Section N_8 INSTALLED

This is the Title PhL 3.26.05 N.8 Magnetic Ohm's Law for Arbitrary B In Section N.4 we developed a Magnetic Ohm's Law for B = B . We now allow B(x) to point in a general direction with components B1, B2 and B3 and we consider first a DC static situation. Equation (N.1.5) then says κ v = E + v x B where κ ≡ 1/μ (N.8.1) or κv1 = E1 + v2B3-v3B2 κv2 = E2 + v3B1-v1B3 κv3 = E3 + v1B2-v2B1 . (N.8.2) Notice that dim(κ) = dim(B) = Tesla. Maple solves this equation for the velocity components vi : From (N.1.8) and (N.1.9) we know that J = nqv = (σ/μ)v = κσ v . Extracting the vi from the above Maple solution and multiplying by κσ we get (N.8.3) which is our new and very complicated tensor Magnetic Ohm's Law in the presence of an arbitrary E and B field. That is to say, we have J = Σ E where Σ is a 3x3 matrix which is a function of the Bi. If we could ignore the three Bi components (set them to zero in the above equations), the equations would reduce to the regular Ohm's Law Ji = σEi. This is in effect the case if Bi << κ for all three components of B. So a condition for the tensor Ohm's Law reducing to the regular Ohm's law is this: Bi << κ κ = 1/μ μ = (qτ/m) κ = (m/qτ) (N.8.4) so we need then Bi << (m/qτ) . (N.8.5) For copper, we compute κ = m/qτ using numbers from Section N.1, Our conclusion is that "regular Ohm's Law" is applicable as long as Bi << 569 Tesla. Even the largest practical B fields are far below this number. From wiki: So even the Large Hadron Collider designers and frog levitators can use regular Ohm's Law (along with the writer of Section 1 and Appendix D of this document). So why do we need to use the Magnetic Ohm's Law when dealing with the Hall Effect which has a relatively small B field? Recall (N.4.10), Jx = σ (Ex + ωcτ Ey) ωc ≡ (qB/m) Jy = σ (Ey - ωcτ Ex) B = B Jz = σEz . σ = (nq2τ/m) (N.4.10) In the Hall experiment of Fig N.1 we must have Jy = 0 and that means we cannot ignore the second term in the Jy expression above, even though it is much smaller than the first term. We get Ey = ωcτ Ex as in (N.5.3) which is the tiny delicate Hall field. One can repeat the above analysis to get a tensor Magnetic Ohm's Law for a monochromatic AC situation by replacing κ → κ [ 1 + jωτ ] in (N.8.1), based on (N.1.5). As shown at the end of Section N.1, for copper we have ωτ << 1 for f << 16,000 GHz, so this κ replacement has a miniscule effect and our conclusions above still apply.