How to calculate the time taken by an electron to complete one revolution in any given orbit.
OR
The actual question in the Qpaper was:
In an atom two electrons move around the nucleus in circular orbits of radii R and 4R. Calculate the ratio of the time taken by them to complete one revolution.
I couldnt solve it. :(
OR
The actual question in the Qpaper was:
In an atom two electrons move around the nucleus in circular orbits of radii R and 4R. Calculate the ratio of the time taken by them to complete one revolution.
I couldnt solve it. :(

Speed = Distance / time
So, time = Distance / speed
Distance is the circumference of the bohr orbit = 2 x pi x r
Now, as per BOHR's theory Fcoulomb = F centripetal for the electron
ie KZe^2 / r^2 = (m x v^2) / r OR KZe^2 / r = m x v^2
So, v = [ KZe^2 / m x r] ^1/2
time taken by first electron = (2 x pi x R) / [ KZe^2 / m x R] ^1/2
time taken by second electron = (2 x pi x 4R) / [ KZe^2 / m x 4R] ^1/2
So, t1 / t2 = 1/4 x [1/4]^1/2 = 1/8
So, time taken by first electron : time taken by Second electron = 1 : 8
So, time = Distance / speed
Distance is the circumference of the bohr orbit = 2 x pi x r
Now, as per BOHR's theory Fcoulomb = F centripetal for the electron
ie KZe^2 / r^2 = (m x v^2) / r OR KZe^2 / r = m x v^2
So, v = [ KZe^2 / m x r] ^1/2
time taken by first electron = (2 x pi x R) / [ KZe^2 / m x R] ^1/2
time taken by second electron = (2 x pi x 4R) / [ KZe^2 / m x 4R] ^1/2
So, t1 / t2 = 1/4 x [1/4]^1/2 = 1/8
So, time taken by first electron : time taken by Second electron = 1 : 8

Assuming that they are moving at the same speed, you would just need to calculate the relative distance traveled between the two electrons.
They are moving in a circle so the distance the go would be the circumference of each circle.
The equation for the circumference of a circle is C=πd d = diameter which is 2 x R
Electron with radius R would have a circumference of C=2Rπ
Electron with radius 4 R would have a circumference of C=8Rπ
Since π is constant you can ignore it... now you compare 8R vs. 2R
Amazingly, its the same ratio as the radii given!
The electron with 4R will take 4 times as long to complete a revolution as R
Ratio is 4 to 1
This is all a hypothetical question because , in the real world of chemistry, the position of an electron can't be determined at a given point in time because of quantum theory and the Heisenberg uncertainty principle. But, I'm sure your teacher doesn't want that answer.
They are moving in a circle so the distance the go would be the circumference of each circle.
The equation for the circumference of a circle is C=πd d = diameter which is 2 x R
Electron with radius R would have a circumference of C=2Rπ
Electron with radius 4 R would have a circumference of C=8Rπ
Since π is constant you can ignore it... now you compare 8R vs. 2R
Amazingly, its the same ratio as the radii given!
The electron with 4R will take 4 times as long to complete a revolution as R
Ratio is 4 to 1
This is all a hypothetical question because , in the real world of chemistry, the position of an electron can't be determined at a given point in time because of quantum theory and the Heisenberg uncertainty principle. But, I'm sure your teacher doesn't want that answer.

You couldn't solve it, because it can't be done. According to Heisenberg's uncertainty principle, if you know the velocity of an election you cannot know its position. And if you know the position of an election you cannot know its velocity.