Comparing Electrical and Mechanical Quantities

Most students of physics will have seen the diagrams I've redrawn in Figure 1 below:—

Fig 1. Showing i) the exchange between kinetic energy of a mass and energy stored in a spring, in an oscillating mass-spring system; and ii) the exchange between energy stored in the magnetic field of an inductor and energy stored in the electric field of a capacitor, in an oscillating inductor-capacitor system.

Correspondences

By comparing the energy flow in an oscillating mass-spring system with that in an oscillating inductor-capacitor system, we see that there is a correspondence between certain electrical and mechanical quantities. For example:—

   charge corresponds to distance      (q corresponds to x)
   current corresponds to velocity      (i corresponds to v)
   capacitance corresponds to the      (C corresponds to 1/k)
      inverse of spring constant
   inductance corresponds to mass    (L corresponds to m)

Let's now look more closely at the correspondence between a stretched spring and a charged capacitor.

Fig 2. Energy added to a stretched spring, and to a charged capacitor

Energy triangles

In Figure 2A, a spring of spring constant k has been initially stretched over a distance x, by exerting an ever-increasing force F on it. The energy stored in the spring is given by the triangle (light blue) of base x and height kx  i.e. 

    U_S  =  ½ × x × kx  =  ½kx² 

Now, suppose we wish to stretch the spring a bit more, i.e. by Δx, and thus gain the extra energy represented by the darker blue area. The only possible way to do that is by increasing the exerted force F by ΔF, from an already high value to a new and higher one. Thus we have to pay a full energy penalty in terms of the force being exerted on the spring through the distance Δx, to gain our extra spring stored energy.

At first sight, the corresponding case with a capacitor seems to be exactly analogous. In Figure 2B, a capacitor of capacitance C has been initially charged with charge q, by subjecting it to an ever-increasing voltage V. The energy stored in the capacitor is given by the triangle (light red) of base q and height (1/C).q  i.e.

    U_E  =  ½ × q × (1/C).q  =  ½q²/C

Now, suppose we wish to add a bit more charge to the capacitor, i.e. Δq, and thus gain the extra energy represented by the darker red area. By analogy with the spring, we could do that by increasing the voltage V by ΔV, from an already high value to a new and higher one. Then we would once again pay a full energy penalty in terms of the voltage being applied across the capacitor to add the charge Δq, to gain our extra electrical energy.

Another way — but only for a capacitor

However, we could do something else, which as far as I know has no mechanical analogue. We could make our capacitor as a spherical conducting shell (or a major portion of one) which, by Gauss's Law, cannot retain any charge on its inner surface. Any net charge that is generated within the shell, or brought into it from outside, can then all be added to the shell at its inner surface for no added energy penalty, no matter what voltage the shell has already reached. Thus we might be able to obtain any desired increase in charge Δq without having to directly increase the applied voltage at all. That voltage increase, and the associated energy gain, would then occur incidentally, and automatically, as the charge was added.

Next time, I'll look further into the questions that I've now raised concerning a charged conducting shell:—

a) generating net charge within the shell

b) bringing an externally generated charge into the shell without paying the full expected energy penalty for doing so.