A wide slab of thickness is centred on the plane and carries the uniform charge…

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

A wide slab of thickness 2t2t is centred on the plane x=0x = 0 and carries the uniform charge density ρ=5.0×10−8\rho = 5.0\times10^{-8} C m−3^{-3}. Inside the slab (−t≤x≤t-t \le x \le t) the potential is claimed to beV(x)=−ρx22ε0,V(x) = -\frac{\rho x^2}{2\varepsilon_0},independent of yy and zz, with the reference V=0V = 0 chosen on the mid-plane x=0x = 0 (infinity will not do for a slab of unlimited extent). Take ε0=8.854×10−12\varepsilon_0 = 8.854\times10^{-12} F m−1^{-1}. ((a)) Show that this VV satisfies Poisson's equation for the stated density, stating each derivative you take. ((b)) Evaluate ∇2V\nabla^2V numerically. ((c)) Compute E=−∇V\mathbf{E} = -\nabla V and its value (magnitude and direction) at x=+2.0x = +2.0 mm. ((d)) Verify that your E\mathbf{E} satisfies ∇⋅E=ρ/ε0\nabla\cdot\mathbf{E} = \rho/\varepsilon_0, and say why parts (a) and (d) are the same statement.

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