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Hence, Ar = A · er , etc. The volume element in this coordinate system is d3 r = r dr dθ dz. 164) respectively. Here, V(r) is a general vector field, and A (r) a general scalar field. Finally, the Laplacian is written ∇2 V = 1 ∂ r ∂r r ∂V ∂r + 1 ∂2 V ∂2 V + . 165) In the spherical polar coordinate system the Cartesian coordinates x, y, and z are replaced by r = x2 + y2 + z2 , θ = cos−1 (z/r), and φ = tan−1 (y/x). 26. Note that r and θ in the spherical polar system are not the same as their counterparts in the cylindrical system.

In vector notation. 6726 × 10−11 N m2 kg−2 . 4) Note that Coulomb’s law has the same mathematical form as Newton’s law of gravity. , | f2 | ∝ 1 . 5) However, these laws differ in two crucial respects. Firstly, the force due to gravity is always attractive (there is no such thing as a negative mass). Secondly, the magnitudes of the two forces are vastly different. Consider the ratio of the electrical and gravitational forces acting on two particles. This ratio is a constant, independent of the relative positions of the particles, and is given by |felectrical | |fgravitational | = |q1 | |q2 | 1 .

This tells us that the summation is equivalent to ∇ · A dV over the whole volume. Thus, the integral of A · dS over the outer surface is equal to the integral of ∇ · A over the whole volume, which proves the divergence theorem. 126) for any closed surface S. 21. (Note that the direction of the surface elements on S1 has been reversed relative to those on the closed surface. 21: Two surfaces spanning the same rim (right), and the equivalent closed surface (left). ) Thus, if ∇ · A = 0 then the surface integral depends on the rim but not the nature of the surface which spans it.

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