PHY.6:5.3 - Detect an incompatible ideal connection before attempting a transient calculation

Two ideal linear capacitors have positive capacitances C1 and C2. Their lower terminals share a reference conductor; their upper terminals are connected through a resistance R>0. The effective description neglects leakage, inductance and radiation. Let v1 and v2 be the upper-terminal potentials relative to the common reference, and let current I flow from capacitor 1 to capacitor 2.

Charge balances and the resistive response give

C1*v1' = -I; C2*v2' = I; R*I = v1-v2.

The total upper-plate charge Q=C1*v1+C2*v2 is constant. The difference delta=v1-v2 obeys

delta' = -(1/C1+1/C2)*delta/R.

Thus the difference decays with time constant tau=R*C1*C2/(C1+C2), and both potentials approach

v_final = (C1*v1(0)+C2*v2(0))/(C1+C2).

For C1=1 F, C2=3 F, R=2 ohm, v1(0)=8 V and v2(0)=0 V, the settled potential is 2 V, tau=1.5 s, and I(t)=4*exp(-t/1.5) A, with time measured in seconds.

Stored electrical energy is H=(C1*v1^2+C2*v2^2)/2. Substitution gives H'=-R*I^2. Its initial and final values are 32 J and 8 J, so 24 J is converted into other energy, here heat in the ideal resistance. More generally, the energy difference is

C1*C2*(v1(0)-v2(0))^2/(2*(C1+C2)).

Changed connection. Set R=0 as an ideal connection from the initial instant. Its constraint v1=v2 conflicts with the supplied unequal initial potentials. The smooth evolution above cannot simply start from them under that constraint. Recover the interaction that establishes the common potential when its current, duration or energy conversion matters. Resistance, inductance or electromagnetic emission may matter depending on the actual arrangement and interval; the ideal connection alone does not specify them.

If only the settled potential is needed, conserved charge together with the premise that the connected system settles can already supply it. A transient description is needed for a different question, such as peak current. Taking the positive-resistance time constant to zero does not remove the finite energy conversion. This is why changing an idealization can require revisiting both the preparation and the requested consequence.