How Is the Radius r Related to Quantum Number n in Bohr's Model?

[Saitama]

Homework Statement

A small particle of mass m moves in such a way that P.E.=-1/2mk(r)^2, where k is a constant and r is the distance of the particle from the origin. Assuming Bohr's model of quantization of angular momentum and circular orbit, r is directly proportional to:
(a)n²
(b)n
(c)$\sqrt{n}$
(d)none of these

Homework Equations

$P.E.=-\frac{KZe^2}{r}$

The Attempt at a Solution

I don't understand how should i start?

 

[tiny-tim]

Hi Pranav-Arora!

(try using the X² icon just above the Reply box )

Find the force, then use https://www.physicsforums.com/library.php?do=view_item&itemid=27" to find the relation between v and r, then … ​

 

[Saitama]

Hi Pranav-Arora!

(try using the X² icon just above the Reply box )

Find the force, then use https://www.physicsforums.com/library.php?do=view_item&itemid=27" to find the relation between v and r, then … ​

How do i calculate the force?

 

[I like Serena]

The force is the derivative of the potential energy.
That is, the derivative of the P.E. given in the problem statement, not the one you listed as relevant equation.

 

[Saitama]

That is, the P.E. given in the problem statement, not the one you listed as relevant equation.

Why i have to find the derivative of P.E. given in the problem statement, not the one in the relevant equations?
And with what respect do i have to find the derivative?

Did you get that i refer P.E. as "Potential Energy"?

 

[I like Serena]

Why i have to find the derivative of P.E. given in the problem statement, not the one in the relevant equations?
And with what respect do i have to find the derivative?

Did you get that i refer P.E. as "Potential Energy"?

The P.E. in the relevant equations is for charges attracting each other, which is what Bohr used in his model of the atom (btw, there should be a minus sign included).

The P.E. in the problem statement is about a different system, for instance a system with a mass on a spring, or a charge inside a sphere with homogeneous charge density.
Since the problem is about this P.E., it is the one you should use.Edit: The derivative is with respect to the only variable you have, which is r.

 

[Saitama]

The P.E. in the relevant equations is for charges attracting each other, which is what Bohr used in his model of the atom (btw, there should be a minus sign included).

The P.E. in the problem statement is about a different system, for instance a system with a mass on a spring, or a charge inside a sphere with homogeneous charge density.
Since the problem is about this P.E., it is the one you should use.

But how do i find the derivative?
And how do i calculate the force?

 

[I like Serena]

But how do i find the derivative?
And how do i calculate the force?

You do know what a derivative is?

Start by listing the P.E. and the derivative of the P.E. with respect to r?
The latter is the force.

 

[Saitama]

You do know what a derivative is?

Start by listing the P.E. and the derivative of the P.E. with respect to r?
The latter is the force.

Yes, i know what a derivative is.

$$\frac{d}{dr}(P.E.)=-mkr$$

Am i right..?

 

[I like Serena]

Yes! :)

 

[Saitama]

Yes! :)

But what next?

 

[I like Serena]

What did tiny-tim suggest?

 

[Saitama]

What did tiny-tim suggest?

But i have never studied centripetal acceleration...

 

[I like Serena]

All right, but then you will also get into trouble with angular momentum and with quantization...

What do you know about the mechanics of circular motion?

 

[Saitama]

All right, but then you will also get into trouble with angular momentum and with quantization...

What do you know about the mechanics of circular motion?

Nothing.
I haven't still reached to the circular motion.

 

[Saitama]

I read circular motion on Wikipedia, i have learned about uniform circular motion.

 

[I like Serena]

So can you follow up on tiny-tim's suggestion?

"use centripetal acceleration to find the relation between v and r"

 

[Saitama]

So can you follow up on tiny-tim's suggestion?

"use centripetal acceleration to find the relation between v and r"

Is the knowledge of uniform circular motion sufficient for centripetal acceleration?

 

[Saitama]

I think i have got the answer now.
I did it like this:-
$$\frac{d}{dr}(P.E.)=-mkr$$
$$ma_c=-mkr$$
$$-\frac{mv^2}{r}=-mkr$$
$$v^2=kr^2$$

Using Bohr's model of quantization of angular momentum,
$$mvr=\frac{nh}{2\pi}$$
$$v=\frac{nh}{2\pi mr}$$

Substituting this value of v in our previous equation, i get:-
$$\frac{n^2h^2}{4\pi^2m^2r^2}=kr^2$$
$$\frac{n^2h^2}{4\pi^2m^2}=kr^4$$Therefore,
$$r^4\propto n^2$$
$$r^2\propto n$$
$$r\propto\sqrt{n}$$

 

[Saitama]

*bump*

 

[I like Serena]

(:zzz: I guess you haven't seen what the time is where I live!)

Yep!

So now you know classical mechanics and quantum mechanics.
What's next?
Ready for Maxwell's equations?
Or would you rather do the Schrödinger equation?

 

[Saitama]

(:zzz: I guess you haven't seen what the time is where I live!)

Yep!

So now you know classical mechanics and quantum mechanics.
What's next?
Ready for Maxwell's equations?
Or would you rather do the Schrödinger equation?

(Oh sorry, i thought you got busy. I too just had a nap. Just woke up.:zzz:)

No, i don't think i know much about quantum mechanics. There's only a small article on it in my textbook.
And Schrödinger equation is not of my level, i think this because Schrödinger equation is mentioned in my book and after that it is written that solving this equation is not in the scope of this book. Yesterday i started a thread on Schrödinger equation. https://www.physicsforums.com/showthread.php?t=518047
czelaya said those topics which needs to be covered before solving the Schrödinger equation. And i think, i haven't completed most of the topics stated in the reply by czelaya.

I just saw Maxwell's equations on Wikipedia. And i think it involves the knowledge of partial differentiation. I talked to my physics teacher about the partial differentiation. He said that this is not of my level.

(Btw, i think i should not learn about these today because tomorrow is my chemistry exam on mole concept and atomic structure )

 

[I like Serena]

Ah well, I didn't really think you'd want to go for those yet.
I just thought I'd give you something to aspire to.

 

[Saitama]

Ah well, I didn't really think you'd want to go for those yet.
I just thought I'd give you something to aspire to.

I would learn them after i have completed with MIT lectures but i don't get time for the lectures too.