EPE at a point due to two point charges

[elemis]

The above equation gives the EPE of two point charges separated by a distance r.

Firstly, I do not understand how this formula gives the TOTAL EPE of the system.

Secondly, let's say I have three deuterium nuclei moving towards one another with initial speed V.

They all stop instantaneously at the same point such that they are all separated from one another by distance r.

How would the above formula change to give the total EPE of the system ? Would you consider each pair of nuclei ?

So total EPE of system = (EPE of 1st+2nd) + (1st+3rd) + (2nd+3rd) ?

Hence, the formula becomes

 

[tiny-tim]

hi elemis!

Firstly, I do not understand how this formula gives the TOTAL EPE of the system.

no work is done bringing the first charge from infinity

so you only need to consider the work done by bringing the second charge

Secondly, let's say I have three deuterium nuclei moving towards one another with initial speed V.

They all stop instantaneously at the same point such that they are all separated from one another by distance r.

How would the above formula change to give the total EPE of the system ? Would you consider each pair of nuclei ?

potential energy is defined as minus the work done (per charge) …

if there's more than one force, how does that affect the total work done?

 

[elemis]

Hi TinyTim ! We meet again !

Please have a look at the above image i.e. part (a)

You are telling me that no EPE is gained by one of the nuclei ?

So we consider the EPE of only one?

EDIT : How is that possible when both are moving towards one another and both are trying to overcome the repulsive force of the other ?

 

[tiny-tim]

work done = force "dot" displacement …

if they're both moving, wouldn't you expect the displacement of one to be halved, and the total distance to remain the same? ​

 

[elemis]

work done = force "dot" displacement …

if they're both moving, wouldn't you expect the displacement of one to be halved, and the total distance to remain the same? ​

Could you elaborate ? Are you saying r in the very first formula at the top of the page is in fact r/2

 

[elemis]

if there's more than one force, how does that affect the total work done?

So is it the sum of the forces of all the nuclei multiplied by their displacement ?

 

[tiny-tim]

Are you saying r in the very first formula at the top of the page is in fact r/2

no, the result will be the same

but you can obtain it either by bringing one charge from infinity (zero work), then bringing the other charge (∫ F(r).dr)

or by bringing both charges from infinity together (∫ F(r).d(r/2) + ∫F(r).d(r/2))

So is it the sum of the forces of all the nuclei multiplied by their displacement ?

yes, but you'd be crazy to try to do it that way …

bring each charge from infinity one at a time ​

 

[elemis]

Before we all get confused, let's deal with the question in post no. 3 first.

Its alright to do that i.e. pretend one nuclei moves to the collision point and calculate the change in EPE (zero for the first one because both are at infinity at feel no repulsion) and calculate the change in EPE for the other ?

 

[tiny-tim]

yes

(and there's no problem with the "fixed" one having speed v … we can bring the other one in as fast or as slow as we like, and the speed makes no difference to the force (if we're ignoring relativity))

 

[jewbinson]

With regards to this setup of two point charges, I was thinking the other day (assuming both charges are the same magnitude and opposite sign...) What is the equation of motion? If we write Coulomb's Force Law then the equation seems hard to solve. Can we use energy/ work done equations to solve the equation of motion more simply? Or use Lagrangian mechanics methods??

 

[elemis]

yes

(and there's no problem with the "fixed" one having speed v … we can bring the other one in as fast or as slow as we like, and the speed makes no difference to the force (if we're ignoring relativity))

Thank you very much TinyTim ! As always, I am much obliged.

You have literally answered a query that my teacher couldn't explain and has been 'thinking' about for the last six months.