Uploaded by Rajdeep Kaur

Assignment 1 week2

Assignment 1, Autumn 2020
Set on Thu 12th March, due on Thu 19th March.
Problem 1 (2/30)
Calculate the divergence and the curl of the following vector functions:
𝑎) ð‘Ģ⃗ = ð‘Ĩ 2 𝑖Ė‚ + 3ð‘Ĩ𝑧 2 𝑗Ė‚ − 2ð‘Ĩ𝑧𝑘Ė‚
𝑏) ð‘Ģ⃗ = ð‘Ĩð‘Ķ𝑖Ė‚ + 2ð‘Ķ𝑧𝑗Ė‚ + 3𝑧ð‘Ĩ𝑘Ė‚
𝑐) ð‘Ģ⃗ = ð‘Ķ 2 𝑖Ė‚ + (2ð‘Ĩð‘Ķ + 𝑧 2 )𝑗Ė‚ + 2ð‘Ķ𝑧𝑘Ė‚
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.
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.
Problem 3 (3/30)
⃗⃗ are vector functions):
Prove the following product rules (f is a scalar function; ðī⃗, ðĩ
𝑎) ∇ ∙ (𝑓ðī⃗) = 𝑓(∇ ∙ ðī⃗) + ðī⃗ ∙ (∇𝑓)
⃗⃗ ) = ðĩ
⃗⃗ ∙ (∇ × âƒ—A⃗) − ⃗A⃗ ∙ (∇ × ðĩ
⃗⃗ )
𝑏) ∇ ∙ (ðī⃗ × ðĩ
Problem 4 (Mark 10/30)
The current density in a certain region is given (in spherical coordinates) by:
⃗J = J0
e 𝜏
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)
A sphere of radius R has decreasing uniform charge density 𝜌(ð‘Ą) = 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