Question

4. Total: 34 pts] Suppose the vector potential in a region of space is given by: 2 2 a) [4 pts| What are the units of the given constants a and Ao? b) [10 ptsj Determine the magnetic field from this vector potential and then determine the current density J from the magnetic field. (Eq. (3) is given in Cartesian coordinates, but you may use another coordinate system, if that would make the calculation easier. Briefly, but clearly justify your choice.) c) 10 pts Integrate the current density to mathematically show that the total current flowing through any infinite plane parallel to the x -y plane is zero. Then, give an argument (without doing any formal integral) why you could have known before calculating that this must be the case. d) [10 pts] Calculate the divergence of J. What does your result imply?

Answer 1

Given that the vector potential in a region of space is given by A = A exp (- x^2 + y^2/2a^2) z The given constant a is same of the unit of x or y i.e. unit of a is metre * the given constant A_0 is same as unit of A (vector potential) unit of A_0 is Tesla metre To determine the magnetic field from this vector potential we know that magnetic field B = nabla times A B = Vertical-bar i partial differential/partial differential x 0 i partial differential/partial differential y 0 i partial differential/partial differential 2 A_0 exp (x^2 + y^2/2a^2) Vertical-bar = i [partial differential/partial differential y A_0 exp (- x^3 + y^2/2a^2)] - i [partial differential/partial differential x A exp (- x^2 + y^2/2a^2)]