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Q9.20P
Expert-verified(a) Show that the skin depth in a poor conductor is (independent of frequency). Find the skin depth (in meters) for (pure) water. (Use the static values of and ; your answers will be valid, then, only at relatively low frequencies.)
(b) Show that the skin depth in a good conductor is (where λ is the wavelength in the conductor). Find the skin depth (in nanometers) for a typical metal in the visible range , assuming and . Why are metals opaque?
(c) Show that in a good conductor the magnetic field lags the electric field by , and find the ratio of their amplitudes. For a numerical example, use the “typical metal” in part (b).
Answer
(a) The skin depth in a poor conductor is and the skin depth for pure water is .
(b) The skin depth in a god conductor is and the skin depth for a typical metal is .
(c) The ratio of the real amplitude is .
Write the expression for the wavenumber for the poor conductor.
…… (1)
Here, K is the imaginary part.
Write the expression for the relativity permittivity.
…… (2)
Here, is the Permittivity of the free space.
(a)
Write the expression for imaginary part K.
Solve the above expression for the poor conductor.
Write the expression for the skin depth.
Substitute in the above expression.
Substitute the value of equation (2) in the above expression.
Take the value of from table 4.2 for pure water.
Substitute role="math" localid="1655714742974" and in the above expression.
Therefore, the skin depth in a poor conductor is and the skin depth for pure water is .
(b)
Write the expression for the wavenumber for a good conductor.
Here, K is the imaginary part.
Hence, the wavelength of the conductors will be,
Calculate imaginary part K for a good conductor.
Hence, the skip depth in a good conductor will be,
…… (3)
Substitute , and in equation (3).
As the fields do not penetrate far into the metal, they are opaque in nature.
Therefore, the skin depth in a god conductor is and the skin depth for a typical metal is .
(c)
For a good conductor:
Calculate the phase angle between the magnetic field and the dielectric field.
Take the ratio of the real amplitude of E and B.
Substitute , and in the above expression.
Therefore, the ratio of the real amplitude is .
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