UNIVERSITY OF HYDERABAD
School of Physics
M.Sc.-2022 & I.M.Sc-2019 Batches
Electromagnetic Theory-II
Problem Set: 1 (Minor Examination-1)
Due Date for Submission: April 04, 2023
Please be precise for answering correctly.
Total Marks/Grade: 20 Marks
N.B.: Symbols have their usual meaning. All the problems carry equal weightage.
Electrostatics:
• 1 A. Find the magnetic field at the center of a square loop which carries a steady current I. Let b
be the distance from the center to a side.
B. Find the magnetic field at the center of a regular n-sided polygon carrying a steady current I.
Again, et b be the distance from the center to a side.
C. Show that the magnetic field obtained for the polygon reduces to the result for a circular loop
for n → ∞.
~ at an arbitrary point ~
• 2 A. Find the magnetic vector potential A
r = xî + y ĵ + z k̂ due to a
circular loop carrying a steady current I. Assume the loop be on the x − y plane, its center be at
the origin, and its radius be b.
~ = Bx~i + By~j + Bz~
~
B. Determine the magnetic field (B
k) from the above vector potential A.
~ r) '
C. Show that for a small loop (b r), the vector potential takes the form A(~
µ0 m×~
~ r
.
4π r 3
~ r ) of a static case can have a local minimum,
• 3 A. Show that the magnitude of a magnetic field B(~
~ = 0 and ∇ × B
~ = 0)1 .
but never a local maximum, in a source free region (where ∇ · B
H
~ r) − ~
~ r )] = (m
~ r ).
B. Show that 2I c [d~
r 0 × (~
r 0 · ∇)B(~
r 0 × (d~
r 0 · ∇)B(~
~ × ∇) × B(~
~ r ) be the magnetic field produced by a steady current distribution of the current
• 4 A. Let B(~
density J (~
r ) that lies entirely inside a spherical volume V of radius R. Show that the magnetic
R
3
~ r )d3~
moment due to the current distribution is m
~ = 2µ
B(~
r.
V
0
B. Using the above result show that the magnetic field at ~
r due to a point magnetic
dipole of the
3(r̂·
m)r̂−
~
m
~
µ
8π
~ r) = 0
+ 3 mδ
dipole moment m
~ located at ~
r = 0 is B(~
~ 3 (~
r ) ∀~
r.
4π
r3
1
Refer to Thomson’s theorem for magnetostatics.