Concept explainers
Starting with Maxwell’s equations for simple, charge free media, derive the Helmholtz equation for H.
The Helmholtz equation for magnetic field
Answer to Problem 5.1P
The Helmholtz equation for magnetic field
Explanation of Solution
Given:
The Maxwell’s equations are simple and the media is free of any charge.
Concept used:
The equation of Ampere’s circuital law for simple and charge free media is shown below.
The equation of Faraday’s law is shown below.
Calculation:
The condition for charge free media is
Take curl on both sides of equation (1) as shown below.
Since, the right side of above equation is a position derivative that acts on time derivative, so the above equation can be written as,
Substitute
The curl of curl of any vector is equal to the divergence and Laplacian, so the above equation can be written as,
Due to charge free medium the value of divergence of magnetic field
Now, the above equation can be written as,
The above equation is known as Helmholtz wave equation for magnetic field
Therefore, the Helmholtz equation is
Conclusion:
Thus, the Helmholtz equation for magnetic field
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Chapter 5 Solutions
Fundamentals of Electromagnetics with Engineering Applications
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