Physics

Physics · electromagnetism

Problem

(b) Laplace's equation is a second order differential equation ∇2ϕ=∂2ϕ∂x2+∂2ϕ∂y2+∂2ϕ∂z2=0(8)\nabla^{2} \phi=\frac{\partial^{2} \phi}{\partial x^{2}}+\frac{\partial^{2} \phi}{\partial y^{2}}+\frac{\partial^{2} \phi}{\partial z^{2}}=0 \tag{8} Solutions to this equation are called harmonic functions. One of the most important properties satisfied by these functions is the maximum principle. It states that a harmonic function attains extremes on the boundary. Using this, prove the uniqueness theorem: Solution to Laplace's equation in a volume VV is uniquely determined if its solution on the boundary is specified. That is, if ∇2ϕ1=0\nabla^{2} \phi_{1}=0, ∇2ϕ2=0\nabla^{2} \phi_{2}=0 and ϕ1=ϕ2\phi_{1}=\phi_{2} on the boundary of VV, then ϕ1=ϕ2\phi_{1}=\phi_{2} in VV. Hint: Consider ϕ=ϕ1−ϕ2\phi=\phi_{1}-\phi_{2}. Given constants: 5. Polarization and Oscillation In this problem, we will understand the polarization of metallic bodies and the method of images that simplifies the math in certain geometrical configurations. Throughout the problem, suppose that metals are excellent conductors and they polarize significantly faster than the classical relaxation of the system. Context question: (a) Explain in words why there can't be a non-zero electric field in a metallic body, and why this leads to constant electric potential throughout the body. Context answer: 开放性回答

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