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A gyromagnetic ratio of the electron revolving in a circular orbit of hydrogen atom is 8.8×1010CKg1. What is the mass of the electron? Given charge of the electron is 1.6×1019C.
A. 1×1029KgB. 0.11×1029KgC. 1.1×1029KgD111×1029Kg

Answer
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Hint: For a revolving electron in circular orbit, magnetic moment and angular momentum is associated with it. These values can be found out by applying general equations of uniform circular motion and charge-current relation of a particle. For calculating Gyromagnetic ratio, the magnetic moment of a particle is divided by the angular momentum associated with its revolution.

Formula used:
Magnetic moment of electron μ=evr2
Angular moment of electron L=mvr
Gyromagnetic ratio of electron γ=e2m

Complete step by step answer:
Let us consider an electron is revolving around in a circular orbit of radius r with velocity v. The mass of an electron is m and the charge on the electron is e, both of which are constant values.

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The time period T of the electron’s circular orbit is given as:

T=CircumferencevelocityT=2πrv

The current i due to the motion of the electron is the charge flowing through that time period,

i=Chargetimei=e2πrv=ev2πr

The current is generated in the opposite direction of the movement of the electron as the electron is a negatively charged particle.
The magnetic moment due to a current loop of currenti enclosing an area A is given by:
μ=iA
The magnetic moment of an electron:

μ=ev2πrπr2=ev2πr×πr2μ=evr2

Let’s divide and multiply the above equation by the mass of the electron,
μ=e(mvr)2m
We know that the angular momentum L of a particle is given by:
L=mvr
Or,
L=μ×2me
Or,
μ=(e2m)L

We are given that the gyromagnetic ratio of an electron revolving in a circular orbit of hydrogen atom is 8.8×1010CKg1 and we have to calculate the mass of revolving electron.
For a revolving electron, the magnitude of magnetic moment is given by,

μ=evr2
Where,
e is the charge on an electron
v is the velocity of revolving electron
r is the radius of the circular orbit
Angular momentum associated with revolving electron is given by,
L=mvr

Where,
m is the mass of electron
v is the velocity of revolving electron
r is the radius of circular orbit
Gyromagnetic ratio is expressed as the ratio of the magnetic moment of the particle to its angular momentum. It is symbolized by γ.
γ=μL
For revolving electron,
γ=evr2mvr=e2m

Given,
γ=8.8×1010CKg1e=1.6×1019C

Putting the above values inγ=e2m
We get,

8.8×1010=1.6×10192mm=1.6×10192×8.8×1010m=111×1029Kg

The mass of the revolving electron is 111×1029Kg
Hence, the correct option is D.

Note:
Students should remember that the electric current has a direction opposite to the direction of the flow of electrons. While calculating the magnetic moment of an electron, the magnetic moment vector is taken in direction opposite to the direction of path of revolving electron since electron carries a negative charge. While calculating the gyromagnetic ratio, we can take the magnitude of the magnetic moment and angular momentum of the electron.