Show that the one-dimensional p(x) = Ae−ikx and three-dimensional expression p(x, y, z) = Ae−ikr is a solution to the Cartesian form of the Helmholtz equation, ∇2p + k2p = 0
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Show that the one-dimensional p(x) = Ae−ikx and three-dimensional expression p(x, y, z) = Ae−ikr is a
solution to the Cartesian form of the Helmholtz equation, ∇2p + k2p = 0
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- In the following set of equations (from a quantum mechanics problem), A and B are the unknowns, k and K are given, and i=√1 .Use Cramer's rule to find A and show that |Af = 1. { Use Cramer's rule to solve for x and t the Lorentz equations of special relativity: A ikA x' = y(x - vt) t' = y(t - vx/c²) B = -1 ik KB where ²(1-²/²) = 1Consider a system of charges and currents. The work done per second by the fields on charge q is: F.ö = qü · (Ë + õ x B) = qū · Ë Where F is the Lorentz force. The work done per second per unit volume is: ngữ · Ẽ = J . Ẽ By conservation of energy, the rate of change of the energy density in the electric and magnetic fields is therefore: -J. Ē. 1) Demonstrate the following equation by using Maxwell's equation: -J.Ē=- (ỹ × B) · Ẽ + €o- Ë. μοLorentz Force is given by F = qE + q v, × B Given to vector, v, = ((0) î + (8) ĵ + (0) k) × 10° m/s and B ((-4) î + (5) ĵ + (5) k) Tesla, If the charge is given by q = 24 nC and E = 0 Then, Find the Force F. x component Give your answer up to at least three significance digits. N y component Give your answer up to at least three significance digits. z component Give your answer up to at least three significance digits.