1. Consider an infinitely long wire carrying the current I, which is elongated along the z-axis....

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1. Consider an infinitely long wire carrying the current I, which is elongated along the z-axis. This wire is placed in vacuum. Away from the current, one can define a magnetic scalar potential Vm which satisfies the Laplace equation, V?Vm = 0. (a) From the symmetry of the system, we can see that Vm only depends on ø, the azimuthal angle around the z-axis. Explain why this is the case. You can use the fact that B = -µoVVm: т (b) Using the cylindrical coordinates, find Vm as a function of o by solving the Laplace equation. You can choose Vm(o = 0) = 0 to fix the zero of the potential (the potential can be shifted by a constant). You can find the coefficients in the general solution by using the Ampere's law, S B•dl = µoI. You will find that Vm is not single-valued, namely Vm(o = 27) # Vm(¢ = 0). This is due to the fact that the wire is located at (x, y) = (0,0), which can be viewed as a singularity. т 1. Consider an infinitely long wire carrying the current I, which is elongated along the z-axis. This wire is placed in vacuum. Away from the current, one can define a magnetic scalar potential Vm which satisfies the Laplace equation, V?Vm = 0. (a) From the symmetry of the system, we can see that Vm only depends on ø, the azimuthal angle around the z-axis. Explain why this is the case. You can use the fact that B = -µoVVm: т (b) Using the cylindrical coordinates, find Vm as a function of o by solving the Laplace equation. You can choose Vm(o = 0) = 0 to fix the zero of the potential (the potential can be shifted by a constant). You can find the coefficients in the general solution by using the Ampere's law, S B•dl = µoI. You will find that Vm is not single-valued, namely Vm(o = 27) # Vm(¢ = 0). This is due to the fact that the wire is located at (x, y) = (0,0), which can be viewed as a singularity. т