Digital systems and design

Energy and charge conservation laws in the circuit

Energy and charge conservation in the circuit

This post tells about energy and charge conservation in the circuit. We will consider conservation laws on the circuit with two capacitors in parallel via the switch.

Let’s consider the circuit depicted below.

Circuit with two capacitors connected in parallel via the switch

Here in accordance to the Kirhhoff Current Law we have dq1(t)dt+dq2(t)dt=d(q1+q2)dt=0.  At the same time v1=v2 here and q1(t)C1=q2(t)C2. At the same time we know that q1(t)+q2(t)=C1+C2.

Then q1(t)=C1C1+C2(Q1+Q2) and q2(t)=C2C1+C2(Q1+Q2).

From here we have v1=v2=Q1+Q2C1+C2.

After the moment of switch closure, charge in the circuit is conserved.

Let’s see what happens with energy in the circuit before and after switch closure. Before switch closure the total energy in the circuit wE=Q212C1+Q222C2. After the switch is closed wE1=q12(t)2C1+q22(t)2C2. It means that energy in the circuit after switch is closed is not conserving. More over, it is easy to see that wEwE1>0.

Educational content can also be reached via Reddit community r/ElectronicsEasy.