Educación familiar y desarrollo de valores

What is the magnetic flux density at a distance of 0.1 m from a wire that carries a current of 0.25 A?

Solution

Substituting the numerical values into Eq. (2.53) yields

B=4π×10⁻⁷ Wb A·m

2π ×0.25 A

0.1 m

=5×10⁻⁷Wb

m² =5×10⁻⁷T.

In comparison, the magnetic flux density at the equator is about 3×10⁻⁵T.

Any region in which there is a magnetic flux is called a magnetic field, and the field intensity (or field strength) is directly proportional to the magnetic flux density.

Since magnetic flux has direction as well as magnitude, magnetic field strength is a vector quantity. Michael Faraday, a Scottish experimental physicist, found in 1831 that electricity could be generated from a magnetic field and that electricity and magnetism were related.

In 1873, James Clerk Maxwell, a Scottish physicist, published a general theory that related the experimental findings of Oersted and Faraday. He deduced the quanti-tative relationships among moving charged particles, magnetic fields, and electric fields and formulated an electromagnetic theory that described quantitatively the interaction between moving electric charges and magnetic fields. His theory states that a changing electric field is always associated with a changing magnetic field and a changing magnetic field is always associated with a changing electric field. He showed that an oscillating electric circuit will create an electromagnetic wave as the flowing electrons in the circuit undergo continuous acceleration and deceleration as the current oscillates. When this happens, some of the energy of the charged particle is radiated as electromagnetic radiation. This phenomenon is the basis of radio transmission, in which electrons are accelerated up and down an antenna that is connected to an oscillator. The electromagnetic wave thus generated has a fre-quency equal to that of the oscillator and a velocity of 3×10⁸ m/s in free space.

The waves consist of oscillating electric and magnetic fields that are perpendicular to each other and are mutually perpendicular to the direction of propagation of the wave (Fig. 2-16). The energy carried by the waves depends on the strength of the associated electric and magnetic fields.

In dealing with electromagnetic waves, it is more convenient to describe the mag-netic component in terms of magmag-netic field strength H rather than in terms of magnetic flux density B. Hcan be considered as the magnetizing force that leads to the magnetic field of flux density B. In free space, magnetic field strength His related to magnetic flux densityBby

H= BWb m² μ0

Wb A·m

= B μ0

A

m. (2.54)

Since B has dimensions of Wb/m² andμ0 has dimensions of Wb/(A·m), the di-mensions of magnetic field strength,H, are A/m. In any medium other than air, the magnetic flux density is equal to the product of the magnetic field strength Hand the magnetic permeability of that medium, μ. Safety standards for magnetic field strength are listed in terms of amperes per meter. Thus, in Example 2.15, the mag-netic flux density of 5×10⁻⁷T corresponds to a magnetic field strength of 0.4 A/m.

H

c

H_o

ε ε

Figure 2-16. Schematic representation of an electromagnetic wave. The electric intensityεand the magnetic intensityHare at right angles to each other, and the two are mutually perpendicular to the direction of propagation of the wave. The velocity of propagation isc, the electric intensity isε=ε0

sin 2π /λ(x₋ct), and the magnetic intensity isH₌H_{0sin 2π /λ}(x₋ct). The plane of polarization is the plane containing the electric field vector.

The relationship between the peak magnetic and electric field intensitiesH0and ε0depends on the magnetic permeabilityμand the electrical permittivity∈of the medium through which the electromagnetic wave is propagating. This relationship is given by

H0

√μ=ε0

√∈, (2.55)

where∈is the permittivity of the medium. Permittivity is a measure of the capacity for storing energy in a medium that is in an electric field. The permittivity of free space is∈0=8.85×10⁻¹²C²/N· m², and the permittivity of any other medium is the product of the relative permittivity, k_∈ and the permittivity of free space, ∈0:

∈=k_∈× ∈0. The greater the value of∈, the greater is its interaction with theεfield and the greater is its ability to store energy. Permittivity is frequency dependent and generally decreases with increasing frequency. If the wave is traveling through free space, then

H0

√μ0=ε0

√∈0. (2.56)

Radio waves, microwaves (radar), infrared radiation, visible light, ultraviolet light, and X-rays are all electromagnetic radiations. They are qualitatively alike but differ in wavelength to form a continuous electromagnetic spectrum.

All these radiations are transmitted through the atmosphere (which may be con-sidered, for this purpose, as free space) at a speed very close to 3×10⁸m/s. Since the speed of all electromagnetic waves in free space is a constant, Eq. (2.47), when applied to electromagnetic waves in free space, becomes

c =3×10⁸m/s= f ×λ. (2.57)

Specifying either the frequency or wavelength of an electromagnetic wave in free space is equivalent to specifying both. Free-space wavelengths may range from 5× 10⁶m for 60-Hz electric waves through visible light (green light has a wavelength of about 500 nanometers, or nm, and a frequency of 6×10¹⁴Hz) to short-wavelength X- and gamma radiation (whose wavelengths are on the order of 10 nm or less).

There is no sharp cutoff in wavelength at either end of the spectrum nor is there a sharp dividing line between the various portions of the electromagnetic spectrum.

Each portion blends into the next, and the lines of demarcation, shown in Figure 2-17, are arbitrarily placed to show the approximate wavelength span of the regions of the electromagnetic spectrum.

Figure 2-17. The electromagnetic spectrum.

Generally, the speed of an electromagnetic wave in any medium depends on the electrical and magnetic properties of that medium: on the permittivity and the permeability. The speed is given by

v=

1

∈μ. (2.58)

In free space,

μ=μ0 =4π×10⁻⁷ N A

∈ = ∈0 = 1

4π×9×10⁹ N·m² C²

.

Substituting the values above into Eq. (2.58) gives the speedcof an electromagnetic wave in free space:

v=c =

⎛

⎜⎜

⎝

4π×9×10⁹ N·m² C² 4π×10⁻⁷N

A²

⎞

⎟⎟

⎠

1/2

=3×10⁸ m s .

An electromagnetic wave travels more slowly through a medium than through free space. The frequency of the electromagnetic wave is independent of the medium through which it travels. The wavelength is decreased, however, so that the relation-ship (Eq. [2.47]):

v= f ×λ

is maintained. The wavelength in a medium is given by λ=λ0

1

K_eK_m, (2.59)

where λ0 is the free-space wavelength and K_e and K_m are the relative permittivity (dielectric coefficient) and relative permeability of the medium, respectively. Since the relative permeability is≈1 for most biological materials and dielectrics, we can approximate Eq. (2.59) for most biological materials by

λ=λ0

1 Ke

. (2.60)

If the medium is alossy dielectric(a lossy dielectric is one that absorbs energy from an electromagnetic field; all biological media are lossy), the wavelength of our electro-magnetic wave within the medium is given by

λ=√λ0

K_e

1 2+1

2

1+ σ ω∈

_2−1/2

(2.61)

or, in terms of theloss tangent, λ=√λ0

Ke

!1 2+1

2 +

1+tan²δ

"₋1/2

, (2.62)

where

λ0 =free space wavelength, Ke =dielectric coefficient,

σ =conductivity, in reciprocal ohm meters, or siemens per meter, ω=2π×frequency,

∈ =Ke∈0, and

tan²δ = σ ω∈

2

=loss tangent. (2.63)