Using Rutherford model of atom,derive the expression for the total energy of the electron in hydrogen atom.What is the significance of total negative energy possessed by the electron?

Dear student,

he kinetic energy of an object in orbit can easily be found from the following equations:
The centripetal force on a satellite of mass m moving at velocity v in an orbit of radius r = mv2/r 

But this is equal to the gravitational force (F) between the planet (mass M) and the satellite: 

F =GMm/r2 and so mv2 = GMm/r

But kinetic energy = ½mv2 and so:
the kinetic energy of the satellite = ½ GMm/r

  Kinetic energy in orbit = ½ mv2 = + ½GMm/r

All satellites have to be given a tangential velocity (v) to maintain their orbit position and this process is called orbit injection. 

 

The total energy of a satellite in orbit = -GMm/2r  

However, the total energy INPUT required to put a satellite into an orbit of radius r around a planet of mass M and radius R is, therefore, the sum of the gravitational potential energy (GMm[1/R-1/r]) and the kinetic energy of the satellite ( ½GMm/r).


Energy of launch = GMm[1/R – 1/2r] 

 

Now the significance of this negative energy is, when a satellite is revolving around the earth then due to friction with the atmosphere of the earth or due to the collisions with the meteors in the space the total energy(−GMm2r)-GMm2r reduces so the (GMm2r)GMm2r is increasing .Resultantly the value of r is decreasing.So the satellite is coming close to the earth.So from the surface of the earth as the distance of the satellite decreases the orbital velocity and kinetic energy increase though the total energy of the satellite decreases.

Regards

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