Final answer:
To rewrite a diagram in three different equations, Kirchhoff's junction and loop rules are applied to a circuit to derive equations that relate currents and voltages in the circuit. The three loop equations are formulated from the loop rule by considering voltage sources and resistances, then simplified and solved for the unknowns.
Step-by-step explanation:
When addressing the task of rewriting a diagram in the form of three different equations, we typically use Kirchhoff's circuit laws: the junction rule (current law) and the loop rule (voltage law). These laws facilitate the formation of equations based on a circuit diagram.
Junction Rule Application
The junction rule states that the total current entering a junction must equal the total current leaving the junction. This yields at least one equation when there is at least one unknown current in the circuit.
Loop Rule Application
The loop rule is used to write loop equations, and it states that the sum of potential differences (voltage) around any closed circuit loop must be zero. For our loop equations, we consider the voltage sources and the resistances in each loop of the circuit.
For example, the loop equation for loop abcda could be written as:
Loop abcda: -IR₁ - V₁ - IR₂ + V₂ - IR₃ = 0.
After writing all loop equations, one can simplify them by isolating variables or by dividing by a common factor, like the resistor's value, if it simplifies the equation further.
Combining and Solving Equations
Lastly, these equations are combined and manipulated to solve for the unknowns. If current I₂ is one of the unknowns, it can be eliminated by combining equations strategically. For instance:
Eliminating I₂: Add Eq. (1) times R₂ to Eq. (2) to obtain Eq. (4), which does not include I₂.
Ultimately, selecting the right equations and correctly manipulating them leads to the solution of all unknown variables in the circuit.