The voltage drop Va b across each of resistors A and B was...

The voltage drop $V_{a b}$ across each of resistors $A$ and $B$ was measured as a function of the current $I$ in the resistor. The results are shown in the table: $$ \begin{array}{l|llll} \text { Resistor } A & & & & \\ I(\mathrm{~A}) & 0.50 & 1.00 & 2.00 & 4.00 \\ V_{a b}(\mathrm{~V}) & 2.55 & 3.11 & 3.77 & 4.58 \\ & & & & \\ \begin{array}{l} \text { Resistor } B \\ I(\mathrm{~A}) \end{array} & 0.50 & 1.00 & 2.00 & 4.00 \\ V_{a b}(\mathrm{~V}) & 1.94 & 3.88 & 7.76 & 15.52 \end{array} $$ (a) For each resistor, graph $V_{a b}$ as a function of $I$ and graph the resistance $R=V_{a b} / I$ as a function of $I$. (b) Does resistor $A$ obey Ohm's law? Explain. (c) Does resistor $B$ obey Ohm's law? Explain. (d) What is the power dissipated in $A$ if it is connected to a $4.00 \mathrm{~V}$ battery that has negligible internal resistance? (e) What is the power dissipated in $B$ if it is connected to the battery?

The voltage drop $V_{a b}$ across each of resistors $A$ and $B$ was measured as a function of the current $I$ in the resistor. The results are shown in the table:
$$
\begin{array}{l|llll}
\text { Resistor } A & & & & \\
I(\mathrm{~A}) & 0.50 & 1.00 & 2.00 & 4.00 \\
V_{a b}(\mathrm{~V}) & 2.55 & 3.11 & 3.77 & 4.58 \\
& & & & \\
\begin{array}{l}
\text { Resistor } B \\
I(\mathrm{~A})
\end{array} & 0.50 & 1.00 & 2.00 & 4.00 \\
V_{a b}(\mathrm{~V}) & 1.94 & 3.88 & 7.76 & 15.52
\end{array}
$$
(a) For each resistor, graph $V_{a b}$ as a function of $I$ and graph the resistance $R=V_{a b} / I$ as a function of $I$. (b) Does resistor $A$ obey Ohm's law? Explain. (c) Does resistor $B$ obey Ohm's law? Explain. (d) What is the power dissipated in $A$ if it is connected to a $4.00 \mathrm{~V}$ battery that has negligible internal resistance? (e) What is the power dissipated in $B$ if it is connected to the battery?

Key Concepts

Ohm's Law

Ohm's Law is a fundamental principle in circuit theory which states that the voltage across a resistor is directly proportional to the current flowing through it (V = IR). This relationship holds true for materials and components that are considered ohmic, meaning that their resistance remains constant regardless of the applied voltage or current within certain limits.

Resistance Determination

Resistance is determined by the ratio of the voltage drop across a resistor to the current passing through it, expressed as R = V/I. Analyzing how this ratio behaves over a range of currents can help identify whether a resistor maintains a constant resistance (ohmic behavior) or varies with current (non-ohmic behavior).

Graphical Analysis in Circuit Theory

Graphical analysis involves plotting voltage versus current and resistance versus current to visually assess the behavior of electrical components. A straight line through the origin in a voltage-current graph indicates proportionality as described by Ohm's Law, while deviations suggest changes in resistance with current.

Power Dissipation in Resistive Components

The power dissipated in a resistor can be calculated using the relationship P = VI or alternatively P = I²R, where V is the voltage, I is the current, and R is the resistance. Understanding power dissipation is important for ensuring that components operate within safe thermal limits and for energy management in electrical systems.

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