11.3 Resistance, Resistivity, and Ohm's Law

Resistance, resistivity, and Ohm's Law form the foundation of electrical circuit analysis. These concepts help us understand how current flows through materials and how voltage and resistance are related.

Resistance is a measure of how strongly an object opposes the motion of electric charge through it. Ohm's Law states that voltage equals current times resistance (V = IR). This simple equation is crucial for predicting circuit behavior and designing electrical systems. Understanding these principles is essential for tackling more complex electrical problems.

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Resistance from Physical Properties

For an object with uniform geometry, resistance depends on both the material and the object's dimensions:

$R=\frac{\rho \ell}{A}$

This means resistance increases when the resistivity $\rho$ increases, increases when the length $\ell$ increases, and decreases when the cross-sectional area $A$ increases. Physically, a longer path gives charges more material to move through, while a larger area provides more room for charge to flow.

• Resistivity ($\rho$) is a fundamental property of a material that depends on its atomic and molecular structure and quantifies how strongly the material opposes the motion of electric charge.
  • Resistivity is measured in ohm-meters (Ω·m)
  • For conductors, resistivity typically increases as temperature increases, so the resistance of a metal wire usually increases when it gets hotter.
• If the resistor has uniform cross-sectional area $A$ but the resistivity varies along its length, the total resistance is found by summing small pieces:

$R = \int \frac{\rho(\ell)\, d\ell}{A}$

Electrical Characteristics of Circuit Elements

Ohm's Law in Circuits

Ohm's Law is a fundamental principle that describes the relationship between voltage, current, and resistance in electrical circuits. It states that the current flowing through a conductor is directly proportional to the voltage applied across it, with resistance being the proportionality constant.

• The mathematical expression of Ohm's Law is $V = IR$, where:
  • $V$ is the potential difference (voltage) measured in volts (V)
  • $I$ is the current measured in amperes (A)
  • $R$ is the resistance measured in ohms (Ω)
• Equivalently, Ohm's law can be written as $I=\frac{\Delta V}{R}$, where $\Delta V$ is the potential difference across the circuit element.
• Ohmic materials maintain a constant resistance regardless of the current passing through them
  • For an ohmic material, current is proportional to potential difference, so the material has constant resistance and a constant resistivity. Under the AP Physics C model for this topic, the resistivity of an ohmic material is treated as constant regardless of temperature.
  • Their current-voltage ($I$-$V$) graph is a straight line passing through the origin
  • Examples include most metals over limited temperature ranges
• Non-ohmic materials have resistance that varies with current or voltage
  • Their $I$-$V$ graph is non-linear
  • Examples include diodes, transistors, and light bulbs (when temperature changes)
• The resistance of an ohmic circuit element can be determined from the slope of a graph of current as a function of potential difference:
  • On a graph of current $I$ as a function of potential difference $\Delta V$ for an ohmic element, the slope is $\frac{\Delta I}{\Delta V} = \frac{1}{R}$. Therefore, the resistance is the reciprocal of the slope: $R = \frac{1}{(\Delta I / \Delta V)}$.
  • A steeper slope on an $I$ vs. $\Delta V$ graph means lower resistance.
  • Equivalently, if voltage is graphed as a function of current (a $V$ vs. $I$ graph), the slope is $\frac{\Delta V}{\Delta I} = R$.

Experimental Determination of Resistance

To determine whether a circuit element is ohmic and to find its resistance, vary the potential difference across the element, measure the current for several trials, and plot current $I$ (in amperes) versus potential difference $\Delta V$ (in volts).

• If the graph is linear and passes through the origin, the element is ohmic under those conditions.
• The slope of an $I$ versus $\Delta V$ graph is $\frac{1}{R}$, so the resistance is the reciprocal of the slope.
• To make the conclusion reliable, keep physical conditions such as temperature as constant as possible while collecting the data.
• Resistors convert electrical energy to thermal energy (heat)
  • This energy conversion follows Joule's heating law: $P = I^2R = \frac{V^2}{R} = VI$
  • This heating can change the temperature of the resistor and its surroundings
  • Example: A light bulb filament heats up and glows when current flows through it

Practice Problem 1: Ohm's Law Application

Solution

To find the resistance, we can apply Ohm's Law: $V = IR$

Rearranging to solve for resistance: $R = \frac{V}{I} = \frac{12\text{ V}}{2\text{ A}} = 6\text{ Ω}$

When the length of the resistor is doubled while keeping the same material and cross-sectional area, we can use the relationship $R = \rho \frac{L}{A}$:

• The original resistance is $R_1 = \rho \frac{L_1}{A}$
• The new resistance is $R_2 = \rho \frac{L_2}{A} = \rho \frac{2L_1}{A} = 2 \times \rho \frac{L_1}{A} = 2R_1$

So the resistance doubles to 12 Ω.

Using Ohm's Law again with the new resistance: $I_2 = \frac{V}{R_2} = \frac{12\text{ V}}{12\text{ Ω}} = 1\text{ A}$

Therefore, when the length doubles, the resistance doubles and the current is reduced by half.

Practice Problem 2: I-V Graph Analysis

Solution

To determine if this is an ohmic material, we need to check if the relationship between voltage and current is linear.

Let's calculate the resistance at each data point using $R = \frac{V}{I}$:

• At (2V, 0.4A): $R = \frac{2\text{ V}}{0.4\text{ A}} = 5\text{ Ω}$
• At (4V, 0.8A): $R = \frac{4\text{ V}}{0.8\text{ A}} = 5\text{ Ω}$
• At (6V, 1.2A): $R = \frac{6\text{ V}}{1.2\text{ A}} = 5\text{ Ω}$
• At (8V, 1.6A): $R = \frac{8\text{ V}}{1.6\text{ A}} = 5\text{ Ω}$

Since the resistance is constant (5 Ω) at all voltage/current values, and the I-V relationship is linear (current doubles when voltage doubles), this is an ohmic material.

The resistance of this circuit element is 5 Ω. On a graph of $I$ vs. $\Delta V$, the slope is $\frac{\Delta I}{\Delta V} = \frac{1.6\text{ A} - 0.4\text{ A}}{8\text{ V} - 2\text{ V}} = \frac{1.2\text{ A}}{6\text{ V}} = 0.2\text{ A/V}$. Since the slope equals $\frac{1}{R}$, the resistance is $R = \frac{1}{0.2\text{ A/V}} = 5\text{ Ω}$.