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Unformatted text preview: 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 effects 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” effects 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 • 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 > is flowing to the right as shown in the figure....
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This note was uploaded on 06/20/2011 for the course ECE 329 taught by Professor Kim during the Spring '08 term at University of Illinois, Urbana Champaign.
- Spring '08