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Chapter 9: Problem 23

Prove that the magnetic flux density, B, satisfies the wave equation.

Short Answer

Expert verified
Both the electric and magnetic flux densities, D and B, satisfy the wave equation by the application of vector calculus and Maxwell's equations, and therefore, the magnetic flux density B does as well.

Step by step solution

01

Start with Maxwell's equations

The useful Maxwell's equations here are \(\nabla × E = - \frac{\partial B}{\partial t}\)and \(\nabla × H = J + \frac{\partial D}{\partial t}\) . Here, D is electric flux density, E and H are electric and magnetic field strength respectively, J is current density and \( \frac{\partial}{\partial t} \) defines a partial derivative with respect to time.
02

Apply Vector Identity

Use the relation \(\nabla × (\nabla × A) = \nabla (\nabla \cdot A) - \nabla^2 A \) with A being E and B in respective Maxwell's equations.
03

Apply to Maxwell's equations

For \(\nabla × E\) (let's call this equation 1) and \(\nabla × H\) (equation 2), use the vector Identity. For equation 1, this gives \(\nabla × (\nabla × E) = -\nabla × (\frac{\partial B}{\partial t})\). And for equation 2, we get \(\nabla × (\nabla × H) = \nabla × J + \nabla × (\frac{\partial D}{\partial t})\)
04

Simplify equations

\(\nabla × (\nabla × E) = \nabla (\nabla \cdot E) - \nabla^2 E\). Since \(\nabla \cdot E = 0\) (another Maxwell's equation), drop off that term. Also, \(\nabla × (\nabla × H) = \nabla (\nabla \cdot H) - \nabla^2 H\). Since \(\nabla \cdot H = 0\), drop off that term here also. Substituting B for E and D for H in respective equations, we get \(-\nabla^2B = -\frac{\partial^2 B}{\partial t^2}\) and \(-\nabla^2D = \nabla ×J + \frac{\partial^2 D}{\partial t^2}\). Compared with general wave equation \(\nabla^2 \phi = \frac{\partial^2 \phi}{\partial t^2}\), both B and D are solutions of the wave equation.
05

Conclusion

It's proven that both magnetic and electric flux densities satisfy the wave equation through application of vector manipulation and Maxwell's equations.

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Most popular questions from this chapter

Use Stokes' Theorem to derive the integral form of Faraday's law, $$ \int_{C} \mathbf{E} \cdot d \mathbf{s}=-\frac{\partial}{\partial t} \iint_{D} \mathbf{B} \cdot d \mathbf{S} $$ from the differential form of Maxwell's equations.

Let \(S\) be a closed surface and \(V\) the enclosed volume. Prove Green's first and second identities, respectively. a. \(\int_{S} \phi \nabla \psi \cdot \mathbf{n} d S=\int_{V}\left(\phi \nabla^{2} \psi+\nabla \phi \cdot \nabla \psi\right) d V\). b. \(\int_{S}[\phi \nabla \psi-\psi \nabla \phi] \cdot \mathbf{n} d S=\int_{V}\left(\phi \nabla^{2} \psi-\psi \nabla^{2} \phi\right) d V\).

Use Stokes' Theorem to evaluate the integral $$ \int_{C}-y^{3} d x+x^{3} d y-z^{3} d z $$ for \(C\) the (positively oriented) curve of intersection between the cylinder \(x^{2}+y^{2}=1\) and the plane \(x+y+z=1\)

Compute the following determinants using the permutation symbol. Verify your answer. a. \(\left|\begin{array}{ccc}3 & 2 & 0 \\ 1 & 4 & -2 \\ -1 & 4 & 3\end{array}\right|\) b. \(\left|\begin{array}{ccc}1 & 2 & 2 \\ 4 & -6 & 3 \\ 2 & 3 & 1\end{array}\right|\)

Compute the gradient of the following: a. \(f(x, y)=x^{2}-y^{2}\) b. \(f(x, y, z)=y z+x y+x z\). c. \(f(x, y)=\tan ^{-1}\left(\frac{y}{x}\right)\). d. \(f(x, y, z)=2 y^{x} \cos z-5 \sin z \cos y\).
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