/latex], where Q is the charge on a capacitor with a voltage V applied. (Note that the free energy is not QV, but [latex]\frac{QV}{2}\\[/latex].) Charge and voltage are related to the capacitance C of a capacitor by Q =CV, and so the expression for E _cap tin be algebraically manipulated into three equivalent expressions:

[latex]\displaystyle{E}_{\text{cap}}=\frac{QV}{2}=\frac{CV^2}{2}=\frac{Q^2}{2C}\\[/latex],

where Q is the accuse 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.

Free energy Stored in Capacitors

The free energy stored in a capacitor can be expressed in three ways:

[latex]\displaystyle{Eastward}_{\text{cap}}=\frac{QV}{2}=\frac{CV^two}{ii}=\frac{Q^two}{2C}\\[/latex],

where Q is the accuse, V is the voltage, and C is the capacitance of the capacitor. The free 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 flare-up to a fix of paddles beyond a person's chest can be a lifesaver. The person's heart set on might accept arisen from the onset of fast, irregular beating of the heart—cardiac or ventricular fibrillation. The awarding of a large daze of electric free energy can end the arrhythmia and let the torso's pacemaker to resume normal patterns. Today it is mutual for ambulances to carry a defibrillator, which also uses an electrocardiogram to clarify the patient's heartbeat blueprint. Automatic external defibrillators (AED) are plant in many public places (Figure ii). These are designed to be used past lay persons. The device automatically diagnoses the patient's centre condition and and then applies the daze with appropriate energy and waveform. CPR is recommended in many cases before use of an AED.

Figure 2. Automated external defibrillators are found in many public places. These portable units provide verbal instructions for use in the of import commencement few minutes for a person suffering a cardiac attack. (credit: Owain Davies, Wikimedia Commons)

Instance ane. Capacitance in a Center Defibrillator

A center defibrillator delivers 4.00 × 10² J of free energy by discharging a capacitor initially at 1.00 × x⁴ V. What is its capacitance?

Strategy

We are given E _cap and 5, and we are asked to find the capacitance C. Of the three expressions in the equation for Eastward _cap, the most user-friendly relationship is [latex]E_{\text{cap}}=\frac{CV^2}{two}\\[/latex].

Solution

Solving this expression for C and entering the given values yields

[latex]\begin{array}{lll}C&=&\frac{2E_{\text{cap}}}{V^two}=\frac{2\left(iv.00\times10^two\text{ J}\right)}{\left(1.00\times10^4\text{ V}\correct)^2}=8.00\times10^{-6}\text{ F}\\\text{ }&=&8.00\mu\text{F}\end{assortment}\\[/latex]

Discussion

This is a fairly large, but manageable, capacitance at one.00 × 10^iv 5.

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 exist expressed in iii ways: [latex]{Eastward}_{\text{cap}}=\frac{\text{QV}}{2}=\frac{{\text{CV}}^{2}}{2}=\frac{{Q}^{2}}{2C}\\[/latex], whereQ is the charge, Five 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

1. How does the energy independent in a charged capacitor change when a dielectric is inserted, bold the capacitor is isolated and its charge is abiding? Does this imply that work was washed?
2. What happens to the free energy stored in a capacitor connected to a battery when a dielectric is inserted? Was work washed in the process?

Bug & Exercises

1. (a) What is the energy stored in the x.0 μF capacitor of a heart defibrillator charged to
   9.00 × 10³ Five? (b) Detect the amount of stored charge.
2. In open up heart surgery, a much smaller amount of free energy will defibrillate the heart. (a) What voltage is applied to the 8.00 μF capacitor of a middle defibrillator that stores 40.0 J of energy? (b) Detect the amount of stored charge.
3. A 165 μF capacitor is used in conjunction with a motor. How much energy is stored in it when 119 V is applied?
4. Suppose you have a 9.00 V battery, a 2.00 μF capacitor, and a 7.twoscore μF capacitor. (a) Find the charge and energy stored if the capacitors are connected to the battery in series. (b) Exercise the same for a parallel connection.
5. 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 accept area ane.00 × 10^two one thousand² and are 0.200 thou autonomously? (b) What is the voltage between them if reverse charges of magnitude two.00 nC are placed on them? (c) To show that this voltage poses a small risk, calculate the energy stored.
6. Show that for a given dielectric material the maximum energy a parallel plate capacitor can shop is direct proportional to the volume of dielectric (Book = A · d). Note that the applied voltage is express by the dielectric strength.
7. Construct Your Ain Problem. Consider a heart defibrillator like to that discussed in Instance 1. Construct a problem in which you examine the charge stored in the capacitor of a defibrillator equally a function of stored energy. Amidst the things to exist considered are the applied voltage and whether information technology should vary with energy to exist 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 up-heart surgery every bit a variation on this problem.
8. Unreasonable Results.(a) On a particular twenty-four hour period, it takes 9.60 × 10ⁱⁱⁱ J of electric energy to offset a truck'due south engine. Calculate the capacitance of a capacitor that could store that amount of free energy at 12.0 V. (b) What is unreasonable about this upshot? (c) Which assumptions are responsible?

Glossary

defibrillator: a automobile used to provide an electrical shock to a heart assail victim's eye in order to restore the heart'southward normal rhythmic design

Selected Solutions to Problems & Exercises

1. (a) 405 J; (b) 90.0 mC

two. (a) 3.xvi kV; (b) 25.3 mC

iv. (a) 1.42×10^−v C, 6.38×10⁻⁵ J; (b) 8.46×10⁻⁵ C, 3.81×ten^−four J

5. (a) 4.43 × 10^–12 F; (b) 452 V; (c) 4.52 × ten^–vii J

8. (a) 133 F; (b) Such a capacitor would be too large to acquit with a truck. The size of the capacitor would be enormous; (c) It is unreasonable to presume that a capacitor can store the corporeality of energy needed.

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Source: https://courses.lumenlearning.com/physics/chapter/19-7-energy-stored-in-capacitors/

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