In the circuit in Fig. 29-5, I1=0.20 A and R=5.0 Ω. Find ε. We write the loop equation for loop

step 1 In this video, the concept we will be looking at is Kierkhoff's Laws. Okay, so we have this circ

In the circuit in Fig. 29-5, I1=0.20 A and R=5.0 Ω. Find ε....

In the circuit in Fig. 29-5, $I_{1}=0.20 \mathrm{~A}$ and $R=5.0 \Omega$. Find $\varepsilon$. We write the loop equation for loop cdefc: $$ -R I_{1}+12.0-8.0-7.0 I_{2}=0 \quad \text { or } \quad-(5.0)(0.20)+12.0-8.0-7.0 I_{2}=0 $$ from which $I_{2}=0.43$ A. We can now find $I_{3}$ by applying the node rule at $e$ : $$ I_{1}+I_{3}=I_{2} \quad \text { or } \quad I_{3}=I_{2}-I_{1}=0.23 \mathrm{~A} $$ Now apply the loop rule to loop cdeac: $$ -(5.0)(0.20)+12.0+(2.0)(0.23)+\varepsilon=0 $$ from which $\varepsilon=-11.5 \mathrm{~V}$. The minus sign tells us that the polarity of the battery is actually the reverse of that shown.

In the circuit in Fig. 29-5, $I_{1}=0.20 \mathrm{~A}$ and $R=5.0 \Omega$. Find $\varepsilon$.
We write the loop equation for loop cdefc:
$$
-R I_{1}+12.0-8.0-7.0 I_{2}=0 \quad \text { or } \quad-(5.0)(0.20)+12.0-8.0-7.0 I_{2}=0
$$
from which $I_{2}=0.43$ A. We can now find $I_{3}$ by applying the node rule at $e$ :
$$
I_{1}+I_{3}=I_{2} \quad \text { or } \quad I_{3}=I_{2}-I_{1}=0.23 \mathrm{~A}
$$
Now apply the loop rule to loop cdeac:
$$
-(5.0)(0.20)+12.0+(2.0)(0.23)+\varepsilon=0
$$
from which $\varepsilon=-11.5 \mathrm{~V}$. The minus sign tells us that the polarity of the battery is actually the reverse of that shown.

Key Concepts

Sign Conventions in Circuit Analysis

Adhering to proper sign conventions when applying Kirchhoff's laws is crucial, as it determines the correct polarity for voltage drops and gains. Correct assignment of signs prevents errors in the setup of loop and node equations, ensuring that calculated values for current and voltage reflect the true behavior of the circuit.

Ohm's Law

Ohm's Law establishes a relationship between voltage, current, and resistance, expressed as V = IR. It is a fundamental concept for calculating voltage drops across resistive elements in a circuit and is typically used in conjunction with Kirchhoff's laws to solve for unknown quantities in the circuit.

Kirchhoff's Voltage Law (Loop Rule)

This law states that the algebraic sum of all voltages around any closed loop in a circuit is zero. It allows one to set up equations based on the potential drops across resistors and the electromotive forces of batteries or other sources, facilitating the analysis of voltage relationships in complex circuits.

Kirchhoff's Current Law (Node Rule)

This principle declares that the total current entering a node (or junction) in an electrical circuit equals the total current leaving that node. It is essential for setting up equations that ensure conservation of charge and is widely used to analyze the distribution of currents in different branches of a circuit.

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