20.3 Resistance and Resistivity

Electrical Resistance and Resistivity

Electrical resistance and resistivity describe how materials oppose the flow of electric current. Resistance is a property of a specific object (like a particular wire), while resistivity is a property of the material itself (like copper in general). Understanding the difference between these two concepts, and how temperature affects both, is central to analyzing circuits.

Resistivity vs Resistance

Resistivity ($\rho$) is an intrinsic property of a material that quantifies how strongly it resists current flow. A copper wire and a copper pipe have the same resistivity because they're made of the same stuff. Resistivity depends only on the material's composition and its temperature, and it's measured in ohm-meters ($\Omega \cdot m$). Resistivity is inversely related to conductivity: good conductors like copper have very low resistivity (about $1.72 \times 10^{-8} \, \Omega \cdot m$), while insulators like rubber have extremely high resistivity.

Resistance ($R$) is the opposition to current flow in a specific object or component. Unlike resistivity, resistance depends on the object's shape and size in addition to the material. A long, thin copper wire has more resistance than a short, thick one, even though both are made of copper. Resistance is measured in ohms ($\Omega$).

Resistance Calculations Using Resistivity

The resistance of a uniform object is calculated with:

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

• $R$ = resistance in ohms ($\Omega$)
• $\rho$ = resistivity of the material in ohm-meters ($\Omega \cdot m$)
• $L$ = length of the object in meters ($m$)
• $A$ = cross-sectional area in square meters ($m^2$)

This formula makes intuitive sense: a longer path (larger $L$) gives charge carriers more material to push through, increasing resistance. A wider cross-section (larger $A$) gives them more room to flow, decreasing resistance.

Finding the cross-sectional area:

• For a cylindrical wire (the most common case): $A = \pi r^2$, where $r$ is the radius. This applies to standard electrical wiring in homes and buildings.
• For a rectangular cross-section: $A = w \times h$, where $w$ is the width and $h$ is the height. This applies to thin-film resistors in electronic circuits.

Example: Suppose you have a 2.0 m length of copper wire ($\rho = 1.72 \times 10^{-8} \, \Omega \cdot m$) with a radius of 0.50 mm ($5.0 \times 10^{-4}$ m).

1. Calculate the area: $A = \pi (5.0 \times 10^{-4})^2 = 7.85 \times 10^{-7} \, m^2$
2. Plug into the formula: $R = (1.72 \times 10^{-8}) \frac{2.0}{7.85 \times 10^{-7}} = 0.044 \, \Omega$

That's a very small resistance, which is why copper is used for wiring.

Combining resistances (a preview of topics covered more fully later):

• Series: Resistances add directly: $R_{total} = R_1 + R_2 + \ldots + R_n$
• Parallel: Reciprocals add: $\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots + \frac{1}{R_n}$

Temperature Effects on Resistance

For most conductors (metals), resistivity increases as temperature rises. This happens because higher temperatures cause atoms in the material to vibrate more, which means charge carriers collide with them more frequently.

The temperature coefficient of resistivity ($\alpha$) quantifies this relationship. It's measured in inverse degrees Celsius ($°C^{-1}$) or inverse kelvins ($K^{-1}$), and it tells you the fractional change in resistivity per degree of temperature change.

The resistivity at a new temperature is:

$\rho_T = \rho_0[1 + \alpha(T - T_0)]$

• $\rho_T$ = resistivity at the new temperature $T$
• $\rho_0$ = resistivity at the reference temperature $T_0$ (usually 20°C)
• $\alpha$ = temperature coefficient of resistivity

Since resistance is directly proportional to resistivity (and assuming the object's dimensions don't change significantly), the same form works for resistance:

$R_T = R_0[1 + \alpha(T - T_0)]$

For most metals, $\alpha$ is positive (resistance goes up with temperature). For example, copper has $\alpha \approx 3.9 \times 10^{-3} \, °C^{-1}$. This property is used in thermistors, which are temperature sensors that measure temperature by detecting changes in resistance.

Electric Current and Voltage

These concepts connect directly to resistance through Ohm's Law.

Electric current ($I$) is the flow of charge carriers (usually electrons) through a conductor, measured in amperes (A). Voltage ($V$) is the electric potential difference between two points, measured in volts (V). Voltage is what drives current through a material; it creates an electric field inside the conductor that pushes charge carriers along.

Ohm's Law ties these together:

$V = IR$

This says the voltage across a resistor equals the current through it multiplied by its resistance. If you know any two of these quantities, you can find the third.

The charge carriers don't actually move very fast through the wire. Their average speed, called the drift velocity, is typically on the order of millimeters per second. What travels fast is the electric field itself, which is why a light turns on almost instantly when you flip the switch, even though individual electrons are barely crawling along.