PHYS325/PHY9325 Assignment 1, Autumn 2020 Set on Thu 12th March, due on Thu 19th March. TOTAL RECALL! Problem 1 (2/30) Calculate the divergence and the curl of the following vector functions: ð) ðĢâ = ðĨ 2 ðĖ + 3ðĨð§ 2 ðĖ − 2ðĨð§ðĖ ð) ðĢâ = ðĨðĶðĖ + 2ðĶð§ðĖ + 3ð§ðĨðĖ ð) ðĢâ = ðĶ 2 ðĖ + (2ðĨðĶ + ð§ 2 )ðĖ + 2ðĶð§ðĖ Problem 2 (8/30) A conducting sphere of radius R1 = 5 cm is charged with Q1 = 10-6 C. A conducting spherical shell with internal radius R2 = 10 cm and external radius R3=12cm is concentric with the sphere and it is charged with Q2 = 10 Q1. Calculate: a) the superficial charge density σ2 of the internal surface of the shell; b) the potential difference V(R1)-V(R2) between the sphere and the shell. R3 R2 R1 Problem 3 (3/30) ââ are vector functions): Prove the following product rules (f is a scalar function; ðīâ, ðĩ ð) ∇ â (ððīâ) = ð(∇ â ðīâ) + ðīâ â (∇ð) ââ ) = ðĩ ââ â (∇ × âAâ) − âAâ â (∇ × ðĩ ââ ) ð) ∇ â (ðīâ × ðĩ Problem 4 (Mark 10/30) The current density in a certain region is given (in spherical coordinates) by: t âJ = J0 − e ð r rĖ . Find the total current that leaves a spherical surface whose radius is r = a at t = τ; and the charge density at the surface. Problem 5 (Mark 7/30) ðĄ A sphere of radius R has decreasing uniform charge density ð(ðĄ) = ð0 ð −ð with radial velocity distribution. At the surface, find: a) current density J, b) displacement current density JD, c) electric and magnetic field intensity (E and B). Please note that symbols and letters in bold represent a vector and vector notation MUST be used.