Obtain an expression for energy of an electron in Bohr orbit. Hence obtain the expression for its binding energy.
A rectangular coil having 100 turns each of length 1 cm and breadth `0.5 cm` is suspended in radial magnetic induction `0.002` T. The torsional constant of suspension fiber is `2 xx 10^(-8) Nm/degree. Calculate the current sensitivity of a moving coil galvanometer.

Explanation

Energy of electron in Bohr orbit
According to Bohr's postulate, electron revolves around nucleus in circular orbit. As this motion has two energies.
`:.` Total energy possessed by an electron is given by
Total energy`(E ) = P.E. +K.E."...."(i)`
Consider the electron revolving in the `n^(th)` orbit in hydrogen nucleus . Let `m` and -e be the mass and charge on the electron, r be the radius of orbit and v be linear speed of electron.
The electric potential at a distance'r' from a charg `+e` is given by :
`V = 1/(4 piepsi_(0)) .(e )/r`
P.E. = Potential `xx` charge of electron
P.E. `= ((1)/(4piepsi_(0)). (e)/(r ))(-e)`
P.E. `= (-e^(2))/(4piepsi_(0)r)`
By Bohr's `1^(st)` postulate,
`(mv^(2))/(r) = (1)/(4piepsi_(0)) .(e^(2))/(r^(2))`
`1/2 mv^(2) = (e^(2))/(8piepsi_(0)r)`
`:.` K.E. `= (e^(2))/(8piepsi_(0)r)`
`:.` Substituting the values K.E. and P.E. in equation (i),
`E = (-e^(2))/(4piepsi_(0)r) + (e^(2))/(8piepsi_(0)r)`
`E = (e^(2))/(8repsi_(0)r)`
`E = (-me^(4))/(8epsi_(0)^(2)h^(2)n^(2))`
Energy of electron in `n^(th)` orbit is -
`E_(n) = -((me^(4))/(8epsi_(0)^(2)h^(2))).(1)/(n^(2))`
The binding enegy of electron is the minimum energy required to make it free from the nucleus.
i.e, `E_(n) = -((me^(4))/(8epsi_(0)^(2)h^(2))).1/(n^(2))`
`:.` It must receive energy `= + ((me^(4))/(8epsi_(0)^(2)h^(2)). (1)/(n^(2))`
to make `E_(oo) = 0`
Numerical :
Given : `n = 100, l = 1.0 cm = 0.01m`
`b = 0.5 cm = 0.005 m, B = 0.002T`
`S_(i) = ?`
As we know, `S_(i) = (nBA)/(c)`
`S_(i) = (100 xx 0.002 xx 0.01 xx 0.005)/(2 xx10^(-8))`
`S_(i) = 0.5 xx 10^(3)` div/A
`S_(i) = 500` div/A