Chapter_23 - Chapter 23: Electric Potential In this...

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27 Chapter 23: Electric Potential In this chapter… ± Discuss potential energy associated with electric interactions ± Introduce the concept of electric potential ( voltage ) ± Equipotential surfaces Electric Potential Energy (23-1) ± A force F G on an object is conservative if it depends only on position r G (not on velocity v G , time t , …) and for any two points A and B , the work done by F G (line integral) is the same for all paths from A B ± Examples: Newtonian gravity, elastic spring but not friction! (depends on velocity, path,…) ± For a conservative force work done by F G can be expressed as a change in potential energy U path in direction of F G leads to positive value for line integral, so A B UU > which means B is at lower potential energy total mechanical energy = kinetic energy + potential energy is conserved A ABB K UK U +=+ work done is reversible work done by an external agent to overcome F G is ± The electrostatic force is conservative! (later, we will see non-static E G not conservative) depends only on location (for given charge) work done is path in dependent cos BB AB A A WF d l F d l θ =⋅ = ∫∫ G G F G dl G B A (final) (initial) B AB A B A WU U F d l U =−= = Δ G G B A F dl G G
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28 ± Move point charge q from point A at (0,a,0) to point B at (b,0,0) in electric field of point charge Q at origin (0,0,0) by different paths Force on q is Path 1 (green) : ¾ 0 F dl ⋅= G G all along circular arc of radius a in yz -plane ¾ 0 F dl G G all along circular arc of radius b in xz -plane ¾ On line segment along z -axis Line element ˆ dl k dz = G Force along path Work done Total work done along Path 1: Path 2 (blue) : For an arbitrary path from A Æ B Q B A a b 1 2 x y z ( ) () 23 / 2 222 00 ˆ ˆˆ 1 ˆ 44 xi yj zk Qq Qq Fr r xyz πε ++ == G 2 0 ˆ 4 qQk F z = G 2 0 2 0 ˆ ˆ 4 4 11 1 b a b a b a qQk Fd l kd z z qQ dz z qQ qQ za b = ⎤⎛ ⎜⎟ ⎢⎥ ⎦⎝ ∫∫ G G Path 1 0 4 AB qQ W ab ⎛⎞ =− ⎝⎠ 2 0 2 0 1 cos 4 1 co 4 s BB AB AA b b a a dl dr qQ WF d l r qQ qQ qQ ra b r θ ε π = F G dl G B A Q dr G r
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29 ± Work done as point charge q moves from A Æ B (distances , A B rr away from point charge Q ) along any path is ± Identify the electric potential energy of two point charges q and Q separated by a distance r as given by The integration constant is arbitrary since it is physically irrelevant ¾ work done (or energy needed) in moving from one distance to another depends only on difference in U Æ constant always cancels Often choose constant to be zero so as 0 Ur →→ Shared property of the two charges since consequence of their interaction Depends only on distance between them and does not matter which one was held fixed or whether they were both moved ± Electric total potential energy of several point charges Due to superposition, the potential energy (scalar) of several point charges
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Chapter_23 - Chapter 23: Electric Potential In this...

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