Oct. 4, From last time(s)… Work, energy, and (electric) potential Electric potential and charge Electric potential and electric field. Electric charges, forces, and fields Motion of charged particles in fields. Today… No honors lecture this week

Oct. 4, Forces, work, and energy Particle of mass m at rest Apply force to particle - what happens? Particle accelerates Stop pushing - what happens? Particle moves at constant speed Particle has kinetic energy

Oct. 4, Work and energy Work-energy theorem: Change in kinetic energy of isolated particle = work done Total work

Oct. 4, Electric forces, work, and energy Consider bringing two positive charges together They repel each other Pushing them together requires work Stop after some distance How much work was done? + +

Oct. 4, Calculating the work E.g. Keep Q 2 fixed, push Q 1 at constant velocity Net force on Q 1 ? Force from hand on Q 1 ? + + Q1Q1 Q2Q2 Zero Total work done by hand Force in direction of motion R + x initial x final

Oct. 4, Conservation of Energy Work done by hand Where did this energy go? Energy is stored in the electric field as electric potential energy for pos charges

Oct. 4, Electric potential energy of two charges Define electric potential energy U so that Work done on system Change in kinetic energy Change in electric potential energy Works for a two-charge system if Define: potential energy at infinite separation = 0 Then Units of Joules for two charges

Oct. 4, Quick Quiz Two balls of equal mass and equal charge are held fixed a distance R apart, then suddenly released. They fly away from each other, each ending up moving at some constant speed. If the initial distance between them is reduced by a factor of four, their final speeds are A.Two times bigger B.Four times bigger C.Two times smaller D.Four times smaller E.None of the above

Oct. 4, More About U of 2 Charges Like charges  U > 0 and work must be done to bring the charges together since they repel (W>0) Unlike charges  U < 0 and work is done to keep the charges apart since the attract one the other (W<0)

Oct. 4, Electric Potential Energy of single charge Work done to move single charge near charge distribution. Other charges provide the force, q is charge of interest q q1q1 q2q2 q3q3 Superposition of individual interactions Generalize to continuous charge distribution.

Oct. 4, Electric potential Electric potential V usually created by some charge distribution. V used to determine electric potential energy U of some other charge q V has units of Joules / Coulomb = Volts Electric potential  U energy proportional to charge q Electric potential

Oct. 4, Electric potential of point charge Consider one charge as ‘creating’ electric potential, the other charge as ‘experiencing’ it Q q

Oct. 4, Electric Potential of point charge Potential from a point charge Every point in space has a numerical value for the electric potential y x Distance from ‘source’ charge +Q +Q+Q

Oct. 4, Potential energy, forces, work U=q o V Point B has greater potential energy than point A Means that work must be done to move the test charge q o from A to B. This is exactly the work to overcome the Coulomb repulsive force. Electric potential energy= q o V A B q o > 0 Work done = q o V B -q o V A = Differential form:

Oct. 4, Quick Quiz Two points in space A and B have electric potential V A =20 volts and V B =100 volts. How much work does it take to move a +100µC charge from A to B? A.+2 mJ B.-20 mJ C.+8 mJ D.+100 mJ E.-100 mJ

Oct. 4, V(r) from multiple charges Work done to move single charge near charge distribution. Other charges provide the force, q is charge of interest. q q1q1 q2q2 q3q3 Superposition of individual electric potentials

Oct. 4, Quick Quiz 1 At what point is the electric potential zero for this electric dipole? +Q+Q-Q-Q x=+ a x=- a A B A.A B.B C.Both A and B D.Neither of them

Oct. 4, Superposition: the dipole electric potential +Q+Q-Q-Q x=+ a x=- a Superposition of potential from +Q potential from -Q Superposition of potential from +Q potential from -Q += V in plane

Oct. 4, Electric Potential and Field for a Continuous Charge Distribution If symmetries do not allow an immediate application of the Gauss’ law to determine E often it is better to start from V! Consider a small charge element dq The potential at some point due to this charge element is To find the total potential, need to integrate over all the elements This value for V uses the reference of V = 0 when P is infinitely far away from the charge distribution

Oct. 4, Quick Quiz Two points in space have electric potential V A =200V & V B =150V. A particle of mass 0.01kg and charge C starts at point A with zero speed. A short time later it is at point B. How fast is it moving? A.0.5 m/s B.5 m/s C.10 m/s D.1 m/s E.0.1 m/s

Oct. 4, E-field and electric potential If E-field known, don’t need to know about charges creating it. E-field gives force From force, find work to move charge q q Electric potential Non-constant potential Non-zero E-field

Oct. 4, Potential of spherical conductor Zero electric field in metal -> metal has constant potential Charge resides on surface, so this is like the spherical charge shell. Found E = k e Q / R 2 in the radial direction. What is the electric potential of the conductor? Integral along some path, from point on surface to inf. difficult path easy path Easy because is same direction as E,

Oct. 4, Electric potential of sphere Conducting spheres connected by conducting wire. Same potential everywhere. So conducting sphere of radius R carrying charge Q is at a potential R1R1 R2R2 Q1Q1 Q2Q2 But  not same everywhere

Oct. 4, Connected spheres Since both must be at the same potential, Surface charge densities? Charge proportional to radius Surface charge density proportional to 1/R Electric field? Since Local E-field proportional to 1/R (1/radius of curvature)

Oct. 4, Varying E-fields on conductor Expect larger electric fields near the small end. Can predict electric field proportional to local radius of curvature. Large electric fields at sharp points, just like square Fields can be so strong that air is ionized and ions accelerated.

Oct. 4, Quick Quiz Four electrons are added to a long wire. Which of the following will be the charge distribution? A) B) C) D)

Oct. 4, Conductors: other geometries Rectangular conductor (40 electrons) Edges are four lines Charge concentrates at corners Equipotential lines closest together at corners So potential changes faster near corners. So electric field is larger at corners.

Oct. 4, E-field and potential energy

Oct. 4, Zero What is electric potential energy of isolated charge?

Oct. 4, The Electric Field is the Electric Field It is independent of the test charge, just like the electric potential It is a vector, with a magnitude and direction, When potential arises from other charges, = Coulomb force per unit charge on a test charge due to interaction with the other charges. We’ll see later that E-fields in electromagnetic waves exist w/o charges!

Oct. 4, Electric field and potential Electric field strength/direction shows how the potential changes in different directions For example, Potential decreases in direction of local E field at rate Potential increases in direction opposite to local E-field at rate potential constant in direction perpendicular to local E-field Said before that

Oct. 4, Potential from electric field Electric field can be used to find changes in potential Potential changes largest in direction of E-field. Smallest (zero) perpendicular to E-field V=VoV=Vo

Oct. 4, Quick Quiz 3 Suppose the electric potential is constant everywhere. What is the electric field? A)Positive B)Negative C)Zero

Oct. 4, Electric Potential - Uniform Field Constant E-field corresponds to linearly increasing electric potential The particle gains kinetic energy equal to the potential energy lost by the charge-field system E cnst A B x +

Oct. 4, Electric field from potential Said before that Spell out the vectors: This works for Usually written

Oct. 4, Equipotential lines Lines of constant potential In 3D, surfaces of constant potential

Oct. 4, Electric Field and equipotential lines for + and - point charges The E lines are directed away from the source charge A positive test charge would be repelled away from the positive source charge The E lines are directed toward the source charge A positive test charge would be attracted toward the negative source charge Blue dashed lines are equipotential

Oct. 4, Quick Quiz 1  C  C  C  m 1.W = mJ 2.W = 0 mJ 3.W = mJ Question: How much work would it take YOU to assemble 3 negative charges? Likes repel, so YOU will still do positive work!

Oct. 4, Work done to assemble 3 charges W 1 = 0 1C 3C 2C  m W 2 = k q 1 q 2 /r W 3 = k q 1 q 3 /r + k q 2 q 3 /r (9  10 9 )(1  )(3  )/5 + (9  10 9 )(2  )(3  )/5 =16.2 mJ W = mJ W E = mJ U E = mJ =(9  10 9 )(1  )(2  )/5 =3.6 mJ q3q3 q2q2 q1q1 Similarly if they are all positive:

Oct. 4, Quick Quiz 2 QQ QQ  Q  m 1.positive 2.zero 3.negative The total work required for YOU to assemble the set of charges as shown below is:

Oct. 4, Why  U/q o ? Why is this a good thing?  V=  U/q o is independent of the test charge q o Only depends on the other charges.  V arises directly from these other charges, as described last time. Last week’s example: electric dipole potential -Q-Q+Q+Q x=+ a x=- a Superposition of potential from +Q potential from -Q Superposition of potential from +Q potential from -Q

Oct. 4, Dipole electric fields Since most things are neutral, charge separation leads naturally to dipoles. Can superpose electric fields from charges just as with potential But E-field is a vector, -add vector components +Q+Q-Q-Q x=+ a x=- a

Oct. 4, Quick Quiz 2 In this electric dipole, what is the direction of the electric field at point A? A) Up B) Left C) Right D) Zero +Q+Q-Q-Q x=+ a x=- a A

Oct. 4, Dipole electric fields +Q+Q -Q-Q Note properties of E-field lines

Oct. 4, Conservative forces Conservative Forces: the work done by the force is independent on the path and depends only on the starting and ending locations. It is possible to define the potential energy U  W conservative  U = U initial - U final = = -(K final - K initial ) = -  K FgFg

Oct. 4, Potential Energy of 2 charges Consider 2 positive charged particles. The electric force between them is The work that an external agent should do to bring q 2 at a distance r f from q 1 starting from a very far away distance is equal and opposite to the work done by the electric force. Charges repel  W>0! r 12 F

Oct. 4, Potential Energy of 2 charges Since the 2 charges repel, the force on q 2 due to q 1 F 12 is opposite to the direction of motion The external agent F = -F 12 must do positive work! W > 0 and the work of the electric force W E < 0 r 12 F dr

Oct. 4, Potential Energy of 2 charges Since W E = -  U = U initial - U final = = -W  W =  U We set U initial = U(  ) = 0 since at infinite distance the force becomes null The potential energy of the system is

Oct. 4, More than two charges?

Oct. 4, U with Multiple Charges If there are more than two charges, then find U for each pair of charges and add them For three charges: