Eample: Electic Potential Enegy What is the change in electical potential enegy of a eleased electon in the atmosphee when the electostatic foce fom the nea Eath s electic field (diected downwad) causes the electon to move vetically upwads though a distance d? 1. U of the electon is elated to the wok done on it by the electic field: Key Idea: ΔU = W 2. Wok done by a constant foce on a paticle undegoing displacement: Key Idea: W = F d 3. Electostatic Foce and Electic Field ae elated: Key Idea: F = q E ΔU = W = qed 9/10/12 1 E F e W = q E d = qed cosθ = qed cos180 = qed Electic potential deceases as electon ises. d Summay Electic Potential fo chage V() kq ote: A positively chaged paticle poduces a positive electic potential. A negatively chaged paticle poduces a negative electic potential. Electic Potential fo positively chaged spheical conducto R V() V(R) = kq R 9/10/12 2 kq R Potential due to a Goup of Point Chages Find the Potential at the cente of the squae. V() = V n () = 1 n=1 4πε o q n n=1 n Chapte 23 Electostatic Potential Enegy of a system of fied point chages is equal to the wok that must be done by an etenal agent to assemble the system, binging each chage in fom an infinite distance. = 12 nc q 2 = -24 nc - Point 1 Point 2 q 3 =31 nc q 3 =17 nc 9/10/12 3 9/10/12 4 1
Eample: Electostatic Potential Enegy Eample: Thee Point Chages Point 1 Point 2 q 2 q 3 Point 3 2,3 1,3 Point 1 1,2 q 2 Point 2 If & q 2 have the same sign, U is positive because positive wok by an etenal agent must be done to push against thei mutual epulsion. If & q 2 have opposite signs, U is negative because negative wok by an etenal agent must be done to wok against thei mutual attaction. 9/10/12 5 9/10/12 6 Electostatic Potential Enegy We can conclude that the total wok equied to assemble the thee chages is the electostatic potential enegy U of the system of thee point chages: kq kq q 2 1 3 1 U = Wtotal = W2 W3 = 1,2 1,3 q kq3q 2,3 2 Calculate Electic Field fom the Potential Electic field always points in the diection of steepest descent of V (steepest slope) and its magnitude is the slope. Potential fom a egative Point Chage Potential fom a Positve Point Chage V( ) The electostatic potential enegy of a system of point chages is the wok needed to bing the chages fom an infinite sepaation to thei final position 9/10/12 7 y y -V( ) 9/10/12 8 2
Calculating the Electic Field fom the Potential Field If we can get the potential by integating the electic field: Calculating the Electic Field fom the Potential Field In the diection of steepest descent E = V î V y ĵ V z ˆk E, E y y, and E z z 9/10/12 9 9/10/12 10 Eample: Calculating the Electic Field fom the Potential Field What is the electic field at any point on the cental ais of a unifomly chaged disk given the potential? Given: V = σ 2ε 0 ( z 2 R 2 z) Potential due to a Goup of Point Chages V() = = = E d l 1 2 3 X 4 E = E y = E z = q 2 q3 q4 9/10/12 11 V() = V n () = 1 n=1 4πε o q n n=1 n 9/10/12 12 3
Potential fom a Continuous Chage Distibution Chage Densities total chage Q small pieces of chage dq Line of chage: = chage pe unit length [C/m] dq = d Suface of chage: = chage pe unit aea [C/m 2 ] dq = da Cylinde: dq = σdθdz Sphee: dq = σ 2 sinθdθdφ 9/10/12 13 Volume of Chage: = chage pe unit volume [C/m 3 ] dq = dv Cylinde: dq = ρddθdz Sphee: dq = ρ 2 d sinθdθdφ 9/10/12 14 Calculate Potential on the cental ais of a chaged ing Calculate Potential on the cental ais of a chaged disk dq = σ A = σ 2πa da V = k dq # 9/10/12 15 9/10/12 16 4
Calculate Potential on the cental ais of a chaged disk (anothe way) V = E z = σ 2ε o E d l Fom Lectue 3: 1 z z 2 R 2 # Calculate Potential due to an infinite sheet dv = E d l dv = 2πkσ î (d î d ĵ d ˆk) V = 2πkσ d = 2πkσ C 0 0 9/10/12 17 9/10/12 18 E due to an infinite line chage Equipotentials Definition: locus of points with the same potential. Geneal Popety: The electic field is always pependicula to an equipotential suface. Coona dischage aound a high voltage powe line, which oughly indicates the electic field lines. 9/10/12 19 9/10/12 20 5
Equipotentials: Eamples Point chage V() = k q infinite positive chage sheet V() = 2πkσ V o electic dipole Equipotential Lines on a Metal Suface Locally E = σ ε 0 Gauss: E = 0 at electostatic equilibium in electostatic equilibium all of this metal is an equipotential; i.e., it is all at the same voltage 9/10/12 21 9/10/12 22 Potential inside & outside a conducting sphee Summay If you know the functional behavio of the potential V at any point, you can calculate the electic field. The electic potential fo a continuous chage distibution can be calculated by beaking the distibution into tiny pieces of dq and then integating ove the whole distibution. V ef = 0 at =. The electic field is zeo inside a conducto. The electic potential is constant inside a conducto. 9/10/12 23 Finally no wok needs to be done if you move a chage on an equipotential, since it would be moving pependicula to the electic field. The chage concentates on a conducto on sufaces with smallest adius of cuvatue. 9/10/12 24 6