LEC 33 - Problem Solving-4

This lecture focuses on solving problems related to the magnetic vector potential. The instructor begins by reviewing the definition of the magnetic field as the curl of the vector potential. The first problem involves finding the magnetic field given a vector potential of the form A = r × C, where C is a constant vector. The solution involves computing the curl in Cartesian coordinates, leading to a constant magnetic field B = -2C. The second problem provides a vector potential A = kx ĵ and asks for the current density. The magnetic field is found to be constant, B = k k̂, and since it is uniform, the current density is zero. The third problem involves a vector potential in cylindrical coordinates, A = μ₀ I z / s ŝ, and requires showing that its divergence is zero and finding the magnetic field. The divergence is zero, and the magnetic field is B = μ₀ I / s φ̂. A fourth problem gives another vector potential, A = μ₀ I ln(a/s) k̂, and again shows divergence is zero and finds the same magnetic field. The instructor explains that the non-uniqueness of the vector potential arises because boundary conditions are not specified. The final problem involves a rotating charged sphere and asks for the magnetic field at a point outside. Using the vector potential for a magnetic dipole, the magnetic field is derived in spherical coordinates, yielding B = (μ₀ ω σ R⁴ / (3 r³)) (2 cos θ r̂ + sin θ θ̂). The lecture concludes with the magnetic field at a specific point.