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ECE 329 - Lecture 12 - 12 Magnetic force and fields and...

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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 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. These observations preclude the possibility that a particle velocity u could appear as u = u- v (Newtonian) to an observer moving with a velocity v in the same direction as the particle — instead, u must transform to the observer frame like u = u- v 1- uv c 2 , (relativistic) so that if u = c , then u = c also, as first pointed out by Albert Einstein. This “relativistic” velocity transformation in turn requires that positions and times of physical events transform like 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 , a consequence of dx = γdx at a fixed t : since γ > 1 , dx < dx , indicating that moving objects appear contracted (shorter) when viewed instantaneously from other reference frames. A second one is time dilation , a consequence of dt = γdt at a fixed x : since γ > 1 , dt < dt , indicating that moving clocks tick slower — with dilated (longer) intervals dt — than a clock stationed at some x . 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...
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ECE 329 - Lecture 12 - 12 Magnetic force and fields and...

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