By J.E. Parton
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Extra resources for Applied Electromagnetics
Work done by charge = qE. dl. Hence work done on charge is -qE . dl that is dW = -qE . dl joules. Now suppose +q is moved from B to A. 13b) A WB-+A = -q A f E. 13 Movement of a test charge +q in an electric field E. if E is constant in space. This is clearly independent of the actual path taken between B and A, and depends only on the start and fmish. In general the electric field strength E is not constant in space; even so the work done is still independent of the actual path. In this general case If q =+1 coulomb, the work done in moving +q from B to A is known as the potential of A with reference to B, or the potential difference between points A andB VAB =potential of A with reference to B =VA - VB =- JE.
SF) = S V. F + V S• F and rearrange it as (V. DV) devol) -1 f (D. dS ~l S ~l divergence theorem This can be used on the first of the two volume integrals giving WE =1 f (DV) . dS -1 f (D. V V) devol) S vol The first of these integrals is zero for, taking the surface approaching infmity, D is proportional to l/r'2, V is proportional to l/r and S is proportional to r2; the . integral therefore tends to zero. Finally we are left with the second volume integral in which we can replace VV by -E giving WE = 1 f D.
It will be the same for all ct>. The potential V will be that due to the difference between potentials arising from +Q ana - Q. p) 47teOr In this we have introduced the dipole moment p = Qd. It has the direction of d and the separation -Q to +Q. It can also be used in the expression for potential V. V =Qd cos 0 =Q d. or 47teor2 47teor2 = p. or 47teor2 It should be fairly clear that these results are only approximate and hold when r ~ d.. They are not correct when r -+ 0, that is for points near the dipole itself.