I had been working on a problem recently and i stumbled upon something I did not quite get. So the relevant part of the problem is like this:
Suppose we have two conducting spheres with radii a and b separated by a distance, r, much greater than either radius( refer to diagram below). A total charge Q is shared between the spheres. we show that when the electric potential energy of the system has a minimum value, the potential difference between the spheres is zero.<The total charge Q is equal to q1 + q2, where q1 represents the charge on the first sphere and q2 the charge on the second>
I started this part as follows:
--> Firstly because the spheres are so far apart, I assume an uniform charge distribution on either one. Next, I calculate the energy associated with a single conducting sphere by starting from Gauss's Law $\Phi_E=\iint_sE.dA=\frac{q}{\epsilon_0}$. The field due to a sphere carrying charge q is then $E=\frac{\kappa_e*q}{R^2}$ with R being the sphere's radius. From this i get the potential on the surface by using $\Delta V=-\int_cE.dr$ where the path of integration would be from R to $\infty$(we assume an infinite conductor with charge -q surrounding the sphere in question) from which i get $\Delta V=\frac{\kappa_e}{R}$; with the charge and the potential known i get the associated energy as $U_E=\frac{\kappa_eq^2}{2R}$
-->Applying all of the above to my system(from the problem), i get that the total energy corresponding to my system of two conducting spheres is: $$U_{E,total}=\frac{\kappa_e}{2}[\frac{\kappa_eq_2^2}{2b} +\frac{\kappa_eq_1^2}{2a}]=\frac{\kappa_e}{2}[\frac{\kappa_e*(Q-q_1)^2}{2b} +\frac{\kappa_eq_1^2}{2a}]$$
-->Now differentiating with respect to $q_1$ and setting the derivative to 0, i get $q_1=\frac{Qa}{a+b}$ and using the given information Q=$q_1+q_2$ i get that $q_2=\frac{Qb}{a+b}$ using these in the expression for the potential of a single sphere i get $V_1 ,V_2$ the difference of which really is 0. Now i must ask:
- what are the physical interpretations of what i did just now? that is, why is the potential difference 0 when the energy is minimized?
- Is there a more fundamental approach to determining this relation? 3)can any one give me an intuition behind this? why does it work like it does? why does a minimum electric potential energy mean no potential difference?.