P25_041 - U = Qq/ 4 r , and the total energy is E = K + U =...

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41. The particle with charge q has both potential and kinetic energy, and both of these change when the radius of the orbit is changed. We Frst Fnd an expression for the total energy in terms of the orbit radius r . Q provides the centripetal force required for q to move in uniform circular motion. The magnitude of the force is F = Qq/ 4 πε 0 r 2 . The acceleration of q is v 2 /r ,where v is its speed. Newton’s second law yields Qq 4 πε 0 r 2 = mv 2 r = mv 2 = Qq 4 πε 0 r , and the kinetic energy is K = 1 2 mv 2 = Qq/ 8 πε 0 r . The potential energy is
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Unformatted text preview: U = Qq/ 4 r , and the total energy is E = K + U = Qq 8 r Qq 4 r = Qq 8 r . When the orbit radius is r 1 the energy is E 1 = Qq/ 8 r 1 and when it is r 2 the energy is E 2 = Qq/ 8 r 2 . The dierence E 2 E 1 is the work W done by an external agent to change the radius: W = E 2 E 1 = Qq 8 1 r 2 1 r 1 = Qq 8 1 r 1 1 r 2 ....
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