PHY PHY 184 184 Spring 2007 Lecture 10 Title The Electric Potential V x 1 24 07 184 Lecture 10 1 Review Electric Potential V x a scalar function of position The change in electric potential energy U of a charge q that moves in an electric field is related to the change in electric potential V U V q or U q The unit of electric potential is the volt V The unit of electric field is V m For reference state at infinity V x x 1 24 07 184 Lecture 10 E ds 2 Electric Potential for a Point Charge We ll derive the electric potential for a point source q as a function of distance R from the source That is V R Remember that the electric field from a point charge q at a distance r is given by The direction of the electric field from a point charge is always radial We integrate from distance R distance from the point charge along a radial to infinity 1 24 07 184 Lecture 10 3 Electric Potential of a Point Charge 2 The electric potential V from a point charge q at a distance r is then kq V r r Negative point charge Positive point charge 1 24 07 184 Lecture 10 4 Electric Potential from a System of Charges We calculate the electric potential from a system of n point charges by adding the potential functions from each charge n n kqi V Vi i 1 i 1 ri This summation produces an electric potential at all points in space a scalar function Calculating the electric potential from a group of point charges is usually much simpler than calculating the electric field because it s a scalar 1 24 07 184 Lecture 10 5 Example Superposition of Electric Potential Assume we have a system of three point charges q1 1 50 C q2 2 50 C q3 3 50 C q1 is located at 0 a q2 is located at 0 0 q3 is located at b 0 a 8 00 m and b 6 00 m What is the electric potential at point P located at b a 1 24 07 184 Lecture 10 6 Example Superposition of Electric Potential 2 The electric potential at point P is given by the sum of the electric potential from the three charges r1 r2 r3 q1 q1 q2 q3 kqi q2 q3 V k k 2 2 a r1 r2 r3 b a b i 1 ri 3 1 50 10 6 C V 8 99 10 N C 6 00 m 9 2 50 10 6 C 2 2 8 00 m 6 00 m 3 50 10 6 C 8 00 m V 562 V 1 24 07 184 Lecture 10 7 Clicker Question Electric Potential Rank a b and c according to the net electric potential V produced at point P by two protons Greatest first A b c a B all equal C c b a D a and c tie then b 1 24 07 184 Lecture 10 8 Clicker Question Electric Potential Rank a b and c according to the net electric potential V produced at point P by two protons Greatest first B all equal 2qd V a 1 24 07 184 Lecture 10 9 Calculating the Field from the Potential We can calculate the electric field from the electric potential starting with We V q Which allows us to write If we look at the component of the electric field along the direction of ds we can write the magnitude of the electric field as the partial derivative along the direction s V ES s 1 24 07 184 Lecture 10 10 Math Reminder Partial Derivatives Given a function V x y z the partial derivatives are act on x y and z independently Example V x y z 2xy2 z3 Meaning partial derivatives give the slope along the respective direction 1 24 07 184 Lecture 10 11 Calculating the Field from the Potential 2 We can calculate any component of the electric field by taking the partial derivative of the potential along the direction of that component We can write the components of the electric field in terms of partial derivatives of the potential as V V V Ex E y Ez x y z In terms of graphical representations of the electric potential we can get an approximate value for the electric field by measuring the gradient of the potential perpendicular to an equipotential line E V 1 24 07 184 Lecture 10 12 Example Graphical Extraction of the Field from the Potential Assume a system of three point charges q1 6 00 C q2 3 00 C q3 9 00 C x1 y1 1 5 cm 9 0 cm x2 y2 6 0 cm 8 0 cm x3 y3 5 3 cm 2 0 cm 1 24 07 184 Lecture 10 13 Example Graphical Extraction of the Field from the Potential 2 We calculate the magnitude of the electric field at point P To perform this task we draw a line through point P perpendicular to the equipotential line reaching from the equipotential line of 1000 V to the line of 1000V The length of this line is 1 5 cm So the magnitude of the electric field can approximated V 0 V as 5 V be 2000 ES 1 24 07 s 1 5 cm 1 3 10 V m The direction of the electric field points from the positive equipotential line to the negative potential line 184 Lecture 10 14 Clicker Question E Field from Potential Pairs of parallel plates with the same separation and a given V of each plate The E field is uniform between plates and perpendicular to them Rank the magnitude of the electric field E between them Greatest first A 1 2 3 B 3 and 2 tie then 1 C all equal D 2 then 1 and 3 tie 1 24 07 184 Lecture 10 15 Clicker Question E Field from Potential Pairs of parallel plates with the same separation d and a given V of each plate The E field is uniform between plates and perpendicular to them Rank the magnitude of the electric field E between them Greatest first D 2 then 1 and 3 tie Use 1 24 07 and take the magnitude only 184 Lecture 10 16 Electric Potential Energy for a System of Particles So far we have discussed the electric potential energy of a point charge in a fixed electric field Now we introduce the concept of the electric potential energy of a system of point charges In the case of a fixed electric field the point charge itself did not affect the electric field that did work on the charge Now we consider a system of point charges that produce the electric potential themselves We begin with a system of charges that are infinitely far apart Reference state U 0 To bring these charges into proximity with each other we must do work on the charges which changes the electric potential energy of the system 1 24 07 184 …
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