Electric Circuits

Electromotive Force

Electrical energy is transferred from a power source, such as a battery, to a device, such as a light bulb. Inside a battery, a chemical reaction separates positive and negative charges, creating a potential difference. This potential difference is equivalent to the battery's voltage, or $emf\ (\epsilon)$ (electromotive force, which is a potential, not a force).

Because of the $emf$ on the battery, an electric field is produced within and parallel to the wires. This creates a force on the charges in the wire and moves them around the circuit. This flow of charge in a conductor is called the electrical current ( $I$ ).

Electrical current: $\boxed{I=\dfrac{\Delta q}{\Delta t}}$

units - $\left[ \dfrac{\text{Charge}}{\text{time}} \right]=\left[ \dfrac{C}{s} \right]=[\text{Ampere}]=[A]$

If the current only moves in one direction, like with batteries, it's called direct current ( $DC$ ). If the current moves in both directions, like in a house, it's called alternating current ( $AC$ ).

Electric current is due to the flow of moving electrons, but we will use the positive conventional current in circuit diagrams. $I$ shows the direction of "positive" charge flow from high to low potential.

Ohm's Law

The battery pushing the current acts like a water pump pushing the water through a pipe. The voltage of the battery is like pump pressure - the larger the voltage, the greater the current. $V \varpropto I$ .

Ohm's Law: $\boxed{V=IR}$

$R=$ electrical resistance
units - $R=\left[ \dfrac{\text{Volts}}{\text{Amp}} \right]=\left[ \dfrac{V}{A} \right]=[\text{Ohm}]=[\Omega]$

Resistor: component of an electrical circuit that offers resistance to the flow of electric current.

Resistivity

The electrical resistance of a conductor depends on its shape. Longer wires have more resistance; fatter wires have less. Thus, $R \varpropto \frac{L}{A}$ .

Resistivity: $\boxed{R=\rho \dfrac{L}{A}}$

$\rho=$ electrical resistivity (proportionality constant)
units - $\rho=R\dfrac{A}{L} \left[ \dfrac{\Omega*m^2}{m} \right] \rightarrow [\Omega*m]$

Resistivity is an intrinsic property of materials, like density. Every piece of copper has the same resistivity, but the resistance of any one piece depends on its size and shape.

Temperature Dependence of Resistivity

The resistance of most materials changes with temperature. For good conductors (metals), the resistance decreases with decreasing temperature. For poor conductors (insulators), the resistance increases with decreasing temperature.

Resistance: $\boxed{R=R_0[1+\alpha(T-T_0)]}$

$R=$ resistance at temperature $T$
$R_0=$ resistance at temperature $T_0$
$\alpha=$ temperature coefficient of resistivity ( $\alpha>0$ for metals // $\alpha<0$ for insulators)

Electrical Power

Power: $P=IV=I^2R=\dfrac{V^2}{R}$ .

Electrical Energy

Work and energy have the same units (Joules). So, $\boxed{\text{Energy}=\text{Power}*\text{Time}}$ . Electrical companies measure your monthly energy in this way, using kilowatt*hours ( $kWh$ ).

$\boxed{1\ kWh=3.6*10^6\ J}$

Power Transmission

Electrical power is transmitted at high voltages instead of high currents, because thermal losses (losing electrical power by converting it to heat) is proportional to $I^2$ for a given voltage and proportional to $V$ .

$\boxed{P \varpropto I^2}$ // $\boxed{P \varpropto V}$

Alternating Current

Circuits where the current periodically switches direction use alternating current (AC). The most common type of AC current is one that varies sinusoidally with time.

Voltage in an AC circuit as a function of time: $\boxed{V=V_0\sin 2\pi ft=V-0 \sin \omega t}$ .

$V_0=$ peak (max) voltage
$f=$ frequency (Hz)

Since the voltage in an AC circuit oscillates in time, so does the current. $I=\dfrac{V_0}{R}\sin2\pi ft=I_0 \sin2\pi ft$ , where $I_0=V_0/R$ .

Power in an AC Circuit

Just like in a DC circuit, the power is $P=IV$ , but since both the current and the voltage fluctuate in time, so does the power. $P=IV=(I_0\sin2\pi ft)(V_0\sin2\pi ft)=I_0V_0\sin^22\pi ft$ .

Since the power fluctuates in time, we often just reference the average power in the circuit. $\bar{P}=\frac{1}{2}I_0V_0$ . We can rewrite the equation for this as: $\bar{P}=\left(\dfrac{I_0}{\sqrt{2}}\right) \left(\dfrac{V_0}{\sqrt{2}}\right)=I_{\text{ms}}V_{\text{ms}}$ Unless otherwise stated, the voltage in AC circuits will represent the rms voltage, and the power will represent the average power. $\bar{P}=I_{\text{rms}}V_{\text{rms}}=I_{\text{rms}}^2R=\dfrac{V_{\text{rms}}^2}{R}$

Series Circuits

A series circuit is when the same current runs through two components connected in series. The voltage drop (potential difference) across resistors is different.

Net resistance (equivalent resistance): $\boxed{R_{\text{eq}}=R_1+R_2+\dots+R_n}$

Parallel Circuits

A parallel circuit is when the current is split across different wires. For this type of circuit, the voltage drop (potential difference) is the same.

Net resistance (equivalent resistance): $\boxed{ \dfrac{1}{R_{\text{eq}}}=\dfrac{1}{R_1}+\dfrac{1}{R_2}+\dots+\dfrac{1}{R_n} }$

Series and Parallel Circuits

If resistors are hooked up both in series and in parallel in the same circuit, you must combine resistors to find $R_{\text{eq}}$ .

Internal Resistance

Batteries and generators have some resistance as well. This is called internal resistance ( $r$ ). In batteries it's due to the chemicals, and in generators it's due to the wire resistance. The voltage across the battery (terminal voltage, $V_T$ ) is less than the full voltage $V$ , since some is lost across $r$ .

Terminal voltage: $\boxed{V_T=V-V_r}$

Kirchhoff's Rules

In many circuits, applying the series or parallel methods is not sufficient to analyze them. There are two other rules we can use called Kirchhoff's Rules.

1. Junction Rule

Current into a junction has to equal current out, based on conservation of charge.

2. Loop Rule

Around any closed circuit loop, the sum of the potential difference has to equal the sum of the potential rises. (This is based on conservation of energy.)

Measuring Voltages and Currents

Currents and voltages can be measured with devices called ammeters and voltmeters, respectively. Both these devices rely on the DC galvanometer.

Ammeter Resistance

(A circuit must be "broken" to use an ammeter - it needs to be inserted inside the circuit.)

Inside an ammeter, a small resistor ( $r_s$ ) is wired in parallel with the galvanometer. Almost all of the current passes through the shunt resistor, so the ammeter has very little effect on the circuit, i.e. nearly zero resistance.

Voltmeter

Voltmeter: measures the potential difference between two points in the circuit, i.e. across a resistor. It doesn't have to be inserted in the circuit. A voltmeter has a high resistance, so very little current actually flows through the voltmeter.

Capacitors in Series and Parallel

Capacitors connected in parallel: $\boxed{ C_{\text{eq}}=C_1+C_2+\dots+C_n }$ Capacitors connected in series: $\boxed{ \dfrac{1}{C_{\text{eq}}}=\dfrac{1}{C_1}+\dfrac{1}{C_2}+\dots+\dfrac{1}{C_n} }$

RC Circuits

RC Circuit: a resistor is in series with a capacitor. When the switch closes, charge builds up gradually on the capacitor plates as the current flows and approaches an equilibrium value ( $Q_0$ ). If the capacitor is initially uncharged at $t=0$ , then the charge on the capacitor at some later time $t$ is $\boxed{Q=Q_0\left[1-e^{-t/RC}\right]}$ .

If $t=0$ , then $Q=Q_0\left[1-e^{0/RC}\right]=Q_0[1-1]=0$ . We can get the voltage across the capacitor at any time by dividing by the charge by the capacitance, since $V=Q/C$ : $\boxed{V=V_0\left[1-e^{-t/RC}\right]}$ .

RC

units - $[\text{Resistance}*\text{Capacitance}]=[\Omega*F]=\left[ \left(\dfrac{V*s}{Q}\right) \left(\dfrac{Q}{V}\right) \right]=[\text{s}]$

$RC$ has the units of time, and the product is known as the time constant of the circuit. $\boxed{\tau=RC}$

Charging equation: $\boxed{Q=Q_0\left[1-e^{-t/\tau}\right]}$

Discharging a Capacitor in an RC Circuit

A capacitor is charged all the way to $Q_0$ and then discharged through the resistor. When we close the switch, a current flows through the resistor and charge dissipates from the capacitor plates.

Discharging equation: $\boxed{Q=Q_0e^{-t/\tau}}$ . At $t=0$ , $Q=Q_0e^{0/\tau}=Q_0$ .