PES 1120 Spring 2014, Spendier Lecture 12/Page 1 Lecture today:

1) Electric Potential Energy and Electric Potential Energy 2) Conservation of energy 3) Equipotential Surfaces

Potential Energy, U - saved or stored energy, i.e., a type of energy that can be converted into kinetic energy at a later time. In an electric field, charges acquire potential energy, Uelec [J = Joules] Basic idea:

Moving q from the initial position i to final position b change the charges’ potential energy. The electric force is a conservative force, therefore it does not matter which path the charge takes to go from point i to point f since the work done by the electric force is the same. For any path, the work done by the electric field as q moves from i to f is

i f elecW U

final initial i f i fU U U U W General Calculations When the field isn’t uniform or we wish to follow a curved path from i to f, we have to do a “line" integral.

f

i fi

W qE dl

(This is a line integral and it does not matter which path is chosen)

f

i

U qE dl

PES 1120 Spring 2014, Spendier Lecture 12/Page 2 For a point charge:

1 2q qU r kr

units [J], J..Joules NOTICE: No absolute value signs!

For a straight line on a uniform field:

cosx r f

cosU qEr qE rf

Electric potential: When we move q in an electric field, we just learned that both work and electric potential energy depend upon the value of q. For example, the work done will double if I move a charge of 1C compared to a charge of 2C. If I double q, I double the change in electric potential energy. Calculate how much work is done per C (Coulomb) when moving from point i to point f.

elecUVq

Unit [J/C] = [V] V…Volt

f

i

r

r

U WV E dlq q

Electric potential of a point charge: For a point charge (and for many other distributions of charge), we will choose our point of reference to be at infinity, such that V = 0 at infinity,

2

ˆ 1 1 1f

i

r r

rr r r

r qV E dr E dr kq dr kq k kr r r r

PES 1120 Spring 2014, Spendier Lecture 12/Page 3 And, again, if we have more than one charge involved, then the electric potential due to charges q1, q2, etc. is just the sum of the potential due to each charge: The Sign Electric Potential: The electric potential is a scalar quantity that can be positive or negative. It depends on our choice of points a and b, as well as, the direction of E

Moving “with” E

decreases V

Vb is smaller than Va so

0a bV V If Va = 0 then Vb < 0. Moving “against” E

increases V

Vb is larger than Va so

0a bV V If Va = 0 then Vb > 0.

PES 1120 Spring 2014, Spendier Lecture 12/Page 4 Last lecture problems (Prof. Livesey): Example 1: There is a uniform electric field 3 ˆ(1.3 10 / )E N C i

in a 2D space.

a) What is the change in electric potential moving from (1.0,1.0) to (1.5,1.0)? b) What is the change in electric potential moving from (1.0,1.0) to (1.0,3.0)? c) If a charge q = 6.7μC is moved as suggested in parts a) and c), how does its

electrostatic potential change?

As a positive particle is decelerated across the gap from left to right total energy must be conserved - hence

i) it gains electrostatic potential energy ii) it looses kinetic energy

PES 1120 Spring 2014, Spendier Lecture 12/Page 5

Example 2: Consider a point charge q = 1.0 μC, point A at distance d1 = 2.0 m from q and point B at distance d2 = 1.0m

a) If A and B are diametrically opposite each other as shown in Fig. (a), what is the electric potential difference VA-VB?

b) What is the electric potential difference VA-VB of A and B are located as shown in Fig. (b)

Example 3: A spherical drop of water carrying a charge of 30 pC has a potential of 500V at its surface (with V = 0 at infinity). What is the radius of the drop?

PES 1120 Spring 2014, Spendier Lecture 12/Page 6 Conservation of Energy Conservation of Energy - If only conservative forces do work on an object its total energy cannot change Conservative Force - A force that creates potential energy. Gravity, springs, and the electric force are all conservative Total Energy, E = K + U - the kinetic plus potential energy If the electric force is the only force doing work on a charge:

2 21 1 2 2

1 12 2

mv qV mv qV

(Remember hat v = speed, V = potential) Example 4: A charged particle (either an electron or a proton) is moving rightward between two parallel charged plates separated by distance d = 2.00 mm. The plate potentials are V1 = -70.0 V and V2 =- 50.0 V. The particle is slowing down from the initial speed of 90.0 km/s at the left plate. (a) Is the particle an electron or a proton? (b) What is its speed just as it reaches plate 2?

PES 1120 Spring 2014, Spendier Lecture 12/Page 7

Example 5: For the dipole shown, at which point is the electric potential equal to zero?

Answer: All these are zero.

For two point charges we use the principle of superposition:

1 2

1 2

0qqV V V k k

r r

since r1 = r2. For each point, A, B, and C, r1 = r2!

PES 1120 Spring 2014, Spendier Lecture 12/Page 8 This means that points A, B, and C have equal potential. By joining up all the points with equal potential we construct a diagram of equipotential surfaces. Equipotential Surfaces •Lines or surfaces of constant potential are called equipotential lines or surfaces. •Since a charge moving along an equipotential surface will always have the same potential energy, then the electric field does no work. •Hence the equipotential surface must be perpendicular to the field lines everywhere. •One can think of equipotential surfaces in the same way as contour lines on a topo map. •Notice that the contour lines are everywhere perpendicular to the “line of fall” that a stone would take rolling down a hill.

Equipotential Surface for a uniform field:

PES 1120 Spring 2014, Spendier Lecture 12/Page 9 Equipotential Surface for a point charge

For a point charge, the equipotential surfaces are just spheres. •Notice that the equipotential spheres get closer together as you approach the charge. •Notice that the field and the equipotential surface are perpendicular. Example 5: When an electron moves from A to B along an electric field line as shown below, the electric field does 3.94 x 10-19J od work on it. What are the electric potential differences

a) VB-VA b) VC-VA c) VC-VB?