Lecture11 - Lecture 11 Notes, 95.658, Spring 2012,...

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Lecture 11 Notes, 95.658, Spring 2012, Electromagnetic Theory II Dr. Christopher S. Baird, UMass Lowell 1. Einstein's Principle of Relativity - The goal is to find a principle of relativity between different frames of reference that will make Maxwell's equations have the same form in all frames. - Einstein began with the two postulates: 1. The laws of physics (including electrodynamics) are the same in every inertial frame. 2. The speed of light is the same constant in every inertial frame. - Absent from these postulates is any assumption of universal time, universal mass, or universal lengths. - Consider a railway car (frame K ') moving at some constant speed v with respect to the ground (frame K ). - In frame K ', at time zero, a light bulb on the car's floor is turned on and at time t ' a detector on the ceiling that is a distance x ' forward of the bulb detects its light. - In frame K , the detector moved along with the car and the light hitting it must have traveled in a more diagonal path and taken a time t to do so. - Let us calculate the speed of light in each frame and use the second postulate to force these speeds to be equal to the same universal constant c . - In calculating the speed, we make use of the first postulate and say that speed is just distance over time in both frames, as measured in its own frame. - In frame K ' we have: c = x ' 2 y ' 2 t ' - In the lab frame K we have: c = x 2 y 2 t - Simplify both equations to find: c 2 t ' 2 x ' 2 y ' 2 = 0 and c 2 t 2 x 2 y 2 = 0 v x K' y v K y' x '
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- They both equal zero so they must equal each other: c 2 t ' 2 − x ' 2 y ' 2 = c 2 t 2 − x 2 y 2 - We should note that we neglected the z dimension to make the illustration easier to draw, but we should include it to get the most general case. It is handled exactly as the y dimension was, so that we get: c 2 t ' 2 −( x ' 2 + y ' 2 + z ' 2 )= c 2 t 2 −( x 2 + y 2 + z 2 ) Principle of Einstein's Relativity - This is the fundamental principle of Einstein's relativity relating the coordinates in two different inertial frames. It is essentially a mathematical version of the two postulates. - Historically, Einstein called his theory of relativity for inertial frames the “Special Theory of Relativity” to differentiate it from his theory for non-inertial frames. - Though it is in a fundamental form, the equation above is not particularly useful in practice. - We would like to express the primed coordinates as functions of the unprimed coordinates. - First note that if the motion of frame K ' is in the x dimension, as in this problem, then in the y and z dimensions both frames are at rest relative to each other. This leads to: y ' = y z ' = z - This reduces the relativity equation to: c 2 t ' 2 x ' 2 = c 2 t 2 x 2 Principle of Einstein's Relativity for x-Directional Velocity - We have two unknowns and one equation, so there is no unique solution. There must be another
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This note was uploaded on 02/13/2012 for the course PHYSICS 95.658 taught by Professor Staff during the Spring '11 term at UMass Lowell.

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Lecture11 - Lecture 11 Notes, 95.658, Spring 2012,...

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