The discussion centers on determining the stability of electrostatic equilibrium at the center of a charged ring. Participants analyze the potential energy, electric field, and forces acting on a point charge placed at the center. The electric potential at the center is given by $$ V(0) = \frac{\lambda}{2\epsilon_0} $$, and the stability is assessed using Laplace's equation in cylindrical coordinates. The consensus is that the equilibrium is stable if the electric field points back towards the center upon small displacements.
PREREQUISITESPhysics students, electrical engineers, and anyone interested in electrostatics and stability analysis of charged systems.
I don't think you were supposed to get into detailed calculations. That's why the question said simply "motivate your answer". You're looking for a good approximation for the force on a particle that is perturbed off centre.Angela G said:
You have examined $$- \frac{\partial^2 V}{\partial z^2} = \frac{1}{\rho} \frac{\partial}{\partial \rho} \left( \rho \frac{\partial V}{\partial \rho} \right )$$ This says that:Angela G said:
I think you already have said the way the ring field acts and you know which way it makes a charge with a small displacement go for each sign of ##q##.Angela G said:
It was used as a starting case, that's all. The possibility of different signs were discussed earlier.PeroK said:
You solved the stability for the ##z## axis above and you know the direction and therefore force in the plane (toward the center for positive ring) and so you know whether it is stable in the plane of the ring given the sign of the charges. If you want to be more detailed for example on the ##z## axis just compute the potential along the ##z## axis which is easy because of symmetry and take the second derivative of it.Angela G said:
Looks correct.Angela G said:
