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2. Electromagnetic Scattering Theory: Scattering of a plane electromagnetic wave by a spherical object is perhaps the important problem in electromagnetic scattering theory. The sphere serves as a model of a compact smooth scatterer and is one of only two compact geometries for which Maxwell's equations can be solved in an explicit form. An exact solution is available not only for perfectly electrically conducting (PEC) but also for material, homogeneous or radially stratified, magneto-dielectric spheres. The physical interpretation of scattering from spheres, depending on the electrical size and the degree of absorption in the material of the sphere. Many applications of electromagnetic scatterings can be adopted in electrical engineering, biophysics, astrophysics and nuclear physics. To simplify the problem, we will only study and calculate the electric field inside the hollow metal sphere: E = -∇V. This problem, a boundary-value problem, can be stated as below in which Laplace's equation of V(r, θ, φ) can be solved using the separation of variable method subjected to Dirichlet's (boundary) conditions in spherical coordinates: The potential $V_0(\theta)$ is specified on the surface of a hollow sphere, of radius R. Find the potential inside the sphere. In the spherical system, Laplace's equation reads: $$ \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial V}{\partial r} \right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial V}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 V}{\partial \phi^2} = 0. $$ (3.53) assume the problem has azimuthal symmetry, $\frac{\partial V}{\partial \phi} = 0.$ so that V is independent of $\phi$; in that case, Eq. 3.53 reduces to $$ \frac{\partial}{\partial r} \left( r^2 \frac{\partial V}{\partial r} \right) + \frac{1}{\sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial V}{\partial \theta} \right) = 0. $$ (3.54) subject to the boundary condition, the potential $V_0(\theta)$ is specified on the surface $V_0(\theta) = k \sin^2(\theta/2)$, (3.70) Hollow Conducting Sphere

          2. Electromagnetic Scattering Theory: Scattering of a plane electromagnetic wave by a spherical object is perhaps the important problem in electromagnetic scattering theory. The sphere serves as a model of a compact smooth scatterer and is one of only two compact geometries for which Maxwell's equations can be solved in an explicit form. An exact solution is available not only for perfectly electrically conducting (PEC) but also for material, homogeneous or radially stratified, magneto-dielectric spheres. The physical interpretation of scattering from spheres, depending on the electrical size and the degree of absorption in the material of the sphere. Many applications of electromagnetic scatterings can be adopted in electrical engineering, biophysics, astrophysics and nuclear physics.
To simplify the problem, we will only study and calculate the electric field inside the hollow metal sphere: E = -∇V. This problem, a boundary-value problem, can be stated as below in which Laplace's equation of V(r, θ, φ) can be solved using the separation of variable method subjected to Dirichlet's (boundary) conditions in spherical coordinates:
The potential $V_0(\theta)$ is specified on the surface of a hollow sphere, of radius R.
Find the potential inside the sphere.
In the spherical system, Laplace's equation reads:
$$ \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial V}{\partial r} \right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial V}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 V}{\partial \phi^2} = 0. $$ (3.53)
assume the problem has azimuthal symmetry, $\frac{\partial V}{\partial \phi} = 0.$ so that V is independent of $\phi$; in that case, Eq. 3.53 reduces to
$$ \frac{\partial}{\partial r} \left( r^2 \frac{\partial V}{\partial r} \right) + \frac{1}{\sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial V}{\partial \theta} \right) = 0. $$ (3.54)
subject to the boundary condition, the potential $V_0(\theta)$ is specified on the surface
$V_0(\theta) = k \sin^2(\theta/2)$, (3.70)
Hollow Conducting Sphere

2 electromagnetic scattering theory scattering of a plane electromagnetic wave by a spherical object is perhaps the important problem in electromagnetic scattering theory the sphere serves a 26113

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