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MA330: Assignment 4 Required Reading. • Read Chapter 4, § 42 To be turned in April 12th at the start of class. 1. Recall Maxwell’s equations, as stated in class. ∇·E= ρ 0 ∇·B=0 ∇×E=− ∂B ∂t ∇ × B = µ0 J + µ0 0 ∂E ∂t Assume that {µ0 , 0 } are constants. (a) In class, we showed that the electric field, E, obeys an inhomogeneous wave equation. Using a similar method, show that the magnetic field, B, also obeys an inhomogeneous wave equation. (b) Using Maxwell’s equations, derive the following relationship between current density and charge density. ∂ρ +∇·J=0 ∂t 2. Consider the transformation u= x , x2 + y 2 v= y x2 + y 2 and its inverse u v , y= 2 . u2 + v 2 u + v2 If the point (x, y) corresponds to (r, θ) in polar coordinates, show that the transformed point (u, v) corresponds to (1/r, θ) in polar coordinates. In visual terms, this transformation reflects (x, y) across the unit circle. If f (u, v) is a scalar function, show how to compute grad(f ) in this new coordinate system. If g(u, v) is a vector function, show how to compute div(g) in this new coordinate system. If g(u, v) is a vector function, show how to compute curl(g) in this new coordinate system. If f (u, v) is a scalar function, show how to compute the Laplacian of f in this new coordinate system. x= (a) (b) (c) (d) (e) 3. In class, we discussed 2D irrotational flows, described by scalar and vector potentials, φ and ψ. We showed that both potentials satisfy Laplace’s equation. There is a much easier way to do this. (a) If the flow is two dimensional, and u = grad(φ) and u = curl(ψk), then show that the following equations hold. φy = −ψx . φx = ψy , (b) Supposing that φ and ψ are both twice differentiable, use the result from the previous part to show ∆φ = 0 and ∆ψ = 0. 4. Consider the vector field u = hax + by, cx + dyi, where {a, b, c, d} is a set of nonzero constants. Give conditions on the constants so that u could be the velocity field of a potential flow. Under these conditions, show that the system dx = ax + by dt dy = cx + dy dt has a saddle point at the origin, then sketch the vector field and some trajectories. Why is it not surprising that this equilibrium point isn’t a node, spiral, or center? (Hints: Recall things you learned in DE2. Talk about divergence and curl.) 1