Question

Write Kirchhoff's laws related to electric circuit by drawing suitable circuit diagram.

Hint:

When applying KVL, establish a consistent sign convention. A common convention is: \begin{itemize} \item EMF is positive if traversed from the negative to the positive terminal. \item Potential drop across a resistor (\(-IR\)) is negative if traversed in the same direction as the current. \end{itemize}

Answer 1

Step 1: Understanding the Concept:
Kirchhoff's laws are two rules that deal with the conservation of charge and energy within electrical circuits. They provide a systematic way to analyze complex circuits that cannot be simplified into simple series or parallel combinations.

Step 2: Detailed Explanation:
1. Kirchhoff's First Law (Junction Rule or Kirchhoff's Current Law - KCL)
\begin{itemize} \item Statement: The algebraic sum of all electric currents meeting at any junction (or node) in a circuit is zero. In other words, the total current entering a junction must equal the total current leaving that junction.
\item Principle: This law is based on the law of conservation of charge. Charge cannot accumulate at a junction.
\item Equation: \(\sum I = 0\)
\item Circuit Diagram:
Consider a junction 'A' where currents \(I_1\) and \(I_2\) are entering, and currents \(I_3\) and \(I_4\) are leaving.
\begin{center} \begin{circuitikz}[american currents] \draw (0,0) node[circ] (A) {} node[above left] {A}; \draw (A) -- ++(-1,1) to[short, i<=$I_1$] ++(-1,0); \draw (A) -- ++(-1,-1) to[short, i<=$I_2$] ++(-1,0); \draw (A) -- ++(1,1) to[short, i>=$I_3$] ++(1,0); \draw (A) -- ++(1,-1) to[short, i>=$I_4$] ++(1,0); \end{circuitikz} \end{center} According to KCL, if we consider incoming currents as positive and outgoing currents as negative:
\(I_1 + I_2 - I_3 - I_4 = 0\)
or,
\(I_{in} = I_{out} \implies I_1 + I_2 = I_3 + I_4\)
\end{itemize} \vspace{0.3cm} 2. Kirchhoff's Second Law (Loop Rule or Kirchhoff's Voltage Law - KVL)
\begin{itemize} \item Statement: The algebraic sum of the changes in electric potential (voltage) encountered in a complete traversal of any closed loop in a circuit is zero.
\item Principle: This law is based on the law of conservation of energy. The net energy gained by a charge after moving around a closed loop must be zero.
\item Equation: \(\sum \Delta V = 0\)
\item Circuit Diagram:
Consider a simple closed loop containing a voltage source (EMF \(\mathcal{E}\)) and two resistors \(R_1\) and \(R_2\), with a current \(I\) flowing through them.
\begin{center} \begin{circuitikz}[american voltages] \draw (0,0) to[battery, l=$\mathcal{E}$] (0,3) to[R, l=$R_1$] (3,3) to[R, l=$R_2$] (3,0) to[short] (0,0); \node at (1.5,1.5) [rotate=-90] {\huge$\circlearrowright$}; \node at (2.1,1.5) {$I$}; \end{circuitikz} \end{center} To apply KVL, we choose a direction (e.g., clockwise) and sum the potential changes:
- Traversing the battery from negative to positive terminal: Potential gain of \(+\mathcal{E}\).
- Traversing resistor \(R_1\) in the direction of current: Potential drop of \(-IR_1\).
- Traversing resistor \(R_2\) in the direction of current: Potential drop of \(-IR_2\).
The sum of these potential changes is zero:
\(\mathcal{E} - IR_1 - IR_2 = 0\)
\end{itemize}

Step 3: Final Answer:
Kirchhoff's two laws are the Junction Rule, which states that the sum of currents at a junction is zero (\(\sum I = 0\)), and the Loop Rule, which states that the sum of potential differences around any closed loop is zero (\(\sum \Delta V = 0\)).