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After an introductory bankruptcy eager about the historical past of force-free magnetic fields, and the relation of such fields to hydrodynamics and astrophysics, the ebook examines the boundaries imposed by means of the virial theorem for finite force-free configurations. numerous ideas are then used to discover suggestions to the sphere equations.
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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 . 17 × 1042 . 7) This is a colossal number!
44) THE SCALAR TRIPLE PRODUCT Consider three vectors a, b, and c. The scalar triple product is deﬁned a · b × c. Now, b × c is the vector area of the parallelogram deﬁned by b and c. So, a · b × c is the scalar area of this parallelogram times the component of a in the direction of its normal. 10. This volume is independent of how the triple product is formed from a, b, and c, except that a · b × c = − a · c × b. 10: A vector parallelepiped. “chapter2” — 2007/11/29 — 13:42 — page 17 — #13 18 MAXWELL’S EQUATIONS AND THE PRINCIPLES OF ELECTROMAGNETISM right-hand grip rule by rotating b onto c) and negative if they form a left-handed set.
56) Suppose that a is, in fact, the product of a scalar φ(t) and another vector b(t). What now is the time derivative of a? 57) which implies that dφ da db = . 59) 20 MAXWELL’S EQUATIONS AND THE PRINCIPLES OF ELECTROMAGNETISM and db d da (a × b) = ×b+a× . 60) Hence, it can be seen that the laws of vector differentiation are analogous to those in conventional calculus. 10 LINE INTEGRALS Consider a two-dimensional function f(x, y) which is deﬁned for all x and y. What is meant by the integral of f along a given curve from P to Q in the x-y plane?