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Question
Explain the origin of magnetism in material, hence find a magnetic moment of an electron revolving around the nucleus of an atom.
Solution
- Origin of magnetism in material:
a. Magnetism has its origin in the circulating charges in an atom.
b. Circulating electron is equivalent to a current loop and has a magnetic dipole moment.
c. An atom of any substance consists of a small massive positively charged nucleus surrounded by negatively charged electrons revolving in a circular orbit around the nucleus.
d. The magnetic moment is associated with the motion of an electron in its orbit and is termed an orbital magnetic moment. - Expression for magnetic dipole moment:
a. Consider an electron of mass me and charge e revolving in a circular orbit of radius r around the positive nucleus in the clockwise direction, leading to an anticlockwise current.
U.C.M of an electron around the nucleus
b. If the electron travels a distance 2πr in time T then, its orbital speed v = 2πr/T
c. The magnitude of circulating current is given by,
I = e`(1/"T")`
But, T = `(2pi"r")/"v"`
∴ I = e`(1/(2pi"r""/""v")) = "ev"/(2pi"r")`
d. The orbital magnetic moment associated with the orbital current loop is given by,
morb = IA = `"ev"/(2pi"r") xx pi"r"^2` [∵ Area of current loop, A = πr2]
∴ morb = `"evr"/2` ….(1)
e. The angular momentum of an electron due to its orbital motion is given by, L = mevr
f. Multiplying and dividing the R.H.S of equation (1) by me,
morb = `"e"/(2"m"_"e") xx "m"_"e""vr"`
∴ morb = `"eL"/(2"m"_"e")`
g. This equation shows that orbital magnetic moment is proportional to the angular momentum. But as the electron bears negative charge, the orbital magnetic moment and orbital angular momentum are in opposite directions and perpendicular to the plane of the orbit.
Using vector notation, `vec"m"_"orb" = -("e"/(2"m"_"e"))vec"L"`
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