Electrostatics Problem - Understanding Inertial Systems in Mechanics Discussions

[neelakash]

It is a mechanics discussion regarding a problem of electrostatics.

Couple of months ago I dealt with a problem:Given two electric dipoles, separated at a distance r and are perpendicular to each other.We are to find the torque exerted on each other.
I took p1 as vertical and p2,right to p1,at a distance r,facing towards right.
I used co-ordinate free form of E_dip which gave the torques...

The torques were in the same direction and were not cancelling one another.It resulted from the fact that there were also dipole-forces exerted by one on the other---which I did not need to consider for the purpose of the problem.If one finds the torques due to both the factors---one arising from pxE and the other from (p.grad)E,the total torque on the system P1+P2 can be shown equal to zero.It is OK as the angular momentum of the system does not change.

Everything is clear.What I want to know is that when I used co-ordinate free form,how can I be sure that I am working from an inertial system?

Is my qustion clear?I am not referring to any particular co-ordinate system...but the end result is consistent with that as viewed from an inertial system...
So,I think using a co-ordinate free form somehow makes it possible.

can you try out something to make it clear?

 

[neelakash]

I did not use a frame to avoid two different frames involved in the same problem

 

[Meir Achuz]

If you are doing electrostatics, both dipoles are at rest.
If either dipole is moving, the problem gets very complicated and needs special relativity. In either event, you are in an "inertial system" if the equations you used apply.

 

[neelakash]

Which inertial frame do the formula refer to?
I know it is an obvious answer that all frames are equivalent...Remember that we are to check that external torque (about the origin or any other point) should vanish...
But I do not find such a point.

 

[neelakash]

Oh! Sorry! I got such a point...When I did the explicit calculation I worked from the location of one charge.

What I got is that in STR the co-ordinate free forms have meaning only in inertial co-ordinate systems...

 

[Meir Achuz]

What I got is that in STR the co-ordinate free forms have meaning only in inertial co-ordinate systems...

"co-ordinate free" has nothing to do with it. Anything or any coordinate system is only valid in STR in an inertial system. The definition of "inertial system" is one in which the STR equatins hold.

 

[neelakash]

Then may we conclude:
whenever we encounter an application of co-ordinate free formula,we will get the dynamical result same as that obtained from an inertial frame?

 

[Meir Achuz]

Yes, if you use the inertial frame equations.
For instance, if you use the Coriolis force, then you are not in an iinertial frame.

 

[neelakash]

I did not so far "co-ordinate free" equations for coriolis force.However,for the formulas like the dipole moment/dipole-dipole interaction, we may use co-ordinate free formulas meaning an inertial frame.
I was wondering why this can be done!Now I realize,the formulas can be as well derived from their corresponding co-ordinate based (inertial frame)formulas.