Magnetic induction

1.Explain the physical quantity “stationary magnetic field” and its characteristics.

Solution:

A stationary magnetic field is a time-invariant, constant magnetic field. It arises around a stationary permanent magnet or around a stationary conductor carrying a constant electric current. It is represented by magnetic induction lines (MIL).

1.) Magnetic Induction B

$B = \frac{F_m}{I \cdot l \cdot \sin \alpha} \quad \text{and if } \alpha = 90^\circ \Rightarrow B = \frac{F_m}{I \cdot l}$

$[B] = N \cdot A^{-1} \cdot m^{-1} = 1T = \text{tesla}$

Where:

$F_{m}$ = force with which the magnetic field acts on a conductor carrying current
$I$ = electric current passing through the conductor
$l$ = active length of the conductor
$\alpha$ = angle between the conductor and the magnetic induction vector

2.) Permeability of the Medium μ

\mu = \mu_0 \mu_r \quad \text{where} \quad \mu_0 = 4\pi \cdot 10^{-7} N \cdot A^{-2}

\mu_r = \text{relative permeability}

Medium	μᵣ
air	1.0
water	0.999
iron	15,000
Fe + Ni + Cu	100,000

3.) Ampère’s Law

F_m = B \cdot I \cdot l \cdot \sin \alpha

4.) Some Cases of Magnetic Induction

a) At a distance r from a straight conductor:

B = \frac{\mu I}{2\pi r}

b) At the center of a circular loop with radius r:

B = \frac{\mu I}{2r}

c) Inside a long cylindrical coil (solenoid) with N turns per unit length l:

B = \frac{N \mu I}{l}

2.A conductor placed in a homogeneous stationary magnetic field perpendicular to the direction of induction lines with an active length of 5 cm carries a current of 25 A. The magnetic field exerts a force of 50 mN on the conductor. Determine the magnitude of the magnetic induction.

Solution:

Analysis:

The magnetic induction of the field is B = 40 mT

3.A straight conductor of length 10 cm carrying a current of 2 A is acted upon by a force of 20 mN in a homogeneous magnetic field with magnetic induction of 0.2 T. Determine the angle between the conductor and the direction of the magnetic induction lines.

Solution:

Analysis:

The angle between the conductor and the direction of the magnetic induction lines is 30⁰.

4.A conductor carrying a current of 1 A with a cross-sectional area of 1 mm² moves in a homogeneous magnetic field with constant acceleration of 2 m·s^–2 perpendicular to the direction of induction lines. The density of the material of the conductor is 2500 kg·m³. Determine the magnitude of the magnetic induction.

Solution:

Analysis:

The magnitude of magnetic induction is B = 5 mT.

5.A straight conductor forming an angle α₁ = 90⁰ with the induction lines of a homogeneous magnetic field experiences a force greater by 0.134 N than when it formed an angle α₂ = 60⁰. Determine the magnetic induction of the field if the effective length of the conductor is l = 12.5 cm and it carries a current of 10 A.

Solution:

Analysis:

The magnetic induction of the field is B = 0.8 T.

6.What electric current flows through a very long straight conductor if the magnetic induction at a distance of 20 cm from the conductor is 20 μT?

Solution:

Analysis:

The very long straight conductor carries a current of I = 20 A.

7.What magnetic induction does the magnetic field of a solenoid of length 20 cm with 400 turns have, if the current passing through the solenoid is 5 A? What is the diameter of the wire from which the solenoid is wound, if the single-layer winding has turns placed directly next to each other?

Solution:

Analysis:

The magnetic field of the solenoid has a magnetic induction B = 4π·10^-3 T.
The diameter of the wire is d = 0.1 mm.

8.A straight conductor with current I formed an angle α with the induction lines of a homogeneous magnetic field. After changing its position, the conductor forms an angle α + 18⁰ with the induction lines. The magnitude of the force acting on the conductor increased by 20%. Determine the angle α.

Solution:

Analysis:

The angle α = 51⁰.

9.A conductor of length l = 80 cm and mass m = 0.16 kg is suspended on two thin hanging wires and placed in a homogeneous magnetic field whose induction lines are directed vertically upward. Determine the angle α by which the hanging wires deviate from the vertical direction if a current I = 2 A flows through the conductor and B = 1 T.

Solution:

The wires deviate from the vertical direction by an angle α = 45⁰.

10.A cylindrical coil without a core has the form of a long solenoid wound tightly with insulated wire so that adjacent turns touch each other. A current I = 0.5 A flows through the coil and inside it the magnetic induction has the value
B = 3.15 mT. Determine the diameter d of the wire from which the coil winding is made.

Solution:

The diameter of the wire is about d = 0.2 mm.

11.What is the magnitude of the force acting on a conductor of length 0.2 m placed in a homogeneous magnetic field, forming an angle α = 30⁰ with the direction of the magnetic induction of the magnetic field? The magnetic induction is
B = 0.1 T, and a current I = 10 A flows through the conductor.

Solution:

The force acting on the conductor is F_m = 0.1 N.

12.In a homogeneous magnetic field with magnetic induction B = 2 T, a force of magnitude F = 1.2 N acts on a conductor of length l = 20 cm, perpendicular to the induction lines. Determine the magnitude of the current in the conductor.

Solution:

The current flowing through the conductor is I = 3 A.

13.A force F_m = 1.8 N acts on the rotor winding conductor of an electric motor through which 20 A flows. Determine the magnetic induction at the location where the conductor is situated. The length of the conductor is 15 cm.

Solution:

The magnetic induction is B = 0.6 T.

14.A straight conductor of length l = 50 cm through which a current I = 2 A flows is acted upon in a magnetic field with magnetic induction B = 0.1 T by a force F_m = 0.1 N. Determine the angle α that the conductor forms with the direction of the magnetic induction lines.

Solution:

The conductor is perpendicular to the direction of the magnetic induction lines.

15.Calculate the magnitude of the magnetic induction of the magnetic field in vacuum at a distance of 2 cm from a very long conductor carrying a current of 5 A.

Solution:

The magnitude of the magnetic induction of the magnetic field is B = 5·10^–5 T.

16.A straight conductor of length 0.4 m carrying a current of 21 A lies in a homogeneous magnetic field with magnetic induction 1.2 T in a position perpendicular to the induction lines. Calculate the work that must be done to move the conductor along a path of 25 cm in a direction perpendicular to the induction lines.

Solution:

The work required to move the conductor is W = 2.52 J.

17.In a homogeneous magnetic field, a straight conductor is inserted perpendicular to the induction lines, parallel to the ground. The mass of the conductor is 0.75 kg, its length is 75 cm, and a current of 0.5 A flows through it. How large must the magnetic induction be for the conductor not to fall but to levitate?

Solution:

The magnetic induction must be B = 20 T.

18.We need to wind a long cylindrical coil so that the magnetic field in the center of its cavity has magnetic induction not less than B = 8.2·10^-3 T when a current I = 4.3 A flows through the coil. What should be the turn density of the coil?

Solution:

The turn density of the coil must be 1.5·10³ m^–1.

19.A circular loop surrounded by air has a radius of 1 cm. A current of 2 A flows through the loop. Calculate the magnitude of the magnetic induction at the center of the loop.

Solution:

The magnitude of the magnetic induction at the center of the loop is B = 0.13 mT.

20.A historical model representing the hydrogen atom consisted of a stationary nucleus and an electron orbiting around it on a circular path with radius r = 0.53·10^–10 m and period T = 1.5·10^–16 s. Calculate the magnitude of the magnetic induction of the magnetic field excited by the motion of the electron along the circular path.

Solution:

The magnetic induction excited by the electron's motion in the hydrogen atom is approximately B = 13 T.

21.What is the magnitude of the magnetic field intensity H from which a conductor of length l = 50 cm carrying a current of 10 A, positioned perpendicular to the induction lines, was pushed out by a force of 1 N?

Solution:

The magnitude of the magnetic field intensity is H = 1.6·10⁵ A·m^–1.

Physics

Magnetic induction

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