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Notes on axial vectors and symmetries

Notes on axial vectors and symmetries - Electricity&...

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Unformatted text preview: Electricity & Magnetism March 29, 2011 Robert Haussman Notes Symmetries and Ampere’s Law Margin Notes H Much like what we faced with electrostatics, there does not seem to be a good discussion in most of the literature about symmetries and their associated implications with respect to magnetic field and applications to Ampere’s Law. For the case of the magnetic field, the symmetry arguments become at first glance slightly non-intuitive due to the fact that the magnetic field is not technically a vector, but rather an axial vector or pseudovector . The differences will of course be discussed in what follows. In many books, some of the symmetries of the magnetic field are discussed, but usually in terms of parity and what that means for axial vectors. † However, to properly exploit the symmetries, we need to know how to handle axial vectors under reflections , which is not often mentioned. 1 Motivation To begin, recall Ampere’s Law: The line integral of the magnetic field over any closed loop γ is proportional to the total current enclosed by the loop: I γ ~ B · d ~ ‘ = μ I enc . (1) I Ampere’s Law: It is important to stress that Ampere’s law is true for any closed loop, so we can make our calculations easier by choosing a loop such that the integral is either simple or trivial. ! Key Point: Ampere’s Law is true for any closed loop γ, so we can choose whatever is most convenient. Naturally, we give these loops special names and refer to them as Amperian loops . To determine such a loop, we need to first examine the symmetries of the physical setup and see what the implications are for the magnetic field. In this document, we will only focus on reflections, so it may be useful to review the other set of notes Symmetries and Gauss’ Law . 2 Projections Before discussing reflections, it would first help to elaborate on projections. We’ve already seen projections when we wanted to decompose vectors into components. For instance if we have a vector ~v in the xy-plane an angle θ from the x-axis, we could rewrite it as ~v = | ~v | cos θ ˆ ı + | ~v | sin θ ˆ . The two components are just projections of ~v onto ˆ ı and ˆ respectively. Projections in a more general sense are what you would expect – a projection of a vector ~a onto a vector ~ b , denoted Proj ~ b ( ~a ) , is the vector component of ~a in the direction of ~ b. † Unfortunately, there are a few authors who have confused the terminology and called parity “reflec- tions.” Don’t let them confuse you! Symmetries and Ampere’s Law Figure 1: Projection of a vector ~a on a vector ~ b . Consider the two vectors ~a and ~ b separated by an angle θ as in Figure 1. We can decompose ~a into components that are parallel and perpendicular to ~ b , denoted ~a k and ~a ⊥ respectively. From our knowledge of trigonometry and dot products, we find Proj ~ b ( ~a ) ≡ ~a k = | ~a | cos θ ˆ b = | ~a || ~ b | | ~ b | cos θ ˆ b = ~a · ~ b | ~ b | ˆ b = ( ~a ·...
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