Deck 5: Static Magnetic Fields

Question 1

Refer to Problem P.5-4 and Fig. 5-25. Find the magnetic flux density B 2. at point P 2 ( w + d 2 , 0).

Answer 1

Write the expression for magnetic flux density at a point located at a distance
from the wire in the bisecting plane:
When
Consider that a small element of conducting wire
along x -axis leads to corresponding change in the magnetic flux density
, as shown in Figure 1.
Figure 1
The vector form of the magnetic field due to small element of conducting wire
is:
If we align the conductor along the z-axis, the magnetic flux density
will be
directed in negative direction. Thus,
…… (1)
Integrate equation (1) with limits from
to
on both sides,
Therefore magnetic flux density
at point
is
.

Question 2

What is the relation between vector magnetic potential A and the magnetic flux through a given area

Answer 2

The relation between the vector magnetic potential
and magnetic flux
through a given area is stated by an equation:
Clearly from this expression, the circulation of the vector magnetic potential
around in any closed area is equal to the total magnetic flux which is passing through the enclosed area path.

Question 3

Does the magnetic field intensity due to a current distribution depend on the properties of the medium Does the magnetic flux density

Answer 3

The equation of the magnetic flux density(B) is,
(Wb/m²)
The magnetic flux density is
times the magnetic field intensity.
The equation of the magnetic field intensity (H) is,
(A)
The magnetic field intensity due to current distribution depends on the property of the medium because the magnetic field intensity is mainly based on the distribution of the current in the circuit.
The magnetic flux density is
times the magnetic field intensity,
As, the magnetic field intensity due to current distribution depends on the medium, hence magnetic flux density also depends on the medium.

Question 4

Write the formula expressing the torque on a current-carrying circuit in a magnetic field.

Question 5

Which postulate of magnetostatics denies the existence of isolated magnetic charges

Question 6

A very large slab of material of thickness d lies perpendicularly to a uniform magnetic field of intensity H 0 = a z H 0. Ignoring edge effect, determine the magnetic field intensity in the slab:
a) if the slab material has a permeability , and
b) if the slab is a permanent magnet having a magnetization vector M i = a z M i.

Question 7

The cross section of a long thin metal strip and a parallel wire is shown in Fig. 5-30. Equal and opposite currents I flow in the conductors. Find the force per unit length on the conductors.
Fig. Cross section of parallel strip and wire conductor (Problem P.5-19)

Question 8

Explain the principle of operation of d-c motors.

Question 9

A cylindrical magnet of radius 5 (cm) and length 12 (cm) has an axial magnetization a z 130 (A/cm). Find B at
a) the center of the top face,
b) the center of the bottom face, and
c) the center of the magnet.

Question 10

State Biot-Savart law.

Question 11

Define diamagnetic, paramagnetic, and ferromagnetic materials.

Question 12

What is the relation between the force and the stored magnetic energy in a system of current-carrying circuits under the condition of constant flux linkages

Question 13

An infinitely long, very thin cylindrical conducting tube of radius b carries a uniform surface current J s = a z J s (A/m). Find B everywhere.

Question 14

A current I flows in the inner conductor of an infinitely long coaxial line and returns via the outer conductor. The radius of the inner conductor is a and the inner and outer radii of the outer conductor are b and c, respectively Find the magnetic flux density B for all regions and plot |B| versus r.

Question 15

A circular rod of magnetic material with permeability is inserted coaxially in a very long air solenoid. The radius of the rod, a, is less than the inner radius, b, of the solenoid. The solenoid's winding has n turns per unit length and carries a current I.
a) Find the values of B, H, and M inside the solenoid for r and for a
b) What are the equivalent magnetization current densities J mv and J ms for the magnetized rod

Question 16

The bar AA in Fig. 5-31 serves as a conducting path (such as the blade of a circuit breaker) for the current I in two very long parallel lines. The lines have a radius b and are spaced at a distance d apart. Find the direction and the magnitude of the magnetic force on the bar.
Fig Force on conducting bar (Problem P5-20)

Question 17

A point charge Q with a velocity u = a x u 0 enters a region having a uniform magnetic field B = a x B x + a y B y + a z B z. What E field should exist in the region so that the charge proceeds without a change of velocity

Question 18

State the law of conservation of magnetic flux.

Question 19

What is a magnetic dipole Define magnetic dipole moment. What is its SI unit

Question 20

What is a hysteresis loop

Question 21

What is the expression for the force on a test charge q that moves with velociy u in a magnetic field of flux density B

Question 22

The magnetic field intensity H 1 in medium 1 having a permeability 1 makes an angle 1 with the normal at an interface with medium 2 having a permeability 2. Find the relation between the angle 2 (that H 2 makes with the normal) and 1.

Question 23

A ferromagnetic sphere of radius b is magnetized uniformly with a magnetization M = a z M 0.
a) Determine the equivalent magnetization current densities J mv and J ms.
b) Determine the magnetic flux density at the center of the sphere.

Question 24

A d-c current I = 10 (A) flows in a triangular loop in the xy -plane as in Fig. 5-32. Assuming a uniform magnetic flux density B = a y 6(mT) in the region, find the forces and torque on the loop. The dimensions are in (cm).
Fig. A triangular loop in a uniform magnetic field (Problem P.5-21)

Question 25

Find the magnetic flux density inside a very long cylindrical solenoid with an air core having n turns per meter and carrying a current I.

Question 26

A thin conducting wire of length 3 w forms a planar equilateral triangle A direct current I flows in the wire. Find the magnetic flux density at the center of the triangle.

Question 27

Define magnetization vector. What is its SI unit

Question 28

Define remanent flux density and coercive field intensity.

Question 29

Find the total magnetic flux through a circular toroid with a rectangular cross section of height h. The inner and outer radii of the toroid are a and b, respectively. A current I flows in N turns of closely wound wire around the toroid. Determine the percentage of error if the flux is found by multiplying the cross-sectional area by the flux density at the mean radius. What is the error if b/a = 5

Question 30

State Ampère's circuital law.

Question 31

Consider a plane boundary ( y = 0) between air (region 1, r 1 = 1) and iron (region 2, r 2 = 5000).
a) Assuming B 1 = a x 2 a y 10(mT), find B 2 and the angle that B 2 makes with the interface.
b) Assuming B 2 = a x 10 + a y 2 (mT), find B 1 and the angle that B 1 makes with the normal to the interface.

Question 32

A small circular turn of wire of radius r 1 that carries a steady current I 1 is placed at the center of a much larger turn of wire of radius r 2 ( r 2 r 1 that carries a steady current I 2 in the same direction. The angle between the normals of the two circuits is and the small circular wire is free to turn about its diameter. Determine the magnitude and the direction of the torque on the small circular wire.

Question 33

Verify that tesla (T), the unit for magnetic flux density, is the same as Volt second per square meter (V · s/m 2 ).

Question 34

Express the stored magnetic energy in terms of flux linkage and current I in an inductor having an inductance L

Question 35

What is meant by "equivalent magnetization current densities" What are the SI units for × M and M × a n

Question 36

Discuss the difference between soft and hard ferromagnetic materials.

Question 37

Verify Eq. (5-17) in Cartesian coordinates.

Question 38

Refer to Fig. 5-26; Determine the magnetic flux density at a point P on the axis of a solenoid with radius b and length L, and with a current I in its N turns of closely wound coil. Show that the result reduces to that given in Eq (5-82) when L approaches infinity. Hint: Use Eq. (5-37).

Question 39

Determine the self-inductance of a toroidal coil of N turns of wife wound on an air frame with mean radius r o and a circular cross section of radius b. Obtain an approximate expression assuming b r o.

Question 40

What is curie temperature

Question 41

direct current I flows in a straight filamentary conductor P 1 P 2.
a) Prove that B at a point P , whose location is specified by the perpendicular distance r and the two angles 1 and 2 shown in Fig 5-24 is
(5-135).
b) Verify that Eq. (5-135) reduces to Eq. (5-35) when the wire is infinitely long.

Question 42

In what manner does the B -field of an infinitely long straight filament carrying a direct current I vary with distance

Question 43

Define magnetic field intensity vector. What is its SI unit

Question 44

What are the boundary conditions for magnetostatic fields at an interface between two different magnetic media

Question 45

Write Lorentz's force equation.

Question 46

A current I flows in the N -turn toroidal coil in Fig. 5-15.
a) Obtain an expression for the stored magnetic energy.
b) Use Eq. (5-109) to determine its self-inductance and check your result with Eq. (5-81).

Question 47

Determine the mutual inductance between a very long, straight Wire and a conducting equilateral triangular loop, as shown in Fig. 5-27.

Question 48

Define (a) the mutual inductance between two circuits, and (b) the self inductance of a single coil.

Question 49

An 8 (cm) × 6 (cm) rectangular conducting loop lies in the xy -plane. A direct current of 5 (A) flows in a clockwise direction viewing from the top. Find B at the center of the loop.

Question 50

A direct current I flows in an infinitely long wire of a radius 2( mm ) along the z -axis.
a) Obtain the vector magnetic potential A at r 2(mm) from the expression of B in Eq. (5-12). Choose the reference zero potential at wire surface.
b) If I = 10(A), determine from A the total amount of magnetic flux passing through a square loop specified by z = ±0.3(m) and y = 0.1 (m) and 0.7 (m).

Question 51

Write the two fundamental governing differential equations magnetostatics.

Question 52

What is meant by the internal inductance of a conductor

Question 53

A current I flows lengthwise in a very long, thin conducting sheet of width w, as shown in Fig. 5-25. Assuming that the current flows into the paper, determine the magnetic flux density B 1 at point P 1 (0, d ).

Question 54

Define vector magnetic potential A. What is its SI unit

Question 55

Find the mutual inductance between two coplanar rectangular loops with parallel sides, as shown in Fig. 5-28. Assume that h 1 h 2 ( h 2 w 2 d ).

Question 56

Write the expression for the stored magnetic energy of two coupled current carrying loops.

Question 57

What are the two fundamental postulates of magnetostatics

Question 58

Assume that a current I 2 flows in the rectangular loop in Fig. 5-18 in the clockwise direction. Determine the net force on the loop.

Question 59

Define magnetic susceptibility and relative permeability. What are their units

Question 60

Write the expression for stored magnetic energy in terms of field quantities.

Question 61

Refer to Fig. 5-6. Find B :
a) at the center of a circular loop of radius 5 (cm) carrying a direct current 2(A) and
b) at the center of a semi-circular loop of radius 8 (cm) carrying a direct current 4(A).

Question 62

A d-c surface current with density a x J s 0 flows in an infinite conducting sheet coinciding with the xy -plane.
a) Determine the magnetic flux density B at (0, 0, z) and at (0, 0, z ).
b) Find the vector magnetic potential A at (0, 0, z ) from B. Choose the reference zero potential at an arbitrary point z = z 0.

Question 63

Calculate the force per unit length on each of three equidistant, infinitely long, parallel wires 10 (cm) apart, each carrying a current of 25(A) in the same direction. A cross section of the arrangement is shown in Fig. 5.29 Specify the direction of the force.

Question 64

Give the integral expression for the force on a closed circuit that carries. a current I in a magnetic field B.