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CONCEPT: ELECTRIC POTENTIAL ENERGY If you release 2 charges, they move gain. Where did it come from? - Two charges have a stored energy between them, called - ENERGY CONSERVATION: ΔU = ΔK Electric potential energy between TWO POINT CHARGES: U = q 1 q 2 r EXAMPLE: How far apart do a 3 µc and a 2 µc charge have to be so that their potential energy is 100 mj? Potential energy for a GROUP OF CHARGES: - U = EXAMPLE: How much potential energy is carried by the following system of charges? 1 C 4 m -2 C 3 m 3 C Page 2
CONCEPT: ELECTRIC POTENTIAL ELECTRIC POTENTIAL, also called simply POTENTIAL, is related to, but different from Electric Potential ENERGY. FIELD FORCE POTENTIAL ENERGY - A single charge produces an Electric FIELD - Field tells charges how much to feel - So we think of this as a(n) Field - A single charge also produces an Electric POTENTIAL - Potential tells charges how much to have - So we think of this as a(n) Field - Once there s a second charge, there is F = q E - E is the strength of the field - q is the [ PRODUCING / FEELING ] charge - Once there s a second charge, there is U = - V is the strength of the field - q is the [ PRODUCING / FEELING ] charge Electric FIELD E = FORCE Field Electric POTENTIAL V = ENERGY Field The UNIT of Electric Potential is the ( 1 = 1 / 1 ) - CAREFUL! V is the symbol for both Electric Potential AND its unit. Example:. EXAMPLE: A 5 C and 3 C charge are separated by some distance. If the 5 C charge feels 200 V from the 3 C charge, what is the potential energy of the 5 C charge? Page 3
CONCEPT: MOVEMENT OF CHARGES IN POTENTIAL FIELDS [ + / - ] charges ALWAYS move to low potential, and [ + / - ] charges ALWAYS to high potential - Potential is a field that provides motivation for charges to move gives them potential energy EXAMPLE 1: An electron is at rest between two points, A at 10 V, and B at 0 V. Which point will the electron move to? EXAMPLE 2: A metal rod is placed in a uniform electric field as shown below. Which end of the rod is at a higher potential? Page 4
CONCEPT: POTENTIAL DUE TO A POINT CHARGE Remember: Electric POTENTIAL (POTENTIAL) is an ENERGY field U = q V V = So we also think of POTENTIAL as Electric Potential Energy per. A POINT CHARGE produces an Potential: - V = - Units are VOLTS (1 V) Potential DIFFERENCE = difference in potential between 2 points = (aka ) - CAREFUL! Voltage Volts EXAMPLE: What is the potential 0.5 m away from a 2 C charge? What about 1 m away? What is the potential difference between these two points? The voltage? P 1 P 2 q Voltage is, not. Page 5
PRACTICE: ELECTRIC POTENTIAL DUE TO A POINT CHARGE How far from a 5 C charge will the potential be 100 V? PRACTICE: POTENTIAL BETWEEN TWO POINT CHARGES A -1 C and a 5 C charge lie on a line, separated by 5cm. What is the electric potential halfway between the two charges? EXAMPLE: POTENTIAL DIFFERENCE BETWEEN TWO CHARGES Two charges, q and -3q, lie on a line as shown below. What is the potential difference between point A and point B? s q -3q x A B x Page 6
CONCEPT: WORK DUE TO ELECTRIC FORCE Whenever a charge moves, it changes its position so its changes. - A change in energy is called. Energy conservation: ΔU = ΔK - Work energy theorem: W = ΔK q 2 - U & V relationship: ΔU = qδv W = q 1 W = W = - Work due to the electric force depends ONLY on, NOT on the path. For two point charges, CANNOT use W = Fdcosθ, because electric force changes with distance. EXAMPLE 1: A 2 nc charge is initially 5 mm away from a 10 nc charge. The 5 nc charge is then moved 2 mm closer to the 10 nc charge. What is the work done by the electric force? ONLY case when we can use W = Fdcosθ: EXAMPLE 2: A 1 C charge is placed in a horizontal, uniform electric field of magnitude 1,000 N/C. What is the work done on the charge when it travels a distance of 2 m at an angle of 30 o below the horizontal? Page 7
PRACTICE: WORK DUE TO POTENTIAL DIFFERENCE An electron moves from point A to point B. The potential difference between these two points is 100 V. What is (a) the point of higher potential? (b) the work done on the electron? (c) the final speed of the electron if its initial speed is zero? EXAMPLE: BRINGING TWO CHARGES FROM INFINITY How much work will it take to bring a 5 C charge from infinitely far away to the origin of a coordinate, and then bring a 2 C charge from infinitely far away to a point (3 m, 4 m) on the same coordinate system? Imagine there are no other charges. Page 8
PRACTICE: WORK TO ASSEMBLE A TRIANGLE OF CHARGES What work is needed to assemble an equilateral triangle of side length 5 cm, with a 5 C charge at each vertex? 5 cm 5 cm 5 cm EXAMPLE: SPEED OF ELECTRON IN UNIFORM ELECTRIC FIELD An electron is initially at rest in a uniform, 500 N/C electric field. After traveling 10 cm, what is the electron s speed? Page 9
CONCEPT: RELATIONSHIPS BETWEEN FORCE, FIELD, ENERGY, POTENTIAL. So far we have seen FOUR related terms with similar NAMES and EQUATIONS. Now let s put it all together: r 2 r q1 q2 ELECTRIC FORCE F = k q 1q 2 r 2 ELECTRIC POTENTIAL ENERGY U = k q 1q 2 r q ELECTRIC FIELD (ELECTRIC FORCE FIELD) E = k q r 2 ELECTRIC POTENTIAL (POTENTIAL) (ELECTRIC ENERGY FIELD) V = k q r Remember: Electric Potential DIFFERENCE = Potential Difference = VOLTAGE = ΔV Page 10
EXAMPLE: POTENTIAL AT CENTER OF CHARGES ARRANGED IN A SQUARE What is the potential at the center of the arrangement shown in the following figure? 2 nc 5 mm -1.5 nc 5 mm -3 nc 1 nc PRACTICE: POTENTIAL AT CENTER OF CHARGES ARRANGED IN A CIRCLE 4 identical charges are arranged so that they are evenly spaced in a circle. If the radius of the circle is 10 cm, and the potential at the center of the circle is 100 V, what is the magnitude of each charge? Page 11
PRACTICE: POTENTIAL DIFFERENCE DUE TO A POINT CHARGE A -2 C charge lies at rest. What is the potential difference between point A, which is 1.5 m from the charge, and point B, which is 4 m from the charge? What would the work on a 4 C charge be to move it from A to B? EXAMPLE: POTENTIAL DIFFERENCE DUE TO TWO CHARGES A 5 nc charge and a -3 nc charge lie on a line, separated by 6 mm. What is the potential halfway between the two charges on the line connecting them? What is the potential halfway between the charges, but 4 mm above the line connecting them? How much work would it take to move a 1 nc charge from the first point to the second? PRACTICE: STOPPING A POINT CHARGE A 5 g, 3 µc point charge is moving with an initial speed of 20 m/s away from a 5 µc charge. If they are initially 5 cm apart, how far can the 3 µc travel before stopping? Page 12
CONCEPT: POTENTIAL DIFFERENCE WITH CALCULUS The potential difference between two points can be defined as ΔV = - For a constant electric field, this reduces to the familiar equation E = - For a DISTRIBUTION of charges, we will V. Q P Q P EXAMPLE: Calculate the potential difference due to a point charge between a point r away from the charge and infinity. Page 13
EXAMPLE: ELECTRIC POTENTIAL DUE TO LINE OF CHARGE What is the electric potential due a line of charge, Q, with length 2a, at the point depicted in the following figure? y +a Q -a P z Page 14
EXAMPLE: ELECTRIC POTENTIAL DUE TO RING OF CHARGE What is the electric potential due to a ring of charge, Q, and radius, R, at the point indicated in the following figure? y Q P z x Page 15
PRACTICE: ELECTRIC POTENTIAL DUE TO DISK OF CHARGE What is the electric potential due a disk of charge, Q, with some radius, R, at the point shown in the following figure? (hint: see Electric Potential due to Ring of Charge ) y Q R P z x Page 16
CONCEPT: THE ELECTRONVOLT Suppose two plates of equal & opposite charge have a potential difference of 1V - + - Electron moves from one plate to another, the potential difference is Change in Potential Energy is ( ) 1 V 1 ev is called an ELECTRONVOLT 1 ev = (1.6 10 19 C)(1V) = 1.6 10 19 J - Electronvolt = Change in potential energy of ONE electron through ONE volt. - Example: One electron going through TWO volts of potential difference gains ev of kinetic energy. EXAMPLE: What is the speed of an electron with 150 ev of kinetic energy? Page 17
CONCEPT: EQUIPOTENTIAL SURFACES EQUIPOTENTIAL SURFACES are surfaces of potential Relationship between electric field and potential E = Electric field only exists where potential is changing E has to be to equipotential surfaces Work along equipotential surface = - Because W = qδv No ΔV, no W Equipotential surfaces of a POINT CHARGE: Equipotential surfaces of a DIPOLE: EXAMPLE: What is the distance from a 1 C to an equipotential surface of 150 V? Page 18
EXAMPLE: ELECTRIC FIELD DUE TO EQUIPOTENTIAL SURFACES What is the magnitude and direction of the electric field due to the equipotential surfaces shown in the following figure? y 1 cm 1 cm 10 V 15 V 20 V 60 o x PRACTICE: DRAWING EQUIPOTENTIAL SURFACES FROM ELECTRIC FIELD LINES Draw the electric field that corresponds to the equipotential surfaces shown in the following figure. Note that the potential is decreasing in the upward direction. Page 19
CONCEPT: ELECTRIC FIELD AS DERIVATIVE OF POTENTIAL The relationship between the x-component of the Electric Field and the Potential is E x = - The electric field can be defined as: E = V x V V i j k y z ( means partial derivative ) EXAMPLE: A distribution of charges has the following electric potential. What is the electric field associated with it? V(x, y, z) = x 2 y xy 2 z 2 + 3yz Page 20