By Hans J. Weber and George B. Arfken (Auth.)
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14 An electric charge q\ moving with velocity νχ produces a magnetic in duction B given by μο vi x r B = —c —q\—τ,— 4π' (mks units), where f points from qi to the point at which B is measured (Biot and Savart law). (a) Show that the magnetic force on a second charge q2i velocity v 2 , is given by the triple vector product 4π r2 (b) Write out the corresponding magnetic force Fi that q2 exerts on q\. Define your unit radial vector. How do Fi and F2 compare? (c) Calculate Fi and F2 for the case of q\ and q2 moving along parallel trajectories side by side.
7 Verify the identity A x (V x A) = ì V ( A 2 ) - (A · V ) A . Test this identity for a typical vector field, such as A ~ r or r / r 3 . 8 If A and B are constant vectors, show that V(A-Bxr)=AxB. 9 A distribution of electric currents creates a constant magnetic moment m. The force on m in an external magnetic induction B is given by F = V x (B x m ) . Show that F = V(m-B). Note. Assuming no time dependence of the fields, Maxwell's equations yield V x B = 0. Also V · B = 0. 54 CHAPTER 1. 10 An electric dipole of moment p is located at the origin.
97d). We may do this by taking the curl of both sides of Eq. 97d) and the time derivative of both sides of Eq. 97c). Since the space and time derivatives commute, and we obtain I— f- V x (V x E ) = - - 1 d2E 9t 2 58 CHAPTER 1. VECTOR ANALYSIS Application of Eqs. 97b) yields the electromagnetic vector wave equation. Again, if E is expressed in Carte sian coordinates, Eq. 98) separates into three scalar wave equations, each involving a scalar Laplacian. When external electric charge and current densities are kept as driving terms in Maxwell's equations, similar wave equations are valid for the electric potential and the vector potential.