This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 1 Physics 1B Class Notes © Walter Gekelman Sixth Week Spring 2010 Electrical Potential Finally let us do back to a ring of charge. We derived an equation for the electric field along the x axis of the ring E x = 1 4 πε Qx x 2 + a 2 ( ) 3 2 . The ring is in the xz plane. Let us consider the motion of a negative particle released at a small value of x (very close to the center of the ring). In this case x <<a : E x = 1 4 πε Qx a 3 x 2 a 2 + 1 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − 3 2 ≅ 1 4 πε Qx a 3 1 − 3 2 x 2 a 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ . Since x << a we can neglect the second term in the series and explicitly putting Q < 0 E x = 1 4 πε − Qx x 2 + a 2 ( ) 3 2 . The Force is of the form F = − kx . Here k = − Q 4 π a 3 ε . This will behave just like a spring does and the charge will oscillate back and forth about the center of the ring. We can associate a force with a potential : F = − dU dx or U = 1 2 Q 4 a 3 πε x 2 . This is a parabolic potential energy diagram or The same concept we used for the spring applies to the electron! The definition of the electrical potential is: 2 (1) U ba = U b − U a = − E i d l a b ∫ . Notice this is the potential difference between two points a and b. There is a dot product in the integral which means that at each position of the path between a and b only the component of E along the path matters. This makes the potential difference independent of the path chosen to get from a to b. There is an analogy between the electric potential and the work that has to be done raising a mass in a gravitational field. This is shown in the diagram below. The only difference is that the electric potential difference is the work moving a unit or test charge and q does not enter the expression as m does in the work against gravity. a start b end 3 What is the potential of a single charge Q. It is discussed in the text but to make it clearer consider moving a test charge from point b near charge Q out to infinity (point a) The potential difference is V ba = V b − V a = − Q 4 πε E i d l a b ∫ = − Q 4 πε 1 r 2 dr a b ∫ = Q 4 πε 1 r b − 1 r a ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ . If we use our reference point as infinity: V b − V a = Q 4 πε 1 r b − 1 r a ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ; r a = ∞ V b = Q 4 πε 1 r b ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Gravitational Field Electric Field Work against gravity = mgh Work moving test charge W = ! ! F i d ! l " in raising the mass d ! l = dy ˆ j ; ! F = ! mg ˆ j W = mgdy a b " = mg ( y b ! y a ) U b ! U a " W q = ! ! E i d ! l a b # to raise the charge from  to + d !...
View
Full Document
 Spring '10
 gekelman
 Electric Potential, Electrostatics, Electric charge, Fundamental physics concepts, Eidl

Click to edit the document details