# Assignment 1 week2 ```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
⃗⃗ ∙ (∇ &times; ⃗A⃗) − ⃗A⃗ ∙ (∇ &times; 𝐵
⃗⃗ )
𝑏) ∇ ∙ (𝐴⃗ &times; 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.
```