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# Chapter 17: Circuits

Difficulty Level: At Grade Created by: CK-12

A circuit just means a completed loop.  An electrical circuit can be as simple as a few wires held in place by hand.  As long as the electricity can flow in a loop, the circuit is complete.  This chapter covers electrical current, voltage, resistors and resistivity, resistors in series and parallel, and measuring both voltage and current.

Chapter Outline

### Chapter Summary

1. Electric current is defined as the rate at which charges flow within a conducting wire past any point in the wire the \begin{align*}I \rightarrow I = \frac{\Delta Q}{\Delta t}\end{align*}

2. An ampere \begin{align*}(A)\end{align*} is defined as a \begin{align*}\frac{coulomb}{second}\end{align*}.

3. Georg Simon Ohm first proposed that a current \begin{align*}I\end{align*} was directly proportional to a potential difference \begin{align*}V\end{align*} for metal conductors as long as the temperature of the wire did not increase substantially. Under the same conditions, the current is inversely proportional to the resistance. Combining these results we have

\begin{align*}V = IR\end{align*}, which is often referred to as Ohm’s Law

4. The resistivity is a measure of the resistance of a material. For wires, the resistivity is directly proportional to the length \begin{align*}L\end{align*} of the wire and inversely proportional to the cross-sectional area \begin{align*}A\end{align*} of the wire. The constant of proportionality \begin{align*}\rho\end{align*} is known as the resistivity of the material.

\begin{align*}R = \rho \frac{L}{A}\end{align*}

5. The conductivity \begin{align*}\sigma\end{align*} of a material is the reciprocal of the resistivity

\begin{align*}\sigma = \frac{1}{\rho}\end{align*}

6. The equivalent resistance of resistors in series is

\begin{align*}R_{equivalent} = R_1+R_2+...\end{align*}

7. The equivalent resistance of resistors in parallel is

\begin{align*}\frac{1}{R_{equivalent}} = \frac{1}{R_1}+\frac{1}{R_2}+...\end{align*}

8. In a series circuit all resistors have the same current and in a parallel circuit all resistors have the same potential drop.

9. For any electric circuit the electric power \begin{align*}P\end{align*} is the product of current \begin{align*}I\end{align*} and voltage \begin{align*}V\end{align*}

\begin{align*}P = IV\end{align*}

7. Joule heating is the heat dissipated through a resistor. It is equal to the rate at which energy flows through the resistor.

\begin{align*}P &= IV \rightarrow P = I^2 R\\ P &= IV \rightarrow P = \frac{V^2}{R}\end{align*}

10. Kirchhoff’s laws are two statements useful in analyzing circuits.

1. The sum of the currents entering any junction must equal the sum of the currents leaving any junction.

2. The sum of the potential changes around any closed circuit loop must equal zero.

The first is a statement of charge conservation and the second is a statement of the conservation of energy.

The first is a statement of charge conservation and the second is a statement of the conservation of energy.

11. Ammeters are devices that measure the currents within a circuit. They are placed in series with a component in order to measure the current passing through it and have small resistances.

12. Voltmeters are devices that measure the voltages in a circuit. They must be placed in parallel with a component in order to measure the voltage drop across it and have large resistances.

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