1 2 This lecture introduces electric

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This lecture introduces electric potential energy and something called “electric potential. ” If you understand the symbols in the starting equations, and avoid sign and direction mistakes, homework and exams are not difficult. 3

Electric Potential Energy r b dr ds a 4

r b dr ds I did the calculation for a + charge moving away from a – charge; you could do a similar calculation for ++, +, and ++. a 5

The next two slides are intended to draw a parallel between the electric and gravitational forces. Instead of saying “ugh, this is confusing new stuff, ” you are supposed to say, “oh, this is easy, because I already learned the concepts in Physics 103. ” 6

A bit of review: 7

A charged particle in an electric field has electric potential energy. +++++++ It “feels” a force (as given by Coulomb’s law). It gains kinetic energy and loses potential energy if released. The Coulomb force does positive work, and mechanical energy is conserved. 8

The next two slides define electrical potential energy. 9

Now that we realize the electric force is conservative, we can define a potential energy associated with it. r b dr ds a q 10

is equivalent to your starting equation The next two slides use this definition of electrical potential energy to derive an equation for the electrical potential energy of two charged particles. 11

r b dr ds a By convention, we choose electric potential energy to be zero at infinite separation of the charges. q If there any math majors in the room, please close your eyes for a few seconds. We should be talking about limits. 0 0 12

What is the meaning of the + sign in the result? 13

If an external force* moves an object “against” the conservative force, then 14

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From we see that the electric potential of a point charge q is The electric potential difference between points a and b is 17

Things to remember about electric potential: Electric potential difference is the work per unit of charge that must be done to move a charge from one point to another without changing its kinetic energy. 18

Things to remember about electric potential: It is always necessary to define where U and V are zero. Here we defined V to be zero at an infinite distance from the sources of the electric field. Sometimes it is convenient to define V to be zero at the earth (ground). It should be clear from the context where V is defined to be zero, and I do not foresee you experiencing any confusion about where V is zero. 19

Two more starting equations: and so Potential energy and electric potential are defined relative to some reference point, so it is “better” to use 20

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Two conceptual examples. Example: a proton is released in a region in space where there is an electric potential. Describe the subsequent motion of the proton. Example: an electron is released in a region in space where there is an electric potential. Describe the subsequent motion of the electron. 22

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To find the electric potential energy for a system of two charges, we bring a second charge in from an infinite distance away: r before after 24

To find the electric potential energy for a system of three charges, we bring a third charge in from an infinite distance away: before after 25

Collection of charges: Charge distribution: dq P r 26

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Example: find the total potential energy of the system of three charges. 29

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An electron volt (e. V) is the energy acquired by a particle of charge e when it moves through a potential difference of 1 volt. This is a very small amount of energy on a macroscopic scale, but electrons in atoms typically have a few e. V (10’s to 1000’s) of energy. 31

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Example: potential and electric field between two parallel conducting plates. Also, let the plates be separated by a distance d. E d 33

E d 34

Important note: the derivation of did not require rectangular plates, or any plates at all. It works as long as E is uniform. In general, E should be replaced by the component of the displacement vector. along 35

Example: A rod of length L located along the x-axis has a total charge Q uniformly distributed along the rod. Find the electric potential at a point P along the y-axis a distance d from the origin. 36

A good set of math tables will have the integral: 37

Example: Find the electric potential due to a uniformly charged ring of radius R and total charge Q at a point P on the axis of the ring. Every dq of charge on the ring is the same distance from the point P. 38

Could you use this expression for V to calculate E? Would you get the same result as we obtained previously? 39

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We can use the equation for the potential due to a ring, replace R by r, and integrate from r=0 to r=R. 41

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Could you use this expression for V to calculate E? 43

See your text for other examples of potentials calculated from charge distributions, as well as an alternate discussion of the electric field between charged parallel plates. 44

For some reason you think practical applications are important. Well, I found one! 45

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Equipotentials are contour maps of the electric potential. 47

Equipotential lines are another visualization tool. They illustrate where the potential is constant. Equipotential lines are actually projections on a 2 -dimensional page of a 3 dimensional equipotential surface. (“Just like” the contour map. ) The electric field must be perpendicular to equipotential lines. Why? In the static case (charges not moving) the surface of a conductor is an equipotential surface. Why? 48

Here are some electric field and equipotential lines I generated using an electromagnetic field program. Equipotential lines are shown in red. 49

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The electric field vector points from higher to lower potentials. More specifically, E points along shortest distance from a higher equipotential surface to a lower equipotential surface. You can use E to calculate V: You can use the differential version of this equation to calculate E from a known V: 51

For spherically symmetric charge distribution: In one dimension: In three dimensions: 52

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When there is a net flow of charge inside a conductor, the physics is generally complex. When there is no net flow of charge, or no flow at all (the electrostatic case), then a number of conclusions can be reached using Gauss’ “Law” and the concepts of electric fields and potentials… 56

Summary of key points (electrostatic case): The electric field inside a conductor is zero. Any net charge on the conductor lies on the outer surface. The potential on the surface of a conductor, and everywhere inside, is the same. The electric field just outside a conductor must be perpendicular to the surface. Equipotential surfaces just outside the conductor must be parallel to the conductor’s surface. 57

Another key point: the charge density on a conductor surface will vary if the surface is irregular, and surface charge collects at “sharp points. ” Therefore the electric field is large (and can be huge) near “sharp points. ” To best shock somebody, don’t touch them with your hand; touch them with your fingertip. Better yet, hold a small piece of bare wire in your hand gently touch them with that. 58

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Capacitors and Dielectrics A capacitor is basically two parallel conducting plates with air or insulating material in between. E L A capacitor doesn’t have to look like metal plates. 60

The symbol representing a capacitor in an electric circuit looks like parallel plates. Here’s the symbol for a battery, or an external potential. When a capacitor is connected to an external potential, charges flow onto the plates and create a potential difference between the plates. 61

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If the external potential is disconnected, charges remain on the plates, so capacitors are good for storing charge (and energy). +- +- Capacitors are also very good at releasing their stored charge all at once. The capacitors in your tube-type TV are so good at storing energy that touching the two terminals at the same time can be fatal, even though the TV may not have been used for months. High-voltage TV capacitors are supposed to have “bleeder resistors” that drain the charge away after the circuit is turned off. I wouldn’t bet my life on it. 63

assortment of capacitors 64

+Q -Q C + V The magnitude of charge acquired by each plate of a capacitor is Q=CV where C is the capacitance of the capacitor. 65

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We previously calculated the electric field between two parallel charged plates: -Q +Q E This is valid when the separation is small compared with the plate dimensions. This lets us calculate C for a parallel plate capacitor. d A 67

Reminders: Q is the magnitude of the charge on either plate. C is always positive. 68

Parallel plate capacitance depends “only” on geometry. -Q +Q E This expression is approximate, and must be modified if the plates are small, or separated by a medium other than a vacuum. d A 69

An isolated sphere can be thought of as concentric spheres with the outer sphere at an infinite distance and zero potential. We already know the potential outside a conducting sphere: The potential at the surface of a charged sphere of radius R is so the capacitance at the surface of an isolated sphere is 70

Let’s calculate the capacitance of a concentric spherical capacitor of charge Q. In between the spheres 71

alternative calculation of capacitance of isolated sphere 72

We can also calculate the capacitance of a cylindrical capacitor (made of coaxial cylinders). The next slide shows a cross-section view of the cylinders. L 73

Lowercase c is capacitance per unit length: 74

Example: calculate the capacitance of a capacitor whose plates are 20 cm x 3 cm and are separated by a 1. 0 mm air gap. d = 0. 001 area = 0. 2 x 0. 03 75

Example: what is the charge on each plate if the capacitor is connected to a 12 volt* battery? 0 V +12 V 76

Example: what is the electric field between the plates? 0 V d = 0. 001 +12 V 77

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Recall: this is the symbol representing a capacitor in an electric circuit. And this is the symbol for a battery… …or this. 79

Capacitors connected in parallel: The potential difference (voltage drop) from a to b must equal V. 80

Q=CV Now imagine replacing the parallel combination of capacitors by a single equivalent capacitor. By “equivalent, ” we mean “stores the same total charge if the voltage is the same. ” 81

Summarizing the equations on the last slide: Generalizing: 82

Capacitors connected in series: 83

The charges +Q and –Q attract equal and opposite charges to the other plates of their respective capacitors: These equal and opposite charges came from the originally neutral circuit regions A and B. 84

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Let’s replace three capacitors by a single equivalent capacitor. By “equivalent” we mean V is the same as the total voltage drop across the three capacitors, and the amount of charge Q that flowed out of the battery is the same as when there were three capacitors. 86

Collecting equations: Substituting for V: Dividing both sides by Q: 87

Generalizing: (capacitors in series) 88

I don’t see a series combination of capacitors, but I do see a parallel combination. 89

Now I see a series combination. 90

I’ll work this at the blackboard. 91

Electric charge and electric force Coulomb’s Law Electric field calculating electric field motion of a charged particle in an electric field Gauss’ “Law” electric flux calculating electric field using Gaussian surfaces properties of conductors 92

Electric potential and electric potential energy calculating potentials and potential energy calculating fields from potentials equipotentials and fields near conductors Capacitors capacitance of parallel plates, concentric cylinders, (concentric spheres not for this exam) equivalent capacitance of capacitor network Don’t forget the concepts from Physics 103 that were frequently used! 93

Three charges +Q, and –Q, are located at the corners of an equilateral triangle with sides of length a. What is the force on the charge located at point P (see diagram)? 94

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What is the electric field at P due to the two charges at the base of the triangle? You can repeat the above calculation, replacing F by E. Or… 96

An insulating spherical shell has an inner radius a and outer radius b. The shell has a total charge Q and a uniform charge density. Find the magnitude of the electric field for r

An insulating spherical shell has an inner radius a and outer radius b. The shell has a total charge Q and a uniform charge density. Find the magnitude of the electric field for a

An electron has a speed v. Calculate the magnitude and direction of an electric field that will stop this electron in a distance D. 99

Two equal positive charges Q are located at the base of an equilateral triangle with sides of length a. What is the potential at point P (see diagram)? 100

Three equal positive charges Q are located at the corners of an equilateral triangle with sides of length a. What is the potential energy of the charge located at point P (see diagram)? 101

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Let’s calculate how much work it takes to charge a capacitor. + dq + We start with zero charge on the capacitor, and end up with Q, so +q -q 105

The work required to charge the capacitor is the amount of energy you get back when you discharge the capacitor (because the electric force is conservative). Thus, the work required to charge the capacitor is equal to the potential energy stored in the capacitor. Because C, Q, and V are related through Q=CV, there are three equivalent ways to write the potential energy. 106

All three equations are valid; use the one most convenient for the problem at hand. It is no accident that we use the symbol U for the energy stored in a capacitor. It is just another “version” of electrical potential energy. You can use it in your energy conservation equations just like any other form of potential energy! 107

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Example: compare the amount of energy stored in a capacitor with the amount of energy stored in a battery. If a battery stores so much more energy, why use capacitors? 109

Energy is stored in the capacitor: + +Q - -Q The “volume of the capacitor” is Volume=Ad 110

Energy stored per unit volume (u): The energy is “stored” in the electric field! + - We’ve gone from the concrete (electric charges experience forces)… …to the abstract (electric charges create electric fields)… …to an application of the abstraction (electric field contains energy). +Q -Q 111

+ - This is not a new “kind” of energy. It’s the electric potential energy resulting from the coulomb force between charged particles. +Q -Q Or you can think of it as the electric energy due to the field created by the charges. Same thing. 112

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The dielectric is the thin insulating sheet in between the plates of a capacitor. Any reasons to use a dielectric in a capacitor? 115

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Example: how much charge would the capacitor on the previous slide hold if the dielectric were air? The calculation is the same, except replace 6. 7 by 1. Or just divide the charge on the previous page by 6. 7 to get. 117

Conceptual Example V=0 A capacitor connected as shown acquires a charge Q. While the capacitor is still connected to the battery, a dielectric material is inserted. V V Will Q increase, decrease, or stay the same? Why? 118

Example: find the energy stored in the capacitor in slide 13. 119

Example: the battery is now disconnected. What are the charge, capacitance, and energy stored in the capacitor? The charge and capacitance are unchanged, so the voltage drop and energy stored are unchanged. 120

Example: the dielectric is removed without changing the plate separation. What are the capacitance, charge, potential difference, and energy stored in the capacitor? 121

Example: the dielectric is removed without changing the plate separation. What are the capacitance, charge, potential difference, and energy stored in the capacitor? The charge remains unchanged, because there is nowhere for it to go. 122

Example: the dielectric is removed without changing the plate separation. What are the capacitance, charge, potential difference, and energy stored in the capacitor? Knowing C and Q we can calculate the new potential difference. 123

Example: the dielectric is removed without changing the plate separation. What are the capacitance, charge, potential difference, and energy stored in the capacitor? 124

Sure. It took work to remove the dielectric. The stored energy increased by the amount of work done. 125