work and electric potential lecture # physics 2

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WORK AND ELECTRIC POTENTIAL

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Electric Potential Energy

• When a test charge is placed in an electric field, it experiences a force.

• If the test charge is moved in the field by some external agent, the work done by the field is the negative of the work done by the external agent.

• is an infinitesimal displacement vector that is oriented tangent to a path through space. The path may be straight or curved and the integral performed along this path is called either a path integral or line integral

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• The work done within the charge-field system by the electric field on the charge is

• At this, work is done by the field, the potential energy of the charge-field system is changed by

• For a finite displacement of the charge from A to B, the change in potential energy of the system is

- because the force is conservative, the line integral does not depend on the path taken by the charge.

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Electric Potential• It is the potential energy per unit charge.• The potential energy is characteristic of the field only.• The potential energy is characteristic of the charge-field system.• The potential is independent of the value of .• The potential has a value at every point in an electric field.• Thus, electric potential is given by the equation,

• The potential difference between two points exists solely because of a source

charge and depends on the source charge distribution.• For a potential energy to exist, there must be a system of two or more charges.• The potential energy belongs to the system and changes only if a charge is

moved relatively to the rest of the system.

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Work and Electric Potential• Assume a charge moves in an electric field without any change in its

kinetic energy. The work performed on the charge is

• The SI unit of electric potential is

• It takes one Joule of work to move a 1 Coulomb charge through a potential difference of 1V. In addition, 1 N/C = 1 V/m, indicates that the electric field as a measure of the rate of change of the electric potential with respect to position.

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.

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Electric Potential and Potential Energy due to Continuous Charge Distribution

• Consider a small charge element . Treat is as a point charge. The potential at some point due to this charge is

• To obtain the total potential at point P, we integrate the equation above to include contributions from all elements of the charge distribution. Integrating both sides we get,

• The electric potential is taken to be zero when point P is infinitely far from the charge distribution.

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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!

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Potential Energy of a System of Point Charges

• Consider a system of two particles• If V1 is the electric potential due to charge q1 at a point P, then work

required to bring the charge q2 from infinity to P without acceleration is q2V1. If a distance between P and q1 is r, then by definition

• Potential energy is positive if charges are of the same sign and vice versa.

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P A

q1q2

r

Mini-quiz: potential energy of an ion

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Three ions, Na+, 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+ Na+

? Na Cl Na Na Na

e e e Cl Na

q q q q qPE k k k q q

r r r

but : !Cl Naq q

0Nae Na Na

qPE k q q

r

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 surface2. 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

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B AW PE q V V

… but that’s not all!

• Because the electric field is 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!

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0B AW q V V

The Electron Volt• A unit of energy commonly used in atomic, nuclear and particle

physics is electron volt (eV)

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:

1 eV = 1.60´10-19 C·V = 1.60´10-19 J

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Vab=1 V

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.

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Example : Ionization Energy of the Electron in a Hydrogen Atom

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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

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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 mme = 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,

E PE KE

22 2182.18 10 J -13.6 eV

2 2e e

e ee

m k ee eE k k

r m r r

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:

e c cm a F

2 2

,2e e

e vPE k KE m

r with

or2 2

2,e e

v em k

r r or

22 ,e

e

ev k

m r

Thus, total energy is

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

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convenient to represent by drawing equipotential lines

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Sample Problems

1. An automobile headlight is connected to 12 V battery. The amount of energy transformed is proportional to how much charge flows, which in turn depends on how long the light is on. Over a given period, 5 C of charge flows through the light. How much is the total energy transformed?

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2. Two parallel plates are charged to a voltage of 50 V. If the separation distance between the plates is 0.050 m, calculate the electric field between them.

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3. Two parallel plates are separated by 0.5 m. An electric field of 6000 N/C exists between the plates. What is the potential difference between the plates?

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4. System below is composed of three charges. Find the symbolic expression for the total electric potential energy of the system.

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𝑞1=10𝑞

𝑞1=−5𝑞

𝑞1=8𝑞4𝑑

3𝑑

5𝑑

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Ready for this?1. A particle (charge = 50) moves in a region where the only force on it is an

electric force. As the particle moves 25 cm from point A to B, its kinetic energy increases by 1.5 mJ. Determine the electric potential difference .

2. Points A at ( 2, 3)m and B at ( 5, 7)m are in region where the electric field is uniform and given by . What is the potential difference ?3. Calculate the total electric potential at point A and B given

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𝑄1 𝐴𝑄3𝑄2 𝐵

.200𝑚 .200𝑚 .300𝑚 .200𝑚

Reference• Physics 2 – Calculus Based Physics by Stewart• Lecture Guide Calculus Based Physics

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ME31FB1GROUP 1 GROUP 2 GROUP 3

CLASS # 1-5 CLASS # 6-10 CLASS # 11-15

GROUP 4 GROUP 5 GROUP 6

CLASS 16-20 CLASS 21-25 CLASS # 26-31

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1. COPY THIS LECTURE.2. STUDY!!!3. PER GROUP: PLEASE BRING MANILA PAPER, PENTEL PEN NEXT MEETING AND READY FOR “READY FOR THIS”

AR21FB1GROUP 1 Group 2

Ausan, Tantiongco, Mendros, Limtuico, Anacito Dingalan, Decierdo, Baguhin, Villanueva, Cruz

Group 3 Group 4

Estrella, Esguerra, Estrada, Dupaya, Nonsol Campos, Quintil, Nunez, Zabala, Teves

Group 5 Group 6

Bibat, Alcaide, Albaniel, Gargar, Gonzales Dela Cruz, Peredo, Ogania, Toledo, Soriano

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1. COPY THIS LECTURE.2. STUDY!!!3. PER GROUP: PLEASE BRING MANILA PAPER, PENTEL PEN NEXT MEETING AND READY FOR “READY FOR THIS”

EE31FA1GROUP 1 GROUP 2

Acotanza, Calingasan, Canezal, Perolina, Rivera

Morga, Opog, Pontemayor, Roadiel, Tolentino

GROUP 3 GROUP 4

Balde, Castillo, Castro, Cruz, Dumaguit Amlon, Anduque, Fadri, Micua, Morillo

GROUP 5 GROUP 6

Mendez, Pagdanganan, Panol, Peralta, Santos Gatchalian, Narvacan, Ratin, Sibuan, Simon

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1. COPY THIS LECTURE.2. STUDY!!!3. PER GROUP: PLEASE BRING MANILA PAPER, PENTEL PEN NEXT MEETING AND READY FOR “READY FOR THIS”