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 bbe 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→ ∞.
•2 A. Find the magnetic vector potential ~
Aat an arbitrary point ~r =xˆ
i+yˆ
j+zˆ
kdue to a
circular loop carrying a steady current I. Assume the loop be on the x−yplane, its center be at
the origin, and its radius be b.
B. Determine the magnetic field ( ~
B=Bx
~
i+By~
j+Bz~
k) from the above vector potential ~
A.
C. Show that for a small loop (br), the vector potential takes the form ~
A(~r)'µ0
4π
~m×~r
r3.
•3 A. Show that the magnitude of a magnetic field ~
B(~r)of a static case can have a local minimum,
but never a local maximum, in a source free region (where ∇ · ~
B= 0 and ∇ × ~
B= 0)1.
B. Show that I
2Hc[d~r0×(~r0· ∇)~
B(~r)−~r0×(d~r0· ∇)~
B(~r)] = ( ~m × ∇)×~
B(~r).
•4 A. Let ~
B(~r)be the magnetic field produced by a steady current distribution of the current
density J(~r)that lies entirely inside a spherical volume Vof radius R. Show that the magnetic
moment due to the current distribution is ~m =3
2µ0RV~
B(~r)d3~r.
B. Using the above result show that the magnetic field at ~r due to a point magnetic dipole of the
dipole moment ~m located at ~r = 0 is ~
B(~r) = µ0
4π3(ˆr·~m)ˆr−~m
r3+8π
3~mδ3(~r)∀~r.
1Refer to Thomson’s theorem for magnetostatics.