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PHY4221_Homework_Assignment_1_Solution

PHY4221_Homework_Assignment_1_Solution - PHY4221 Homework...

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PHY4221 Homework Assignment 1 Solution (Written by: Leung Shing Chi) 1. (In this problem, we use Einstein’s summation convention, i.e. , also we use the following notation: 3 1 i i i i i A A A A , i i j j A A x ) Given the Lagrangian to be 1 ( , ) ( , ) 2 q L mv v q r t v A r t c   , we apply the Euler-Lagrange Equation (ELE) to L : 0 i i d L L dt v x where 1,2,3 i (3 equations in total). , i i i i j j i i i L q P mv A v c L q q v x x c   A Therefore, we find , i i i A d L d q mv v A dt v dt c t j i j (It should be noted that this result arises as the usage of ordinary derivative) and we take the ELE to be: , , , , , 0 1 i i j i j i j i i i j i j A d q q mv v A q v A dt c t c A d q mv q v A v A dt c t c   , j i j j i This, in fact, is the solution by observing the following relation among the scalar and vector potential and the E and B fields. , , , 1 i i i j k jki i m n mnk ijk j i j j j i i A E c t v B v B v A v A v A   , Certainly, to achieve this result, we need some relations between and , namely: ijk mnk im jn in jm     Finally, it is now obvious to see the equation of motion (EoM) to be:
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