Tutorial questions for basic magnetostatics

1. Anup Kumar Singh (8953976344)
DEPARTMENT OF PHYSICS
MOTILAL NEHRU NATIONAL INSTITUTE OF TECHNOLOGY ALLAHABAD
B.Tech. 1st year [EVEN SEMESTER (2018-19)]
Question 1: What is the magnetic field B produced by a steady line current I at a point having distance r from
it? With the help of same show that magnetic fields do not diverge.
Question 2: (a) A Current density J, in a certain region, is described as:
𝐽 = 5𝑒−2𝑟
𝑧̂
𝐴
𝑚2
for 0 < r < a
𝐽 = 0 , everywhere
Calculate the magnetic field intensity H for given current density.
(b) Show that magnetic field can not change energy of a particle.
(c) Three infinitely long current carrying conductors lie parallel to each other
at 50 cm and carrying a uniform current of 100 Amp in each.
Direction of currents in first and second conductors are same while
reversed in the third one. What is the force acting on these
conductors?
Question 3: A square loop placed as shown in Figure has 2-m sides and carries a
current I1=5 A. If a straight, long conductor carrying a current I2=10 A
is introduced and placed just above the midpoints of two of the loop’s
sides, determine the net force acting on the loop.
Question 4: In a cylindrical coordinate system, a 2-m-long straight wire carrying a
current of 5 A in the positive z-direction is located at r = 4 cm, φ = π/2,
and -1m ≤ z ≤ 1m.
(a) If B = 0.2 cosφ ̑i (T), what is the magnetic force acting on the wire?
(b)How much work is required to rotate the wire once about the z-axis in the negative φ-direction
(while maintaining r = 4 cm)?
(c) At what angleφis the force a maximum?
Question 5: A thick slab extending from z = -a to z = a carries a uniform volume current J =J ̑i . Find the
magnetic field both inside and outside the slab.
Question 6: Suppose that the magnetic field in some region has the form 𝑩 = 𝑘𝑧 𝒙̂ (where k is a constant).
Find the force on a square loop (side a), lying in the y-z plane and centered at the origin, if it
carries a current I, flowing counter clockwise, when you look down the x axis.
Question 7: Find the magnetic field at point P on the axis of a tightly
wound solenoid (helical coil) consisting of n turns per unit
length wrapped around a cylindrical tube of radius a and
carrying current I.
Question 8: A large parallel-plate capacitor with uniform surface charge
σ on the upper plate and -σ on the lower is moving with a constant speed v,
(a) Find the magnetic field between the plates and above and below them.
(b) Find the magnetic force per unit area on the upper plate, including its direction.
(c) At what speed v would the magnetic force balance the electrical force?
Question 9: Find the magnetic field at a point z > R on the axis of (a) the rotating disk and (b) the rotating
sphere,
Question 10: Two infinite conducting planes at z= 0 and z= d carry currents in opposite directions with surface
current density in opposite directions. Calculate the magnetic field ± K𝑖̂ everywhere in space.
Question 11: A long cylindrical wire has a current density flowing in the direction of its length whose density
is J = J0r, where r is the distance from the cylinder’s axis. Find the magnetic field both inside and
outside the cylinder.
50 cm
50 cm

2. Anup Kumar Singh (8953976344)
DEPARTMENT OF PHYSICS
MOTILAL NEHRU NATIONAL INSTITUTE OF TECHNOLOGY ALLAHABAD
B.Tech. 1st year [EVEN SEMESTER (2018-19)]
Question 12: The current density along a long cylindrical wire of radius a is given by 𝐽 = 𝐽0 𝑒−
𝑟
𝑎 𝑘̂, where r is
the distance from the axis of the cylinder. Use Ampere’s law to find the magnetic field both inside
and outside the cylinder.
Question 13: Two infinite conducting sheets lying in x-z and y-z planes intersect at right angles along the z
axis. On each plane a surface current 𝐾𝑧̂ flows. Find the magnetic field in each of the four
quadrants into which the space is divided by the planes.
Question 14: Find the magnetic vector potential of a finite segment of straight wire, carrying a current I.
Question 15: What current density would produce the vector potential, A = k𝑗̂ (where k is a constant), in
cylindrical coordinates?
Question 16: If B is uniform, show that A(r) = -(r x B)/2 works. That is, check that ∇ ∙ 𝑨 = 0 and ∇ × 𝑨 =
𝑩. Is this result unique, or are there other functions with the same divergence and curl?
Question 15: Show that the magnetic field of a dipole can be written in coordinate-free form:
𝑩 𝒅𝒊𝒑(𝒓) =
𝝁 𝟎
𝟒𝝅
𝟏
𝒓 𝟑 [𝟑(𝒎 ∙ 𝒓̂)𝒓̂ − 𝒎]
Question 16: A circular loop of wire, with radius R, lies in the xy plane, centered at the origin and carries a
current I running counter clockwise as viewed from the positive z axis. (a) What is its magnetic
dipole moment? (b) What is the (approximate) magnetic field at points far from the origin?
Question 17: Find the exact magnetic field a distance z above the centre of a square loop of side w, carrying
a current I. Verify that it reduces to the field of a dipole, with the appropriate dipole moment, when
z >> w.
Question 18: Obtain the vector magnetic potential A in the region surrounding an infinitely long, straight,
filamentary current I.
Question 19: Two identical circular current loops of radius r = 3m and I = 20 Amp are in parallel planes,
separated on their common axis by 10 m. Find H at a point midway between two loops.
Question 20: Compute the total magnetic flux ɸ crossing z = 0 plane in cylindrical coordinates for r ≤ 5x10-2
m if 𝐵 =
0.2
𝑟
𝑠𝑖𝑛2
𝜑 𝑧̂.
Question 21: A co-axial conductor with an inner conductor of radius a and an outer conductor of inner and
outer radii b and c, respectively, carries current I in inner conductor. Find the magnetic flux per
unit length, crossing a plane ɸ = constant, between the conductors.
Question 22: Given the vector magnetic potential within a cylindrical conductor of radius a is 𝐴 = −
𝜇0 𝐼𝑟2
4𝜋𝑎2
𝑥̂.
Find the corresponding H.
Referred Books:
1. Introduction to Electrodynamics – David J Griffiths
2. Electromagnetics (Schaum’s outline) – Joseph A Edminister