Inside a wide slab occupying the potential is found to be V(x) = - _0 6 _0 d ,x^3

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Electrostatic fields and potential

Electricity, Magnetism and Electromagnetism, Ch Divergence, curl and the local form of the field laws

Problem

Inside a wide slab occupying 0≤x≤d0 \le x \le d the potential is found to beV(x)=−ρ06ε0d x3,V(x) = -\frac{\rho_0}{6\varepsilon_0 d}\,x^3,independent of yy and zz, with ρ0=4.0×10−6\rho_0 = 4.0\times10^{-6} C m−3^{-3} and d=0.020d = 0.020 m. Take ε0=8.854×10−12\varepsilon_0 = 8.854\times10^{-12} F m−1^{-1}. ((a)) Compute ∇2V=∂2V/∂x2+∂2V/∂y2+∂2V/∂z2\nabla^2 V = \partial^2V/\partial x^2 + \partial^2V/\partial y^2 + \partial^2V/\partial z^2. ((b)) Using Poisson's equation, deduce the charge density ρ(x)\rho(x) inside the slab, and say in one sentence why Poisson's equation is not a new law. ((c)) Evaluate ρ\rho and ∇2V\nabla^2 V at x=0.015x = 0.015 m. ((d)) Check your ρ(x)\rho(x) a second way, by computing E=−∇V\mathbf{E} = -\nabla V and then ∇⋅E\nabla\cdot\mathbf{E}.

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