3sir actually sir i came here to post this qn,fortunately u already posted this qn, i dunno how to solve this one.
Prove that a point charge CANNOT be in the state of stable equilibrium in an electrostatic field!!
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66I have a bad feeling that most of the posts in this section are going to be simply added to the stats.
341if i remember right, it has something to do with Gauss Law using it to prove that all the lines of force do not converge to the point charge as it would have been in equilibrium condition.
But i feel we are talking about a +ve charge here. In that case if all lines of force were inwards, the flux would be negative whereas Gauss Law would require the flux to be +ve (q/4πε0 )
1it will experience a force in only 1 direction or rather a=-kx wont be satisfied ever....m i right kayamat sir??
24V=Potential in teh space
If there is no charge in space
∂2V∂x2+∂2V∂y2
+∂2V∂z2=0 --------(1)
if V is maximum =>
∂2V∂x2+∂2V∂y2
+∂2V∂z2 <0 ------(2)
if V is minimum =>
∂2V∂x2+∂2V∂y2
+∂2V∂z2 >0 ------(3)
but (2) and (3) cant be true due to 1
so no extremum of V in space...
=> Not in equilibrium
plz tell me if it is right.....
and also I couldn't understand prophet sir's hint ..
24ok this is something I found in another book.....
its a lot diff from what I have written and everything went over my head...
If anyone has objection ,I will remove this image..[2][2]
1I don't clearly get what we have to show....
Since electric field has a unique direction at every point so once disturbed the particle can't have the tendency to return to its previous state.
1Earnshaw's theorem...????
Earnshaw's theorem states that a collection of point charges cannot be maintained in a stable stationary equilibrium configuration solely by the electrostatic interaction of the charges. This was first proven by Samuel Earnshaw in 1842. It is usually referenced to magnetic fields, but originally applied to electrostatic fields. It applies to the classical inverse-square law forces (electric and gravitational) and also to the magnetic forces of permanent magnets and paramagnetic materials (but not diamagnetic materials).
(copied)
24Earnshaw's theorem holds well for gravitational,electric and magnetic field
66Yes that's the Earnshaw's theorem....
And the proof is as given by Prophet sir.
Basically one uses contradiction. If the charge we are talking about is positive, for it to be in stable equilibrium, it is required that no matter in what direction we displace the charge, there would be a restoring force forcing it towards its original position. But that will require that the electric field from all directions converge into the point charge. If that be the case, then enclosing the charge within a closed surface, we see that the net flux through that closed surface is negative (since at all points field is inwards while area vector is outward). However, the net enclosed charge is positive, contradicting the Gauss's law.
341Ah, good! That means I am not gonna get Alzheimer's anytime soon :D.
I dont remember anything I learnt in IIT, but things I was taught at Ramaiah a decade-and-a-half ago are etched in memory.