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SUMMARY

The discussion focuses on the application of Maxwell's equations, specifically Gauss's law in differential form and Maxwell-Faraday's law, to analyze electric fields. Participants emphasize the importance of using the identity for the divergence of a product of a scalar function and a constant vector to simplify calculations. The conversation highlights the necessity of substituting the derived expression for the electric field into the equations to achieve accurate results. Key mathematical identities are provided to assist in the calculations.

PREREQUISITES
  • Understanding of Maxwell's equations, specifically Gauss's law and Maxwell-Faraday's law.
  • Familiarity with vector calculus, including divergence and curl operations.
  • Knowledge of scalar functions and their properties in electromagnetic theory.
  • Proficiency in mathematical identities relevant to vector fields.
NEXT STEPS
  • Study the derivation and implications of Gauss's law in differential form.
  • Explore the applications of Maxwell-Faraday's law in electromagnetic induction.
  • Learn about vector calculus identities and their use in simplifying electromagnetic equations.
  • Investigate advanced topics in electromagnetism, such as wave propagation and field interactions.
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Students of physics, electrical engineers, and professionals working in electromagnetic theory who seek to deepen their understanding of Maxwell's equations and their applications.

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Homework Statement
Part a is not an issue, I have that solved. I'm confused with part b (and c since i cant get b), and need someone to explain this to me... I do not even know where to start. Although, I do know that k E and B are orthogonal to one another. Evidently, this means that the dot product between k and E will be 0, as they are perpendicular. I just don't know how to use Maxwells Equations to prove this.
Thanks!
Relevant Equations
Maxwells Equation
Screen Shot 2019-09-16 at 9.35.49 PM.png
 
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The two Maxwell's equations you going to use for this are
1) Gauss's law in differential form $$\nabla\cdot E=0$$ (with no sources)

2)Maxwell-Faraday's law in differential form $$\nabla\times E=-\frac{\partial B}{\partial t}$$

Start by pluging in 1) the expression for electric field you got from doing part a) and try to calculate ##\nabla\cdot E##. You might find the following identity useful ##\nabla\cdot( \phi E_0)=E_0\cdot\nabla\phi## where ##\phi(x,y,z,t)## is a scalar function and ##E_0## is a constant vector independent of x,y,z,t.

When plugging the expression you got from a) for E in 2) you might find useful the following identity
##\nabla\times (\phi E_0)=\nabla\phi\times E_0##
 
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