﻿ Resistance and Ohm's Law
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ECE 110

Course Notes

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A resistor resists current because flowing electrons collide with atoms in the resistor and make them vibrate. In this way, electrical energy in converted into heat energy. The amount by which a resistor resists the flow of charge is quantified as resistance $R$. Resistance is measured in ohms, with symbol $\Omega$.

Figure 1

Fig. 1: Two resistors. When you use resistors in lab, it is good practice to measure their resistances using a multimeter. You can also read off nominal values for their resistances from their color-coded bands. For the resistors shown here, the first three bands indicate the resistance in ohms and the fourth band indicates the tolerance of the actual resistance with respect to the nominal value. These resistors have nominal resistance $\text{330 }\Omega$ (top) and $\text{10 k}\Omega$ (bottom), both with tolerance of $\pm 5\%$. You can verify these values by learning how to read resistor color-coding.

Figure 2

Fig. 2: The symbol for a resistor. The resistor is denoted by the zigzag line. Its label $R$ is the value of its resistance. The voltage drop across $R$ is labeled as $V$ and the current through $R$ is labeled as $I$. Our convention in ECE 110 is to draw the current arrow label from the $+$ label to the $-$ label of the voltage drop. We also label resistors $R_1, R_2, R_3, \ldots$ with voltage drops $V_1, V_2, V_3, \ldots$ and currents $I_1, I_2, I_3, \ldots$, respectively.

Ohm's law is the experimental observation that the voltage drop across a resistor is proportional to the current flowing from the $+$ label to the $-$ label of the voltage drop. In fact, the resistance is defined as ratio of the voltage drop to the current. Using the labeling in Fig. 2, we obtain the following equivalent statements of Ohm's law:

\begin{align}
V &= IR \label{RES-OHV}\\
I &= \frac{V}{R} \label{RES-OHI}
\end{align}

Suppose we reverse the current arrow label in Fig. 2.

Figure 3

Fig. 3: A resistor with voltage drop and current labeled against the usual convention. This labeling is just as valid as the one in Fig. 2, but equations $\eqref{RES-OHV}$ and $\eqref{RES-OHI}$ are not true for this labeling.

To compensate for the reversal of the current arrow label in Fig. 3, Ohm's law in equations $\eqref{RES-OHV}$ and $\eqref{RES-OHI}$ must be restated as follows:

\begin{align}
V &= -IR \label{RES-NOV}\\
I &= -\frac{V}{R} \label{RES-NOI}
\end{align}

Labelings Labeling System Ohm's Law
Standard labeling
"arrow goes from $+$ to $-$"
\begin{aligned} V &= IR \\I &= \frac{V}{R} \end{aligned}
Nonstandard labeling
"arrow goes from $-$ to $+$"
\begin{aligned} V &= -IR \\I &= -\frac{V}{R} \end{aligned}
Table 1: Different labelings of voltage drop polarity and current direction for a resistor.

If the current arrow label starts the $+$ label of the voltage drop and ends at the $-$ label, then Ohm's law is the same as equations $\eqref{RES-OHV}$ and $\eqref{RES-OHI}$ and we say that the labeling follows the standard labeling convention. Otherwise, Ohm's law is the same as equations $\eqref{RES-NOV}$ and $\eqref{RES-NOI}$ and the labeling is nonstandard.

For simplicity, we try to use standard labeling as much as possible but nonstandard labeling is sometimes unavoidable. Later in ECE 110 we will discuss connecting two subcircuits together and it will turn out that one will be labeled in the standard way and the other in the nonstandard way.

The cables and wires you use in lab are typically made of copper, one of the best conductors of electricity. The resistance of a wire is given by:

\begin{align}
R &= \rho\frac{l}{A} \label{RES-WIR}
\end{align}

where

\begin{align}
\rho &= \text{resistivity of the conductor material (}\Omega\cdot\text{m)}\\
l &= \text{length of the wire (m)}\\
A &= \text{cross-sectional area of the wire (m}^2\text{)}
\end{align}

Since copper has a low resistivity $\rho_{copper}=1.7\times 10^{-8} \Omega\cdot$m, the resistance of short lengths of wire (such as the ones in lab or even the electric wiring in a building) is usually negligible compared to the resistance of other components. Therefore, we generally ignore the resistance of wire.

The exceptions to this rule are situations in which wires traverse geographical distances; for example, power lines and telegraph cables. In ECE 329: Fields and Waves I, you will learn how to model losses and time delay in these kinds of transmission lines.

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