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: 𝑎) 𝑣⃗ = 𝑥 2 𝑖̂ + 3𝑥𝑧 2 𝑗̂ − 2𝑥𝑧𝑘̂ 𝑏) 𝑣⃗ = 𝑥𝑦𝑖̂ + 2𝑦𝑧𝑗̂ + 3𝑧𝑥𝑘̂ 𝑐) 𝑣⃗ = 𝑦 2 𝑖̂ + (2𝑥𝑦 + 𝑧 2 )𝑗̂ + 2𝑦𝑧𝑘̂ 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) ⃗⃗ are vector functions): Prove the following product rules (f is a scalar function; 𝐴⃗, 𝐵 𝑎) ∇ ∙ (𝑓𝐴⃗) = 𝑓(∇ ∙ 𝐴⃗) + 𝐴⃗ ∙ (∇𝑓) ⃗⃗ ) = 𝐵 ⃗⃗ ∙ (∇ × ⃗A⃗) − ⃗A⃗ ∙ (∇ × 𝐵 ⃗⃗ ) 𝑏) ∇ ∙ (𝐴⃗ × 𝐵 Problem 4 (Mark 10/30) The current density in a certain region is given (in spherical coordinates) by: t ⃗J = J0 − e 𝜏 r r̂ . 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) 𝑡 A sphere of radius R has decreasing uniform charge density 𝜌(𝑡) = 𝜌0 𝑒 −𝜏 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.