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PHYSICS NYB-21E Winter 2012
Chapter 3: Electric potential energy and electric potential
´ ´ Instructor: Jeremie Vinet Marianopolis College.

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Review
Important points from last lectures:
q A point charge q creates an electric field E = ke 2 r ˆ r

A point charge q0 placed in an electric field E feels a force Fe = q0 E

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NYA flashback: Work
When a net force acts on an object, it accelerates it. When an object accelerates, its velocity changes. When the speed of an object changes, its kinetic energy changes. When an object’s energy changes, work has been done on it.

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NYA flashback: Work
In less cartoonish terms, The amount of work W done by a force F over a displacement ∆r is W = F · ∆r

The amount of work W done by a force F over two displacements ∆r1 and ∆r2 is W = F · (∆r1 + ∆r2 )

Remember the definition of the dot product
A · B = |A||B| cos θ = Ax Bx + Ay By + Az Bz

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NYA flashback: Work: example
What is the work done by gravity on a watermelon dropped from the College’s roof to the parking lot below?

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NYA flashback: Work: example
The force acting on the melon as it falls is Fg = −mgˆ. The j displacement it undergoes is ∆r = −hˆ. We put these j together to find that the work is Wg = Fg · ∆r = (−mgˆ · (−hˆ = mgh(ˆ · ˆ = mgh. So the j) j) j j) amount of work done by gravity on the melon of mass m as it dropped a distance h is mgh. (Remember, ˆ · ˆ = |ˆ ˆ cos θ = 1 × 1 × cos(0) = 1.) j j j||j|

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NYA flashback: Potential energy
We just saw that if we drop an object of mass m from a height h in a gravitational field of magnitude g , the work done by the field on the object will be W = mgh. This expression should remind you of something... It is the gravitational potential energy of an object of mass m held at a height h above the surface of the Earth. The potential energy is equal to the work a field can do on an object if we release the object.

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NYA flashback: Potential energy
Things you should remember about potential energy: Only the difference in the potential energy between two points matters, not the actual values. We can therefore set U to zero wherever we please. The work done by the force is equal to minus the change in the potential energy from point A to point B ; Wg = −∆Ug . For a conservative force (like the gravitational for or the electric force), the path taken from point A to point B makes no difference!

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NYA flashback: Potential energy
What is the work done by gravity on a watermelon dropped on a parabolic path from the College’s roof to the parking lot below?

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NYA flashback: Potential energy
The only thing that matters here is the initial and final heights, since the potential energy is Ug = mgh. So the work done by gravity on the object is again Wg = mgh. Note that this is minus the change in the potential energy ∆Ug = Uf − Ui = 0 − mgh = −mgh = −Wg , as it should. Notice that since we are free to set Ug = 0 wherever we want, we could have said that Ug = 0 at the college’s roof. In this case, however, the potential energy at the ground is Ug = −mgh, and we still find ∆Ug = Uf − Ui = −mgh − 0 = −mgh = −Wg , which confirms that the choice of where Ug = 0 doesn’t change the answer to the problem.

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NYA flashback: Potential energy
Note that the work done doesn’t depend on the path taken from the initial to the final point, so we could find the work by looking at the path shown here,

where clearly the work done on the horizontal stretch by gravity is zero since the angle between force and displacement is 90o , and the work done on the vertical stretch is mgh.

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Work and the electric field
If we place a charge q0 in an electric field E , the electric field can do work on the charge. Indeed, in this case, the force on the charge is F = q0 E , so that the work done on q0 by the electric field is We = Fe · ∆r = q0 E · ∆r

and the change in the potential energy of the charge-field system is ∆Ue = −We...
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