Final answer:
Nodal analysis involves writing equations using Kirchhoff's rules to find voltages and currents in a circuit. By applying these rules systematically, we can determine voltages V1 and V2, using given resistance values and electromotive forces, and by adjusting for changes in the circuit, such as the replacement of a voltage source with a switch.
Step-by-step explanation:
To find v1 and v2 using nodal analysis, we utilize Kirchhoff's current (junction) rule and voltage (loop) rule. First, we would write an equation at a junction point in the circuit using the current rule, like equation (1) showing the sum of currents at junction b equals zero. Then, we apply the loop rule to write equations for the voltage around loops in the circuit, as done in equations (2) and (3).
When specific values of resistance (R1, R2, R3), internal resistances (r1, r2), and electromotive forces (E1, E2) are given, we can substitute these into the loop equations to solve for the loop currents (I1, I2, I3). If components change, such as replacing E2 with a closed switch, we recalculate the currents by adjusting the loop equations appropriately.
Finally, equation (5) exemplifies how to combine equations linearly to solve for a particular current, which then allows for the determination of the others and subsequently calculating the voltages (V1, V2) across nodes or components using the loop equations we initially wrote.