Answer to Question #278381 in Vector Calculus for Mafin

Question #278381

Let E:=Exi^+Eyj^+Ezk^ and H:=Hxi^+Hyj^+Hzk^ be two vectors assumed to have continuous partial derivatives (of second order at least) with respect to position and time. Suppose further that E and H satisfy the equations:∇⋅E=0,∇⋅H=0,∇×E=−1c∂H∂t,∇×H=1c∂E∂tprove that E and H satisfy the equation∇2Ei=1c2∂2Ei∂t2 and ∇2Hi=1c2∂2Hi∂t2Here, i=x,y or z.


1
Expert's answer
2021-12-13T17:46:31-0500

"\u2207\u00d7(\u2207\u00d7E)=-\\frac{1}{c}\\frac{\\partial (\u2207\u00d7H)}{\\partial t}=-\\frac{1}{c^2}\\frac{\\partial E^2}{\\partial t^2}"


"\u2207\u00d7(\u2207\u00d7E)=\u2207(\u2207\u22c5E)\u2212\u2207^2E=\u2212\u2207^2E"


"\u2207^2E=\\frac{1}{c^2}\\frac{\\partial E^2}{\\partial t^2}"


"\u2207^2E_i=\\frac{1}{c^2}\\frac{\\partial E_i^2}{\\partial t^2}"



"\u2207\u00d7(\u2207\u00d7H)=\\frac{1}{c}\\frac{\\partial (\u2207\u00d7E)}{\\partial t}=-\\frac{1}{c^2}\\frac{\\partial H^2}{\\partial t^2}"


"\u2207\u00d7(\u2207\u00d7H)=\u2207(\u2207\u22c5H)\u2212\u2207^2H=\u2212\u2207^2H"


"\u2207^2H=\\frac{1}{c^2}\\frac{\\partial H^2}{\\partial t^2}"


"\u2207^2H_i=\\frac{1}{c^2}\\frac{\\partial H_i^2}{\\partial t^2}"

Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS