Chapter 22: Electric Potential Electric Potential Energy Review of work, potential and kinetic energy Consider a force acts on a particle moving from point a to point b. The work done by the force WAB is given by: If the force is conservative, namely when the work done by the force depends only on the initial and final position of the particle but not on the path taken along the particle’s path, the work done by the force F can always be expressed in terms of a potential energy U.
Electric Potential Energy Review of work, potential and kinetic energy In case of a conservative force, the work done by the force can be expressed in terms of a potential energy U: The change in kinetic energy DK of a particle during any displacement is equal to the total work done on the particle: If the force is conservative, then
Electric Potential Energy Electric potential energy in a uniform field Consider a pair of charged parallel metal plates that generate a uniform downward electric field E and a test charge q0 >0 A + + + + + + + + + + d conservative force - - - - - - - - - - B the force is in the same direction as the net displacement of the test charge In general a force is a vector: m g Fg Note that this force is similar to the force due to gravity:
Electric Potential Energy Electric potential energy in a uniform field (cont’d) In analogy to the gravitational force, a potential can be defined as: When the test charge moves from height ya to height yb , the work done on the charge by the field is given by: U increases (decreases) if the test charge moves in the direction opposite to (the same direction as) the electric force DUAB <0 DUAB>0 DUAB >0 DUAB<0 A B A B + + - - + B + A - B - A
Electric Potential Energy Electric potential energy of two point charges The force on the test charge at a distance r b rb The work done on the test charge q0 r a ra + q
Electric Potential Energy Electric potential energy of two point charges (cont’d) In more general situation tangent to the path B r A Natural and consistent definition of the electric potential
Electric Potential Energy Electric potential energy of two point charges (cont’d) Definition of the electric potential energy Reference point of the electric potential energy Potential energy is always defined relative to a reference point where U=0. When r goes to infinity, U goes to zero. Therefore r= is the reference point. This means U represents the work to move the test charge from an initial distance r to infinity. If q and q0 have the same sign, this work is POSITIVE ; otherwise it is NEGATIVE. U U qq0>0 qq0<0
Electric Potential Energy Electric potential energy with several point charges A test charge placed in electric field by several particles Electric potential energy to assemble particles in a configuration
Electric Potential - + + q1=-e q2=+e q3=+e x=0 x=a x=2a Example : A system of point charges q1=-e q2=+e q3=+e - + + x=0 x=a x=2a Work done to take q3 from x=2a to x=infinity Work done to take q1,q2 and q3 to infinity
Electric Potential Energy Two interpretations of electric potential energy Work done by the electric field on a charged particle moving in the field Work done by the electric force when the particle moves from A to B Work needed by an external force to move a charged particle slowly from the initial to the final position against the electric force Work done by the external force when the particle moves from B to A
Electric Potential Electric potential or potential Electric potential V is potential energy per unit charge 1 V = 1 volt = 1 J/C = 1 joule/coulomb potential of A with respect to B work done by the electric force when a unit charge moves from A to B work needed to move a unit charge slowly from b to a against the electric force
Electric Potential Electric potential due to a single point charge Electric potential or potential (cont’d) Electric potential due to a single point charge Electric potential due to a collection of point charges Electric potential due to a continuous distribution of charge
Electric Potential From E to V Sometimes it is easier to calculate the potential from the known electric field The unit of electric field can be expressed as: 1 V/m = 1 volt/meter = 1 N/C = 1 newton / coulomb
Electric Potential Example : Replace R with r
Electric Potential Example: q1 m, q0 q2 + - A B = 0
Electric Potential Unit: electron volt (useful in atomic & nuclear physics) Consider a particle with charge q moves from a point where the potential is VA to a point where it is VB , the change in the potential energy U is: If the charge q equals the magnitude e of the electron charge 1.602 x 10-19 C and the potential difference VAB= 1 V, the change in energy is: meV, keV, MeV, GeV, TeV,…
Calculating Electric Potential Example: A charged conducting sphere Using Gauss’s law we calculated the electric field. Now we use this result to calculate the potential and we take V=0 at infinity. + + + + + R + + + E the same as the potential due to a point charge r V inside of the conductor E is zero. So the potential stays constant and is the same as at the surface r
Equipotential Surface An equipotential surface is a 3-d surface on which the electric potential V is the same at every point No point can be at two different potentials, so equipotential surfaces for different potentials can never touch or intersect Because potential energy does not change as a test charge moves over an equipotential surface, the electric field can do no work E is perpendicular to the surface at every point Field lines and equipotential surfaces are always mutually perpendicular
Equipotential Surface Examples of equipotential surface
Equipotential Surface Equipotentials and conductors E = 0 everywhere inside a conductor - At any point just inside the conductor the component of E tangent to the surface is zero - The tangential component of E is also zero just outside the surface If it were not, a charge could move around a rectangular path partly inside and partly outside and return to its starting point with a net amount of work done on it. vacuum conductor When all charges are at rest, the electric field just outside a conductor must be perpendicular to the surface at every point When all charges are at rest, the surface of a conductor is always an equipotential surface
Equipotential Surface Equipotentials and conductors (cont’d) Consider a conductor with a cavity without any charge inside the cavity - The conducting cavity surface is an equipotential surface A Take point P in the cavity at a different potential and it is on a different equipotential surface B The field goes from surface B to A or A to B Draw a Gaussian surface which surrounds the surface B inside cavity Guassian surface equipotential surface through P B A P The net flux that goes through this Gaussian surface is not zero because the electric field is perpendicular to the surface Gauss’s law says this flux is zero as there is no charge inside Then the surfaces A and B are at the same potential conductor surface of cavity In an electrostatic situation, if a conductor contains a cavity and if no charge is present inside the cavity, there can be no net charge anywhere on the surface of the cavity
Equipotential Surface Electrostatic shielding
Potential Gradient Potential difference and electric field
Potential Gradient E from V Gradient of a function f Potential gradient (cont’d) E from V Gradient of a function f If E is radial with respect to a point or an axis
Potential Gradient Potential gradient (cont’d)
Exercises Exercise 1
Exercises Exercise 1 (cont’d)
Exercises Exercise 1 (cont’d)
Exercises Exercise 2
Exercises Exercise 3
Exercises Exercise 4
Exercises Exercise 4 (cont’d)
Exercises Exercise 4 (cont’d)
Exercises Exercise 4 (cont’d)
Exercises r r Q -Q line charge density l Exercise 5: An infinite line charge + a conducting cylinder Q -Q Outer metal braid r r Signal wire line charge density l