Electromagnetic Wave Scattering by Conducting Sphere

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

Electromagnetic waves can be scattered by electrical conductors. Consider a linearly polarized plane wave, with electric field in the direction and incident in the direction on a perfectly conducting sphere of radius . The electric and magnetic fields of the incoming wave, expressed in Cartesian coordinates, are and , where ( and are the light velocity and impedance, respectively). The scattered fields and are best expressed in spherical coordinates , using Legendre functions and spherical Hankel functions and taking account of the boundary conditions. The total fields are given by and .


The time-averaged electromagnetic energy density is the sum of the electric and magnetic energy densities: and . (Note that the fields here represent mean-squared values, hence the additional factors 1/2). The total energy density is given by . For a incoming wave with , the total energy density is (). Taking account of scattering, the energy density fluctuates in space and can locally increase up to double the average value: .

This Demonstration shows the three-dimensional distributions of the energy densities and in the vicinity of the sphere, given the sphere radius and the incident wave frequency . The distributions of and are displayed in red and blue, respectively, in the - and - planes. The variation in density depends on the relative dimensions of the conductor radius and the electromagnetic wavelength . The image refreshes rather slowly, even with the image quality decreased.


Contributed by: Y. Shibuya (September 2015)
Open content licensed under CC BY-NC-SA




[1] J. A. Stratton, Electromagnetic Theory, Hoboken, NJ: John Wiley & Sons, 2007.

[2] W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed., Mineola, NY: Dover Publications, 2005.

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.