Figure 27-27 shows a circuit of four resistors that are con-...

Figure $27-27$ shows a circuit of four resistors that are con- nected to a larger circuit. The graph below the circuit shows the electric potential $V(x)$ as a function of position $x$ along the lower branch of the circuit, through resistor $4 ;$ the potential $V_{A}$ is $12.0 \mathrm{~V}$. The graph above the circuit shows the electric potential $V(x)$ versus position $x$ along the upper branch of the circuit, through resistors $1,2,$ and $3 ;$ the potential differences are $\Delta V_{B}=2.00 \mathrm{~V}$ and $\Delta V_{C}=$ $5.00 \mathrm{~V}$. Resistor 3 has a resistance of $200 \Omega$. What is the resistance of (a) resistor 1 and (b) resistor $2 ?$

Key Concepts

Resistors in Series (Voltage Division)

In a series circuit, resistors are connected end-to-end so that the same current flows through each resistor. The total voltage supplied to the series combination is divided among the resistors in proportion to their resistance values. This principle, commonly known as the voltage divider rule, allows for the determination of individual voltage drops (and hence resistances) when the total voltage drop and at least one known resistor are provided.

Graphical Analysis of Potential

Graphical representations of electric potential as a function of position are useful in analyzing circuits, as they visually illustrate how the potential changes along a specific branch. By examining the slopes or changes in the graph, one can infer the voltage drop across various components. This technique helps in understanding complex circuit behavior and is crucial when solving problems that provide potential vs. position data along with the circuit layout.

Electric Potential Difference

Electric potential difference, also known as voltage, represents the work needed per unit charge to move a charge between two points in an electric field. In circuit analysis, knowing the voltage difference across components helps determine how much energy is used or dissipated by a resistor, and it is fundamental for applying Ohm’s law to find unknown quantities in the circuit.

Ohm's Law

Ohm's law is a fundamental relationship in electrical circuits, stated as V = IR, where V is the voltage across a resistor, I is the current flowing through it, and R is the resistance. This law is essential for determining the values of resistive components based on measured voltage drops and inferred current flow, and it underpins the entire analysis of resistive networks.

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