q r 1 4πε Review: Two ways to find V at any point in space: Integrate E dl: Sum or Integrate over charges: q 1 r 1 q 2 r 2 r 3 q 3

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Alban Underwood
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1 Review: Lectue : Consevation of negy and Potential Gadient Two ways to find V at any point in space: Integate dl: Sum o Integate ove chages: q q 3 P V = i 4πε q i i dq q 3 P V = 4πε dq ample of integating ove distibution: line of chage ing of chage disk of chage

2 Consevation of negy The Coulomb foce is a CONSRVATIV foce (i.e., the wok done by it on a paticle which moves aound a closed path etuning to its initial position is ZRO.) Theefoe, a paticle moving unde the influence of the Coulomb foce is said to have an electic potential enegy defined by: U = qv this q is the test chage in othe eamples... The total enegy (kinetic electic potential) is then conseved fo a chaged paticle moving unde the influence of the Coulomb foce.

3 Lectue, ACT 3 3A Two test chages ae bought sepaately to the vicinity of a positive chage Q. chage q is bought to pt A, a distance fom Q. chage q is bought to pt B, a distance fom Q. Compae the potential enegy of q (U A ) to that of q (U B ): Q Q A q B q (a) U A < U B (b) U A = U B (c) U A > U B 3B Suppose chage q has mass m and is eleased fom est fom the above position (a distance fom Q). What is its velocity v f as it appoaches =? (a) v f 4πε Qq m = (b) v f πε Qq m = (c) v = f

4 Lectue, ACT 3 3A Two test chages ae bought sepaately to the vicinity of positive chage Q. chage q is bought to pt A, a distance fom Q. chage q is bought to pt B, a distance fom Q. Compae the potential enegy of q (U A ) to that of q (U B ): (a) U A < U B (b) U A = U B (c) U A > U B Q Q q A q B The potential enegy of q is popotional to Qq/. The potential enegy of q is popotional to Q(q)/(). Theefoe, the potential enegies U A and U B ae QUAL!!!

5 Lectue, ACT 3 3B Suppose chage q has mass m and is eleased fom est fom the above position (a distance fom Q). What is its velocity v f as it appoaches =? (a) v f 4πε Qq m = (b) v f πε Qq m = (c) v = f What we have hee is a little combination of 7 and 7. The pinciple at wok hee is CONSRVATION OF NRGY. Initially: The chage has no kinetic enegy since it is at est. The chage does have potential enegy (electic) = U B. Finally: The chage has no potential enegy (U /R) The chage does have kinetic enegy = K U B =K 4πε Q ( q ) = mv f = v f πε Qq m

6 Detemine V fom : Detemining fom V: ΔV dv = V b V lectical Potential a = Fo an infinitesimal step: = dl = a V = dl ample: V due to spheical chage distibution. dl = dl = b dl cosθ Cases: θ = : dv = - dl θ = 9 o : dv = θ = 8 o : dv = - dl(-) = dl Can wite: dv = dl = ( = ( d y dy iˆ z b a y ˆj dz) z θ dl b a dv a kˆ) ( diˆ dy diectional deivative dv depends on diection maimum change fo θ = o 8 degees F θ dl ˆj dzkˆ) b

7 Potential Gadient Take step in diection: (dy = dz = ) Similaly: And: dv = ( d dy dz) y dv = d V = y y, z const. y z V = V = z V V V = ( iˆ ˆj y z ˆ ( ˆ = i j kˆ) y z z kˆ) = = V d gadient opeato Gadient of V points in the diection that V inceases the fastest with espect to a change in, y, and z. points in the diection that V deceases the fastest. pependicula to equilpotential lines. patial deivative

8 Potential Gadient ample: chage in unifom field U = qy V = U/q = y whee V is taken as at y =. = V = ( iˆ = (iˆ ˆj kˆ) ˆ j y z = ˆj kˆ) y Given o V in some egion of space, can find the othe. Cylindical and spheical symmety cases: Fo adial case and is distance fom point (spheical) o ais (cylindidal): V = o y q ample: of point chage: V q = = ( ) 4πε q q = ( )( ) = 4πε 4πε

9 ample: The electic potential V in a egion of space is given by V(, y, z) = A( 3y z ) whee A is a constant. Deive an epession fo the electic field at any point in this egion. = V = ( iˆ y ˆj z kˆ){ A( 3y z ) )} = (Aiˆ 6Ay ˆj Azkˆ) = A( iˆ 3yj ˆ zkˆ)

10 UI7PF7: This gaph shows the electic potential at vaious points along the -ais. ) At which point(s) is the electic field zeo? A B C D

11 UI7ACT The electic potential in a egion of space is given by V ( ) 3 = 3 The -component of the electic field at = is (a) = (b) > (c) <

12 UI7ACT The electic potential in a egion of space is given by V 3 ( ) = 3 The -component of the electic field at = is (a) = (b) > (c) < We know V() eveywhee To obtain eveywhee, use = V V = = 6 3 ( ) = =

Lecture 11: Potential Gradient and Capacitor Review:

Lecture 11: Potential Gradient and Capacitor Review: Lectue 11: Potentil Gdient nd Cpcito Review: Two wys to find t ny point in spce: Sum o Integte ove chges: q 1 1 q 2 2 3 P i 1 q i i dq q 3 P 1 dq xmple of integting ove distiution: line of chge ing of