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.

Physics

Magnetic induction

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