General Physics PHY 2140 Lecture 2 Electrostatics Electric

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General Physics (PHY 2140) Lecture 2 Ø Electrostatics ü Electric flux and Gauss’s law

General Physics (PHY 2140) Lecture 2 Ø Electrostatics ü Electric flux and Gauss’s law ü Electrical energy ü potential difference and electric potential ü potential energy of charged conductors http: //www. physics. wayne. edu/~alan/ Chapters 15 -16 11/29/2020 1

Lightning Review Last lecture: 1. Coulomb’s law ü the superposition principle 2. The electric

Lightning Review Last lecture: 1. Coulomb’s law ü the superposition principle 2. The electric field Review Problem: A “free” electron and “free” proton are placed in an identical electric field. Compare the electric force on each particle. Compare their accelerations. 11/29/2020 2

Review Solution: Recall For a proton or an electron the size of the force

Review Solution: Recall For a proton or an electron the size of the force is the same! n n n Charges are the same in magnitude. Opposite in sign. The direction of the electric forces are opposite. However the accelerations are different! n n Mass of electron is 9. 11 x 10 -31 Kg Mass of a proton is 1. 67 x 10 -27 Kg So, the acceleration of the proton is smaller by me/mp = 5. 5 x 10 -4 11/29/2020 3

15. 5 Electric Field Lines A convenient way to visualize field patterns is to

15. 5 Electric Field Lines A convenient way to visualize field patterns is to draw lines in the direction of the electric field. Such lines are called field lines. Remarks: 1. 2. Electric field vector, E, is tangent to the electric field lines at each point in space. The number of lines per unit area through a surface perpendicular to the lines is proportional to the strength of the electric field in a given region. E is large when the field lines are close together and small when far apart. 11/29/2020 4

15. 5 Electric Field Lines (2) Electric field lines of single positive (a) and

15. 5 Electric Field Lines (2) Electric field lines of single positive (a) and (b) negative charges. a) b) + q 11/29/2020 - q 5

15. 5 Electric Field Lines (3) Rules for drawing electric field lines for any

15. 5 Electric Field Lines (3) Rules for drawing electric field lines for any charge distribution. 1. 2. 3. 11/29/2020 Lines must begin on positive charges (or at infinity) and must terminate on negative charges or in the case of excess charge at infinity. The number of lines drawn leaving a positive charge or approaching a negative charge is proportional to the magnitude of the charge. No two field lines can cross each other. 6

15. 5 Electric Field Lines (4) Electric field lines of a dipole. + 11/29/2020

15. 5 Electric Field Lines (4) Electric field lines of a dipole. + 11/29/2020 - 7

15. 6 Conductors in Electrostatic Equilibrium Good conductors (e. g. copper, gold) contain charges

15. 6 Conductors in Electrostatic Equilibrium Good conductors (e. g. copper, gold) contain charges (electron) that are not bound to a particular atom, and are free to move within the material. When no net motion of these electrons occur the conductor is said to be in electro-static equilibrium. 11/29/2020 8

15. 6 Conductors in Electrostatic Equilibrium Properties of an isolated conductor (insulated from the

15. 6 Conductors in Electrostatic Equilibrium Properties of an isolated conductor (insulated from the ground). 1. 2. 3. 4. 11/29/2020 Electric field is zero everywhere within the conductor. Any excess charge on an isolated conductor resides entirely on its surface. The electric field just outside a charged conductor is perpendicular to the conductor’s surface. On an irregular shaped conductor, the charge tends to accumulate at locations where the radius of curvature of the surface is smallest – at sharp points. 9

Faraday’s ice-pail experiment +++++ + + - - - + +++++ + + +

Faraday’s ice-pail experiment +++++ + + - - - + +++++ + + + + - + + + + + Demonstrates that the charge resides on the surface of a conductor. 11/29/2020 10

Mini-quiz Question: Suppose a point charge +Q is in empty space. Wearing rubber gloves,

Mini-quiz Question: Suppose a point charge +Q is in empty space. Wearing rubber gloves, we sneak up and surround the charge with a spherical conducting shell. What effect does this have on the field lines of the charge? ? + q 11/29/2020 + 11

Question: Suppose a point charge +Q is in empty space. Wearing rubber gloves, we

Question: Suppose a point charge +Q is in empty space. Wearing rubber gloves, we sneak up and surround the charge with a spherical conducting shell. What effect does this have on the field lines of the charge? Answer: Negative charge will build up on the inside of the shell. Positive charge will build up on the outside of the shell. There will be no field lines inside the conductor but the field lines will remain outside the shell. + + - + q - + + - - + 11/29/2020 + - - + + 12

15. 9 The oscilloscope Changing E field applied on the deflection plate (electrodes) moves

15. 9 The oscilloscope Changing E field applied on the deflection plate (electrodes) moves the electron beam. V 2 d 11/29/2020 V 1 L 13

Oscilloscope: deflection angle (additional) V 2 d V 1 11/29/2020 L 14

Oscilloscope: deflection angle (additional) V 2 d V 1 11/29/2020 L 14

15. 9 Electric Flux and Gauss’s Law A convenient technique was introduced by Karl

15. 9 Electric Flux and Gauss’s Law A convenient technique was introduced by Karl F. Gauss (1777 -1855) to calculate electric fields. Requires symmetric charge distributions. Technique based on the notion of electrical flux. 11/29/2020 15

15. 9 Electric Flux To introduce the notion of flux, consider a situation where

15. 9 Electric Flux To introduce the notion of flux, consider a situation where the electric field is uniform in magnitude and direction. Consider also that the field lines cross a surface of area A which is perpendicular to the field. The number of field lines per unit of area is constant. The flux, F, is defined as the product of the field magnitude by the area crossed by the field lines. Area=A E 11/29/2020 16

15. 9 Electric Flux Units: Nm 2/C in SI units. Find the electric flux

15. 9 Electric Flux Units: Nm 2/C in SI units. Find the electric flux through the area A = 2 m 2, which is perpendicular to an electric field E=22 N/C Answer: F = 44 Nm 2/C. 11/29/2020 17

15. 9 Electric Flux If the surface is not perpendicular to the field, the

15. 9 Electric Flux If the surface is not perpendicular to the field, the expression of the field becomes: Where q is the angle between the field and a normal to the surface. N q q 11/29/2020 18

15. 9 Electric Flux Remark: When an area is constructed such that a closed

15. 9 Electric Flux Remark: When an area is constructed such that a closed surface is formed, we shall adopt the convention that the flux lines passing into the interior of the volume are negative and those passing out of the interior of the volume are positive. 11/29/2020 19

Example: Question: Calculate the flux of a constant E field (along x) through a

Example: Question: Calculate the flux of a constant E field (along x) through a cube of side “L”. y 1 2 E x z 11/29/2020 20

Question: Calculate the flux of a constant E field (along x) through a cube

Question: Calculate the flux of a constant E field (along x) through a cube of side “L”. Reasoning: l Dealing with a composite, closed surface. l Sum of the fluxes through all surfaces. l Flux of field going in is negative l Flux of field going out is positive. l E is parallel to all surfaces except surfaces labeled 1 and 2. l So only those surfaces (1 & 2) contribute to the flux. y 1 2 E x z 11/29/2020 21

Question: Calculate the flux of a constant E field (along x) through a cube

Question: Calculate the flux of a constant E field (along x) through a cube of side “L”. Reasoning: l Dealing with a composite, closed surface. l Sum of the fluxes through all surfaces. l Flux of field going in is negative l Flux of field going out is positive. l E is parallel to all surfaces except surfaces labeled 1 and 2. l So only those surface contribute to the flux. Solution: y 1 2 E x z 11/29/2020 22

15. 9 Gauss’s Law The net flux passing through a closed surface surrounding a

15. 9 Gauss’s Law The net flux passing through a closed surface surrounding a charge Q is proportional to the magnitude of Q: In free space, the constant of proportionality is 1/eo where eo is called the permittivity of of free space. 11/29/2020 23

15. 9 Gauss’s Law The net flux passing through any closed surface is equal

15. 9 Gauss’s Law The net flux passing through any closed surface is equal to the net charge inside the surface divided by eo. Can be used to compute electric fields. Example: point charge 11/29/2020 24

16. 0 Introduction The Coulomb force is a conservative force A potential energy function

16. 0 Introduction The Coulomb force is a conservative force A potential energy function can be defined for any conservative force, including Coulomb force The notions of potential and potential energy are important for practical problem solving 11/29/2020 25

16. 1 Potential difference and electric potential The electrostatic force is conservative As in

16. 1 Potential difference and electric potential The electrostatic force is conservative As in mechanics, work is B A d 11/29/2020 Work done on the positive charge by moving it from A to B 26

Potential energy of electrostatic field The work done by a conservative force equals the

Potential energy of electrostatic field The work done by a conservative force equals the negative of the change in potential energy, DPE This equation n n is valid only for the case of a uniform electric field allows us to introduce the concept of the electric potential 11/29/2020 27

Electric potential The potential difference between points A and B, VB-VA, is defined as

Electric potential The potential difference between points A and B, VB-VA, is defined as the change in potential energy (final minus initial value) of a charge, q, moved from A to B, divided by the charge Electric potential is a scalar quantity Electric potential difference is a measure of electric energy per unit charge Potential is often referred to as “voltage” 11/29/2020 28

Electric potential - units Electric potential difference is the work done to move a

Electric potential - units Electric potential difference is the work done to move a charge from a point A to a point B divided by the magnitude of the charge. Thus the SI units of electric potential In other words, 1 J of work is required to move a 1 C of charge between two points that are at potential difference of 1 V 11/29/2020 29

Electric potential - notes Units of electric field (N/C) can be expressed in terms

Electric potential - notes Units of electric field (N/C) can be expressed in terms of the units of potential (as volts per meter) Because the positive tends to move in the direction of the electric field, work must be done on the charge to move it in the direction, opposite the field. Thus, n n A positive charge gains electric potential energy when it is moved in a direction opposite the electric field A negative charge looses electrical potential energy when it moves in the direction opposite the electric field 11/29/2020 30

Analogy between electric and gravitational fields The same kinetic-potential energy theorem works here A

Analogy between electric and gravitational fields The same kinetic-potential energy theorem works here A A q B d m d B If a positive charge is released from A, it accelerates in the direction of electric field, i. e. gains kinetic energy If a negative charge is released from A, it accelerates in the direction opposite the electric field 11/29/2020 31

Example: motion of an electron What is the speed of an electron accelerated from

Example: motion of an electron What is the speed of an electron accelerated from rest across a potential difference of 100 V? What is the speed of a proton accelerated under the same conditions? Given: DV=100 V me = 9. 11´ 10 -31 kg mp = 1. 67´ 10 -27 kg |e| = 1. 60´ 10 -19 C Vab Observations: 1. given potential energy difference, one can find the kinetic energy difference 2. kinetic energy is related to speed Find: ve=? vp=? 11/29/2020 32

16. 2 Electric potential and potential energy due to point charges Electric circuits: point

16. 2 Electric potential and potential energy due to point charges Electric circuits: point of zero potential is defined by grounding some point in the circuit Electric potential due to a point charge at a point in space: point of zero potential is taken at an infinite distance from the charge With this choice, a potential can be found as Note: the potential depends only on charge of an object, q, and a distance from this object to a point in space, r. 11/29/2020 33

Superposition principle for potentials If more than one point charge is present, their electric

Superposition principle for potentials If more than one point charge is present, their electric potential can be found by applying superposition principle The total electric potential at some point P due to several point charges is the algebraic sum of the electric potentials due to the individual charges. Remember that potentials are scalar quantities! 11/29/2020 34

Potential energy of a system of point charges Consider a system of two particles

Potential energy of a system of point charges Consider a system of two particles If V 1 is the electric potential due to charge q 1 at a point P, then work required to bring the charge q 2 from infinity to P without acceleration is q 2 V 1. If a distance between P and q 1 is r, then by definition q 2 P q 1 r A Potential energy is positive if charges are of the same sign and vice versa. 11/29/2020 35

Mini-quiz: potential energy of an ion Three ions, Na+, and Cl-, located such, that

Mini-quiz: potential energy of an ion Three ions, Na+, and Cl-, located such, that they form corners of an equilateral triangle of side 2 nm in water. What is the electric potential energy of one of the Na+ ions? Cl? Na+ 11/29/2020 Na+ 36

Recall from last chapter: Electric field lines of a dipole. + 11/29/2020 - 37

Recall from last chapter: Electric field lines of a dipole. + 11/29/2020 - 37

16. 3 Potentials and charged conductors Recall that work is opposite of the change

16. 3 Potentials and charged conductors Recall that work is opposite of the change in potential energy, No work is required to move a charge between two points that are at the same potential. That is, W=0 if VB=VA Recall: 1. all charge of the charged conductor is located on its surface 2. electric field, E, is always perpendicular to its surface, i. e. no work is done if charges are moved along the surface Thus: potential is constant everywhere on the surface of a charged conductor in equilibrium … but that’s not all! 11/29/2020 38

Because the electric field in zero inside the conductor, no work is required to

Because the electric field in zero inside the conductor, no work is required to move charges between any two points, i. e. If work is zero, any two points inside the conductor have the same potential, i. e. potential is constant everywhere inside a conductor Finally, since one of the points can be arbitrarily close to the surface of the conductor, the electric potential is constant everywhere inside a conductor and equal to its value at the surface! Note that the potential inside a conductor is not necessarily zero, even though the interior electric field is always zero! 11/29/2020 39

The electron volt A unit of energy commonly used in atomic, nuclear and particle

The electron volt A unit of energy commonly used in atomic, nuclear and particle physics is electron volt (e. V) The electron volt is defined as the energy that electron (or proton) gains when accelerating through a potential difference of 1 V Relation to SI: Vab=1 V 1 e. V = 1. 60´ 10 -19 C·V = 1. 60´ 10 -19 J 11/29/2020 40

Problem-solving strategy Remember that potential is a scalar quantity Superposition principle is an algebraic

Problem-solving strategy Remember that potential is a scalar quantity Superposition principle is an algebraic sum of potentials due to a system of charges Signs are important Just in mechanics, only changes in electric potential are significant, hence, the point you choose for zero electric potential is arbitrary. 11/29/2020 41

Example : ionization energy of the electron in a hydrogen atom In the Bohr

Example : ionization energy of the electron in a hydrogen atom In the Bohr model of a hydrogen atom, the electron, if it is in the ground state, orbits the proton at a distance of r = 5. 29´ 10 -11 m. Find the ionization energy of the atom, i. e. the energy required to remove the electron from the atom. Note that the Bohr model, the idea of electrons as tiny balls orbiting the nucleus, is not a very good model of the atom. A better picture is one in which the electron is spread out around the nucleus in a cloud of varying density; however, the Bohr model does give the right answer for the ionization energy 11/29/2020 42

In the Bohr model of a hydrogen atom, the electron, if it is in

In the Bohr model of a hydrogen atom, the electron, if it is in the ground state, orbits the proton at a distance of r = 5. 29 x 10 -11 m. Find the ionization energy, i. e. the energy required to remove the electron from the atom. Given: r = 5. 292 x 10 -11 m me = 9. 11´ 10 -31 kg mp = 1. 67´ 10 -27 kg |e| = 1. 60´ 10 -19 C Find: E=? The ionization energy equals to the total energy of the electron-proton system, with The velocity of e can be found by analyzing the force on the electron. This force is the Coulomb force; because the electron travels in a circular orbit, the acceleration will be the centripetal acceleration: or or Thus, total energy is 11/29/2020 43

16. 4 Equipotential surfaces They are defined as a surface in space on which

16. 4 Equipotential surfaces They are defined as a surface in space on which the potential is the same for every point (surfaces of constant voltage) The electric field at every point of an equipotential surface is perpendicular to the surface convenient to represent by drawing equipotential lines 11/29/2020 44

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