New kid on the block:

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This is a source of alternating current. It may sound fancy, and soon you will see that it involves some trigonometric functions like sin and cos, but don't worry about that for now. Now I want you to tell me where you can see one of the above objects in your daily life?

I will give you a hint. It lives on walls. It often comes in pairs, and it has three holes.

Yes. Your wall outlet. The mains. Hydro-Quebec, or whatever your local electric company is called. They give you a two prong “voltage source” that keeps changing. Sometimes the voltage on the + terminal is higher than the - terminal and sometimes the polarity changes. If you were to connect a voltmeter accross the two slots in the wall you would see \[ v(t) = 150\sin\left( 120\pi t \right), \] where $\omega=2\pi f$ is the anglular frequency. I am assuming you are in North America and the $f=60Hz$. The alternating current your power company is sending you is changing polarity 60 times per second.

At first you might think, what's the point of having somethign oscillate like that? I mean, the average electricity user probably just wants to run his computer, or heat their house. What good is this wobbly electrycity that keeps alternating?

The reason while AC is better than DC, is that you can convert AC very easily using a transformer. If you have a 1.5V battery it is quite complicated to to make into a 300V battery, but if you have a 1.5V AC source you can turn it into a 300V AC source simply by using a transformer with 200 times more windings on the output side than the inputs side.

• $i(t)$: Current as a function of time. Measured in Amperes $[A]$.
• $v(t)$: Voltate as a function of time. Measured in Volts [V].
• $R$: The resistance value of some resistor. Measured in Ohms [$\Omega$].
• $\omega=2\pi f$: Angular frequency = the coefficient in front ot $t$ inside $\sin$.
• $f$: Frequency of the AC current/voltages
• $p(t)$: power consumed/produced by some component. Measured in Watts [W].

I want to also give you some intuition about the units we normally see for circuit quantities. The voltage that is used for power transport over long distances is on the order of 50000$[V]$ – this is why they call them high-tension lines. The voltage amplitude of the wall outlets in North America is 150V, but the effective voltage as far as power is concerened is $\frac{1}{\sqrt{2}}$ of the maximum amplitude of the sine wave, which gives: \[ V_{rms}=110 \approx \frac{150}{\sqrt{2}}. \]

On a saftery note, note that it is not high voltage that kills, it is high current.

The basic building blocks of of circuits are called electic components.

The most basic are the following:

• Wire: Can carry any current and has no voltage drop accross it.
• Resistor: Can carry any current $I$, and has a voltage accross its terminals of $V=RI$ where $R$ is the resistance measured in Ohmns [$\Omega$]. The energy of the electrons (the voltage) right before netering the resistor is $IR$ [V] higher than when they leave the resistor. It is important to label the positive and negative terminals of the resistor. The positive temrinal is where the current enters, the negative where the current leaves.
• AC voltage source: Provides you with $v(t) = A\sin(2\pi f t)$ [V]

Then there are the energy storing ones:

• Capacitor:

\[ q(t) = \int i(t) dt = Cv(t), \qquad \Leftrightarrow \qquad \frac{dq(t)}{dt} = i(t) = C \frac{dv(t)}{dt} \]

• Inductors:

\[ \int v(t) dt = Li(t), \qquad \Leftrightarrow \qquad \frac{d\Phi(t)}{dt} = v(t) = L \frac{di(t)}{dt} \]