# 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:
ð) ðĢâ = ðĨ 2 ðĖ + 3ðĨð§ 2 ðĖ − 2ðĨð§ðĖ
ð) ðĢâ = ðĨðĶðĖ + 2ðĶð§ðĖ + 3ð§ðĨðĖ
ð) ðĢâ = ðĶ 2 ðĖ + (2ðĨðĶ + ð§ 2 )ðĖ + 2ðĶð§ðĖ
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)
ââ are vector functions):
Prove the following product rules (f is a scalar function; ðīâ, ðĩ
ð) ∇ â (ððīâ) = ð(∇ â ðīâ) + ðīâ â (∇ð)
ââ ) = ðĩ
ââ â (∇ &times; âAâ) − âAâ â (∇ &times; ðĩ
ââ )
ð) ∇ â (ðīâ &times; ðĩ
Problem 4 (Mark 10/30)
The current density in a certain region is given (in spherical coordinates) by:
t
âJ = 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)
ðĄ
A sphere of radius R has decreasing uniform charge density ð(ðĄ) = ð0 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.
```