# 329lect12 - 12 Magnetic force and elds and Amperes law...

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12 Magnetic force and fields and Ampere’s law Pairs of wires carrying currents I running in the same (opposite) direction are known to attract (repel) one another. In this lecture we will explain the I I F F “Things should be made as simple as possible – but no simpler.” Albert Einstein mechanism — the phenomenon is a relativistic 1 consequence of electrostatic charge interactions, but it is more commonly described in terms of magnetic fields. This will be our introduction to magnetic field e ff ects in this course. 1 Brief summary of special relativity: Observations indicate that light (EM) waves can be “counted” like particles and yet travel at one and the same speed c = 3 × 10 8 m/s in all reference frames in relative motion. As first recognized by Albert Einstein, these facts preclude the possibility that a particle velocity u could appear as u = u - v (Newtonian) to an observer approaching the particle with a velocity v ; instead, u must transform to the observer’s frame as u = u - v 1 - uv c 2 , (relativistic) so that if u = c , then u = c also. This “relativistic” velocity transformation in turn requires that positions x and times t of physical events transform (between the frames) as x = γ ( x - vt ) and t = γ ( t - v c 2 x ) , (relativistic) where γ 1 1 - v 2 /c 2 , rather than as x = x - vt and t = t, (Newtonian) so that dx dt = u and dx dt = u are related by the relativistic formula for u given above. Relativistic transformations imply a number of “counter-intuitive” e ff ects ordinarily not noticed unless | v | is very close to c . One of them is Lorentz contraction , implied by dx = dx / γ at a fixed t : since γ > 1 , dx < dx , and moving objects having velocities v appear shorter then they are when viewed from other reference frames where v is determined. A second one is time dilation , implied by dt = dt/ γ at a fixed x : since γ > 1 , dt < dt , and moving clocks having velocities v and fixed x run slower than clocks in other reference frames where v is determined. Consider taking PHYS 325 to learn more about special relativity. 1

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Consider a current carrying stationary wire in the lab frame: the wire has a stationary lattice of positive ions, I v + + + + + + + + λ - λ + = - λ - (a) Neutral wire carrying current I in the "lab frame": I v + + + + + + + + + + _ _ _ _ _ _ _ λ - = λ - / γ λ + = γλ + (b) In the "electron frame" the wire appears positively charged: E = λ 2 π o r ˆ r r _ _ _ _ _ _ _ _ λ λ + v 2 c 2 = Iv c 2 = Ivμ o o electrons are moving to the left through the lattice with an average speed v , and a current I > 0 is flowing to the right as shown in the figure.
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