FWLec8 - A F Peterson Notes on Electromagnetic Fields Waves...

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A. F. Peterson: Notes on Electromagnetic Fields & Waves 10/04 Fields & Waves Note #8 Voltage and Gradient Objectives: Introduce the notion of the voltage between two points in a field around a charge distribution. Show that the line integration used to obtain voltage can be reversed by a derivative operation known as the gradient. Present examples illustrating voltage and gradient calculations. Voltage Figure 1 depicts a system of two conductors, with some surface charge and an associated electric field present. We can define the potential difference between the points A and B in the figure as VV E d Ed AB A B B A -= =- ÚÚ ll (V) (8.1) This quantity is also the work required by an external force to move a unit charge from A to B in the presence of the electric field. The static electric field is conservative (Note #3), so the line integral in (8.1) is independent of the specific path from A to B . The units of potential difference are Volts (V), and in fact the quantity in (8.1) corresponds to our usual notion of the voltage between two points in an electrical system. As in circuit problems, the voltage is a relative quantity — measured between two locations or measured with respect to some reference point or ground. It is often convenient to assign the point B to some reference location, and think of the voltage (at A ) as a scalar field x y z E d A A reference == Ú (,,) l (8.2) For example, consider a point charge Q at the origin. The electric field is given by Er Q r r () ˆ = 4 0 2 pe (8.3) Suppose the reference point B is located at infinity. Then Vr E d r (,, ) qf = Ú l (8.4) Since in this situation dd r r l = ˆ, we obtain

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A. F. Peterson: Notes on Electromagnetic Fields & Waves 10/04 Vr Q r dr Q r rr (,, ) () qf pe = ¢ ¢ = ¢= Ú 4 4 0 2 0 (8.5) We have obtained the voltage field associated with a point charge. The voltage field gives us yet another way of looking at the force field associated with a charge distribution. Since this field is a scalar quantity, it is sometimes easier to work with than the vector fields.

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FWLec8 - A F Peterson Notes on Electromagnetic Fields Waves...

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