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 Circuit Lab (Episodes) & Event Forum Threads 2010 2009 Tests There are no images available for this event There are no question marathons for this event This event was not held recently in Division B This event was not held recently in Division C

## Introduction

Continuing with my trend of the two previous help threads, (which will be linked at the bottom for anyone interested) I've decided to write a new shocking guide. A guide to that energizing event, Circuit Lab. If you were at all like me when you got these rules, you might have screamed in shock. But fear not, my fellow shocked-in-the-fingers-friends, for I've prepared a few notes, definitions, and various helpful items that hopefully won't short circuit (I'll explain that later) your brain. -Demosthenes

## Episode I

First of all, let's answer some questions for the completely new:

-What is a 'circuit'?

Let's take an example of a battery, for now. The battery has a positive (+) end, and a minus ( - ) end. When you touch a wire onto both ends of the battery at the same time, you have created a circuit. What just happened? Current flowed from one end of the battery to the other through your wire. Therefore, our definition of circuit can simply be a never-ending looped pathway for electrons (the battery counts as a pathway!).

-But what's current? What are all these positive and minus things?

This is the next thing we should understand: electron flow. What is an "electron?" To put it simply, an electron is an atomic particle which carries a negative charge. These electrons spin around the nucleus of an atom, which has a positive charge, and is located in the very center of the atom. The concept of "electricity" has to do with these electrons and with their "electron flow." Do you remember the example of our battery? This battery takes these negatively charged electrons from a chemical reaction inside the battery, pushes them out of the negative end of the battery, and into the wire. These electrons will then bump electrons in the atoms of the wire over and over until finally electrons arrive back at the positive end of the battery. Elements which allow this process of "bumping" those electrons on over determines how conductive the element is. So, when there's a current, it's just electrons bumping each other from atom to atom and flowing on.

-Oh, I get it; you could just put a wire onto one end of a battery, and the electrons would still bump each other?

No, you could not. As stated before, in our definition of the circuit, a continuous loop is required. But think about it scientifically: If you did attach the wire to only one end of the battery, where would the electrons go that got bumped to the opposite end of the wire? That is why there needs to be that continuous loop of wire: the electrons need somewhere to go.

-Okay, I gotcha! I know about the flowing thing now, but what about all that "voltage," "resistance," and "ampere," stuff they're going to ask me on the circuit lab tests?

To help everyone understand these 3 concepts, I'm going to use an example from sports.

Imagine you are the coach of the New York Yankees baseball team. Aside from dealing with being the Devil on Earth, you also have to deal with keeping a winning team amidst the ever-so-polite New York fans and cutthroat media. You want to make your team the best it can be by focusing on 2 big things: scoring runs, and preventing runs. If you can do both of these, you'll be an awesome team, but no one is perfect. So naturally, you're going to have to choose one area to focus on.

Say you want to score more runs; let's relate this to the concept of "amperes." The amount of runs you make is your score - the more you get the better your chance of winning. Similarly, amperes measure the amount of current you have flowing per second through an area: is it a lot, or a little bit?

Now, if you want to win the game, you don't necessarily have to score a whole lot of runs, you just need to score more than your opponent. So, maybe your resistance to their scoring of runs will be high - and resistance to current flowing is also one of our important terms we need to know.

Now, how do these concepts of amperes and resistance relate? If you multiply the resistance by the amperes, you have the voltage of a circuit (remember, we're always talking about circuits here, not on a baseball field). This relationship was discovered by Georg Simon Ohm, and it says, simply, that:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): V = I \times R

Or

Voltage = Current times Resistance

If you are having trouble, think back to the baseball example: you can have a high chance of winning (voltage) by either scoring a lot of runs (high current) or having good defense/pitching (resistance).

-But I was the kid in Little League who kicked dirt in the outfield, and I just don't understand 'voltage'...

As for voltage, it's definitely the hardest of the three concepts to understand. Some really smart guys call it "potential," other people use analogies of a water tank. The links at the bottom will explain all this juicy stuff to you: I, again, will develop my own analogy (I won't grill the Yankees this time). Have you ever pumped up a super-soaker in order to blast your little brother? If so, you'll be pleased to understand the next example.

The harder you pump that super-soaker, the harder that stream is going to be when it comes out of the gun. You can think of voltage like that. Voltage is the potential for that water to go very quickly out of the gun: the more you pumped, putting more "voltage" in, the faster that water will go: but sometimes you will have a "multi-functioning" nozzle which even allows you to adjust that water speed even further. Suppose you're new at super-soakers and you don't have a steady arm to hold the gun, so you want the water to go out in a "wider" and "bigger" stream, you might change the nozzle to a bigger opening. What you've just done is changed the amount of space that the water is allowed to go through: the water is now given a much bigger space to flow through. The "voltage," or potential, of the water to go fast and give bruises is still high, but now you've taken away from it's hitting-power by spreading it out. Anyone see where this is going? The bigger your nozzle gets (think of it like the resistance), the smaller the hitting power (current (which is a speed in electricity too!)) is going to be.

-Wow, the water example makes it easier to understand. So it's like if I can get one of those huge guns, with more "voltage," I might be able to get a lot more "amps" out of it (how hard it hits).

If anyone is at all confused by these examples, it would be a good idea to read the links placed throughout the guide, in addition to the links at the bottom of this episode. Note that the analogies given are not perfect and have logical flaws, but I'm just trying to put a model for the relationship of voltage, amps, and current into your heads. If you already understand circuitry, this probably didn't help too much. I'll make a guide for you next, but first thing's first.

1. Solving Resistor Circuits
2. Extensive Site about Circuits
3. "Really Basic Electronics"
4. Analysis of resistive circuits (this link is slightly more advanced)

## Episode II

A little review, perhaps:

The three concepts of Voltage, resistance, and current are all interrelated through this basic formula:

V = I R

or Voltage = Current times Resistance

Okay, whatever Demosthenes, I don't really care, why do I have to learn this formula anyways? Like I care about this 'omg' guy or whoever he is.

You have to learn Ohm's law because it helps you to 'analyze' circuits. That means you can use this law to find voltage, resistance, or current, if you have two of the three. Let's look at how to apply this formula:

The application of this formula is pretty easy, once you get the hang of it. Basically, imagine a wire, a battery, and a resistor somewhere along the wire. If the battery has a voltage of 10 volts, and the resistor has a resistor of 30 ohms, you simply use Ohm's law to find the current:

V= IR .....write your equation, so you know what you're doing here...

10 volts = (I)(30 ohms) ...Set up the equation, plugging in the values....

(10 volts)/(30 ohms) = I ...Divide both sides by '30 ohms' so that you can isolate the variable, I, or the current....

10 volts/30 ohms = 1/3 amperes do out the math....fun!...

But wait Demosthenes, what if they ask for voltage or resistance?

Don't get scared, my young padawan. The equation can be set up so that no matter which two of the three variables you know, you can figure out the other one easily. Suppose there's a circuit with a 6 volt battery and 2 amps of current, how would you set that up? What's your answer? (You try it first, and see if it agrees with mine!!)

Alright, let's see how you did:

V = IR ....okay, first write out the equation so you know what you're doing

R = (V)/(I) .....Manipulate the equation so you have the two knowns on one side

R = 6 volts / 2 amps ....Plug in the values

R = 3 ohms ....solve by dividing 6 by 2.

Here are the 3 general forms of the law you'll need to know:

• V = IR
• R = V/I
• I = V/R

Whichever value you're searching for, simply make that the 'lone' variable, then plug in the values, and see what you get. Pretty simple.

There's some nice little quiz questions at the bottom of the page in the following link which you can test yourself further with...

## Episode III

Let's consider the 'series' and 'parallel' concepts that were mentioned, because they are vitally important.

We have gone over how to calculate Ohm's law in basic circuits, with one resistor, but suppose we have a circuit with multiple resistors. How would the calculation of resistance work?

This is a very important question - because no circuit you'll encounter is going to have just one component. Before we figure out how to calculate, however, we need to look into the concept of components in 'series' and components in 'parallel.'

I have no visual way of showing you how these circuits look on here, so I will refer you to a website for all of the diagrams. Series + Parallel Circuits

Okay, now, looking at this website, we quickly see two diagrams. The first is a series circuit - all of the components are connected in one line, one direct path, from one end of the battery to the other.

The second circuit is a parallel circuit - there are different ways for the current to go from one end of the battery to the other.

Understanding these two things is crucial - because we must recognize whether or not components are 'in series' or 'in parallel' with each other in order to make the correct mathematical calculations. I suggest to all reading this chapter - it is ON POINT with what you need to know for Circuit Lab.

I'll assume that the earlier concepts are understood, and move on to the mathematics part.

To calculate the total resistance of two resistors in series is quite easy - you simply add the resistance of all the resistors in series, and you get the total resistance.

For parallel circuits, however, you can find the inverse of the total resistance by adding the inverse of the resistors in parallel together. This is shown by this formula:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (1/R_T)=(1/R_1)+(1/R_2)+(1/R_N...)

From this formula, we can solve for a direct formula for resistance in parallel.

$total} = \frac {1}{\frac{1}{R_{1}} + \frac{1}{R_{2}}.... + \frac{1}{R_{n}}$

It sounds confusing here, yes, but simply look through this chapter and it will become more clear to you.

Here are some questions for you, from me, and if you want, you can email me your answers and I will check them for you:

1. What is the total resistance of two 3 ohm resistors which are in parallel? What would the current be in this circuit if there is a 6 volt battery?

2. What is the total resistance of three 2 ohm resistors which are in series? What would the voltage of the power source be if the current is 3 amps?

3. What is the total resistance of two 1 ohm resistors in series, and two 2 ohm resistors in parallel? What would the current flow be if there was a 12 volt battery powering the circuit?

## Episode IV

Okay, we've been talking a lot about resistors - but we don't even know what one looks like yet. The Mighty Resistor

There you have it - pretty simple eh? It's just a piece of metal, and the piece in the center there is what provides the resistance.

But if you think about resistance - remember we said it was the force against the flow of the electrons - you must realize an important concept.

Resistors release heat....Don't worry, I'll explain.

Imagine our electrons - merrily flowing along the wire, pushing new electrons to flow on, and so on. This wire is not very hard to flow in - it's made of a material that's very conductive. But what would happen if we placed something in the middle of the wire that was harder for the electrons to flow through? They're going to be bumping into all the atoms in the material, which will cause the atoms to vibrate. This, in turn, will cause nearby air molecules to take some energy.

That energy is in the form of heat. Where did it come from again? From the electrons bumping into atoms inside the resistor.

But, if you think about this even further, wires are matter too - they have atoms. So you can't say that these are perfect conductors either - because they aren't.

Have you ever wondered why the US government wouldn't just put a bunch of solar panels out in New Mexico, Nevada, and Arizona, then ship the electricity everywhere and make it cheaper for us all? The reason is that electrons can't flow in wires for a really long distance without losing a lot of energy. There's just too many atoms along the way; the amount of energy lost is going to be huge.

That's something very important to realize - and the rules sheet also tells us to pay attention to that.

Despite this, during our calculations with Ohm's Law, we usually disregard wire resistance, because in small circuits the amount of heat energy lost is negligible. However, it's still important to consider.

And what about batteries, the sources of the power which are required for our lovely event? Batteries! *cue angel choir music*

Believe it or not - these aren't perfectly efficient either (what kind of world is this?)! The resistance in batteries is also a topic to understand for Circuit Lab. First, let's examine what a battery actually is.

A battery works by producing an excess of electrons through a kind of chemical reaction known as a 'redox' reaction. Basically, without jargon - you have some chemicals inside the battery, they react together, and their reaction creates electrons (we're not concerned with chemistry here, but circuitry). The resistance in a battery comes through the chemicals' ability to react smoothly, and the 'electrode's' ability to get the electrons out smoothly.

In a newer battery, this is not a problem. It is just with older batteries that we find serious internal resistance problems.

These explanations were not meant to have you become the master of science - I just wanted to touch upon some things, because the event rules do. These are some of the more advanced things in the rules - and I'm not looking to explain that in this guide. For further explanations, see the following websites:

## Episode V

Voltage Drops and POWER!

Okay guys, we should all have a firm grasp on our three basic concepts - V, I, and R. Now let's talk about some things that happen in a circuit with these numbers.

Take a look at this circuit - the voltage of the battery is 9 volts, the resistor has a resistance of 100 ohms, so by Ohm's Law, the people who made the picture know that...

9 volts / 100 ohms = A

A = .09 amps

The thing to understand about current - it is ALWAYS the same everywhere you go in a circuit. Before the resistor -- .09 amps. After the resistor -- .09 amps. In Canada -- .09 amps. Always the same!

However, voltage at different points in a circuit is NOT the same. Remember that we said voltage is a measure of 'potential energy,' think of it as the amount of push is behind the electrons to push them forward. So there's 9 volts of push before the resistor - the battery is giving those electrons a real shove.

But, now you have a resistor in the way - it's like trying to bike up a hill. It was easy at first, you were giving the same amount of push to go at, say 10 mph, but to continue to go that speed (think of it as amps), you need to increase the effort. You're going to be tired coming off the hill.

Now, relate that back to voltage drops with resistors - as the current goes past a resistor, it has LESS potential to push it along, because some was lost in going through the resistor.

Just the exact amount of voltage lost can be calculated using Ohm's Law: just take the current at the resistor, and the resistance of each resistor, multiply them, and you have the voltage drop.

Take this example and I will help you through it.

There are three resistors, each with different resistance values. If you remember our rule about adding series resistors, you can just add all three values together to the get the resistance over the whole circuit. Now, you've added these values, and you know the voltage for the whole circuit, so using Ohm's Law, find the current by dividing voltage by resistance.

Next, use the rule as I stated above to find the voltage drop for each resistor (if you did it right, it should add up to 45 volts).

Now what about POWER (I capitalized it just because it looks cooler that way)?

Well, power's a pretty easy concept : to calculate the power of a circuit, just multiply the voltage and the current.

P = IV

## Episode VI

A lot of people have state competitions coming up, so I'm going to broaden things up, make sure everything is covered, and definitely go over all the things we've learned in a big Circuit Lab review sheet!

Circuit Lab Review Sheet

Electron Flow = Electricity is really just electrons flowing from one atom to another in a long chain of molecules in a material. Some materials allow this easily (conductors) and other materials don't really allow this (insulators). Some materials can function both ways, such as Silicon (semi-conductors).

Voltage [(E) or (V)]= A measure of 'potential energy' in electrical circuits. This is a measure of how much 'push' is behind the electrons to slide from atom to atom.

Current [(I) or (A)] = The actual measure of how many electrons are whizzing by a certain spot per second. Basically, this tells you how much flowing is going on in a circuit.

Resistance (R)= Resistance is the force that tries to slow down current, such as friction tries to slow a car down when it travels.

Ohm's Law - V=IR. This is by far the most useful equation that you will need in this event. Using this formula, you will be able to derive one value if you know the other two.

Power (P) = Power can be calculated by multiplying voltage (V) and Current (I) together. In other words, P=IV. This calculation is useful for telling you, for instance, how much heat is dissipated by a resistor (that means power can tell you how much heat is created by the 'electrical friction' of resistance).

Series Components - This connections are when a component in a circuit has a DIRECT link to the other component. If you were to take your finger and move it along the line in the drawing of the circuit, there should be only ONE way to go if the components are in series.

Parallel Components - These components are essentially 'branches' from one main power line. To find out whether or not components are in parallel, try starting from the battery, tracing along the circuit. Are there multiple paths you can take in the circuit? This will give you some parallel components.

"Conventional Current Flow" vs. "Electron Flow" - This has to do with how circuit diagrams are interpreted. Now, remember we said that electrons are 'flowing' in the wires? The question here deals with : Do they 'flow' from the positive end of the battery, or the negative end of the battery?

Conventional current flow, devised by Benjamin Franklin, has the moving particles (later called electrons) positively charged. Therefore, this concept holds that electrons flowed out of the positive end of the battery. Electron flow, on the other hand, deals with the ACTUAL route of the electrons - being negatively charged particles, they go through the negative end of the battery! They then flow around the whole circuit, la la la, and arrive back at the positive end. Capeesh?

RC Time Constant - Don't get scared by this - RC stands for 'resistor-capacitor.' What this value is is how long it takes, in seconds, for a capacitor to be charged to 63.2 percent full charge OR 36.8% of its initial voltage. Don't get scared by this - just know that a capacitor can store charge, and you know what a resistor is. They're not going to slam you on this, just be familiar with what that term means.

Diode - this is a circuit component, and basically it's an electrical gate. It allows current to flow one way through it, but not the other way. An example is the Light Emitting Diode (LED).

Solar Cells - These use beams of light to create electricity (You may notice one on the calculator you're using to do your Ohm's Law calculations). The most common kind is the 'photovoltaic cell,' and basically these use semi-conductors to generate a current flow out of some spare energy hitting them. You might look into 'n-type silicon' and 'p-type' silicon for further reading on this concept...

DC motors - DC motors are little motors that take an electric current and spin really fast. Some can spin as fast as 8000 revolutions per minute! They spin because electricity flowing in the motor creates a magnetic field which pushes the motor output in spins really quickly.

Multimeters, Voltimeters, Ammeters, etc - These are all devices used to measure values in a circuit such as 'voltage' 'amperage' 'resistance' 'capacitance' etc. Try to get your hands on one of these and familiarize yourself with how it works

Resistance Color Code on Resistors - There's a table you have to memorize to help you find out how much a resistor is worth in ohms of resistance....

I'd say if your state competition is coming up, memorize those terms up there, familiarize yourself with all the SI electrical units, and familiarize yourself with a multimeter. If you do all that stuff you should be able to get through most of any test!

## Episode VII

Having trouble with capacitors?

Think about them this way: a capacitor is just something that 'holds a charge.'

If you look on online guides for capacitors, they might say something like 'a capacitor is two metal plates that use electrostatic fields to store a charge of electricity.'

What the heck?:-?

I am going to clarify this for you. First of all, look at this movie to get an idea of what a capacitor does.

I will take you through, step by step, what happens in that movie.

1. At the beginning of the movie, the switch has not been thrown, there is no charge anywhere in the circuit except in the battery. It can't flow out of the battery because it has no where to go! It will take the closing of the switch for it to flow somewhere.

2.Immediately when the switch is thrown, you see the light bulb come on full blast. The current runs through that light bulb, whose filament is a RESISTOR (this ties back to the RC constant we talked about earlier). All of this current wants to go somewhere: that place ends up being the capacitor. This is called the 'charging' of a capacitor.

--But Demosthenes, what happens to the current when it 'goes to the capacitor?' Well, good question. You might remember that electrons are negatively charged, right? So, wherever there is an abundance of them, there will be a negative charge. What a capacitor does is take a whole lot of electrons onto one of it's sides, building up a big electric charge there, due to its' relationship to the battery (the end of the capacitor that's connected to the negative end of the battery will be negative!)

This negative charge building up on the capacitor can NOT go across the capacitor - don't be confused about this. The electrons literally just sit there - they are STORED there. That is the beauty of capacitors: using the attraction of the negative plate and the positive plate, the electrons can literally stay where they are for a very long time.

3. This ability for capacitors to store charge is seen as the switch is closed. You might ask 'well, Demosthenes, how does the light bulb continue to light up even after the battery is removed from the circuit?' Good question, once again, my rhetorical friend.

When the switch is flipped, the only two things to be considered in the new circuit are the capacitor and the light bulb, in series with one another. This capacitor stored a charge from earlier on - the electrons are still sitting there. They are waiting for a chance, once the 'pressure' of the voltage of the battery is gone, to flow back to the other side of the capacitor and make everything neutral and happy. (Note: the reason the electrons stayed on the plate before is because the battery was PUSHING them there. The capacitor was fully charged when the battery didn't have enough push to put even more electrons on the negative plate).

So, what you have in this third step is electrons rushing off the negative plate, through the light bulb, and onto the positive plate. That is called the 'discharging' of a capacitor.

So, now, let's get back to the RC time constant. When the capacitor is either 63.2 percent fully charged, or 36.8 percent discharged, we say that this is 'one rc'.

Don't get scared by the 'one rc constant' language. A constant is just a number - and RC refers to resistance times capitance. RC = t! remember that! this is a time value, which is different for each capacitor (which has a different C value) and a different resistor (which has a different resistance!).

That video is so telling. Everyone who is confused must watch it multiple times. The best way to learn a hard concept like this is to see it in action : and this video is very good for showing this concept of capacitors.

Good luck on capacitors, all!

Diodes:

As covered before, diodes are simply one way gates that allow current flow in one direction. In circuit diagrams they will appear as an triangle, with one point pointing in the direction of current flow (remember that current "flows" from positive to negative, but electrons from negative to positive)

So, given a simple circuit with a resistor, a power supply, and a diode, it is easy to tell if current flows through a diode or not.

But what about more complex circuits?

Well, what I like to do is to start out treating every diode as a conductor, thus, a short circuit. Then figure out the direction of circuit flow at each spot there is a diode. Every time you have current flowing in the opposite direction as a diode, start over with that diode as an open switch.

Light Bulbs

Well A light bulb is a simple device consisting of a filament resting upon or somehow attached to two wires. The wires and filament are conducting materials which allow charge to flow through them. One wire is connected to the ribbed sides of the bulb and the other is connected to the bottom of the base of the bulb. The ribbed edge and the bottom base are separated by an insulating material which prevents the direct flow of charge between the bottom base and the ribbed edge. The only pathway by which charge can make it from the ribbed edge to the bottom base or vice versa is the pathway which includes the wires and the filament. Charge can either enter the ribbed edge, make the pathway through the filament and exit out the bottom base; or it can enter the bottom base, make the pathway through the filament and exit out the ribbed edge. As such, there are two possible entry points and two corresponding exit points.

The Requirement of a Closed Conducting Path

There are two requirements which must be met to establish an electric circuit. The first is clearly demonstrated by the above activity. There must be a closed conducting path which extends from the positive terminal to the negative terminal. It is not enough that there is a closed connecting loop; the loop itself must extend from the positive terminal to the negative terminal of the electrochemical cell. An electric circuit is like a water circuit at a water park. The flow of charge through the wires is similar to the flow of water through the pipes and along the slides of the water park. If a pipe gets plugged or broken such that water cannot make a complete path through the circuit, then the flow of water will soon cease. In an electric circuit, all connections must be made and made by conducting materials capable of carrying charge. Metallic materials are conductors and can be inserted into the circuit to successfully light the bulb. There must be a closed conducting loop from the positive to the negative terminal in order to establish a circuit and to have a current.

The Requirement of an Energy Supply

The second requirement of an electric circuit that is common is that there must be an electric potential difference across the two ends of the circuit. This is most commonly established by the use of an electrochemical cell, a pack of cells, or some other energy source. It is essential that there is some source of energy capable of increasing the electric potential energy of a charge as it moves from the low energy terminal to the high energy terminal. As applied to electric circuits, the movement of a positive test charge through the cell from the low energy terminal to the high energy terminal is a movement against the electric field. This movement of charge demands that work be done on it in order to lift it up to the higher energy terminal. An electrochemical cell serves the useful role of supplying the energy to do work on the charge in order to pump it or move it through the cell from the negative to the positive terminal. By doing so, the cell establishes an electric potential difference across the two ends of the electric circuit.

## Circuits Formulas

Ohm's Law (stated 3 different ways):

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): E = I \times R

$E}{R$

$E}{I$

Power Formulas

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): P = I \times E

$E^2}{R$

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): P = I^2 \times R

### Series Circuits

$total} = E_{1} + E_{2}.... + E_{n$

$total} = R_{1} + R_{2}.... + R_{n$

$total} = I_{1} = I_{2}.... = I_{n$

$total} = P_{1} + P_{2}.... + P_{n$

#### Circuit 1

To learn everything about this circuit we can use a chart. Start by entering what we know:

Using the formula: R(total) = R1 + R2.... + Rn we can find the total resistance:

We can now use Ohm's Law in the form of I = E/R to find the total current in the circuit

From the formula: I(total) = I(R1) = I(R2).... = I(Rn) we can now determine the current in both resistors:

Lastly, we can use Ohm's Law E = I x R to find the voltage used by each of the resistors

Note: Power (P) is measured in Watts (W). The formula is: P = I x E

#### Circuit 2

In Circuit 2 resistor #1 is increased to 9 ohms with all other parameters remaining the same. Calculate the values for this circuit just as in Circuit 1. Results are:

Note: The voltage used by resistor 1 increased and the voltage used by #2, the current and total power decreased.

### Parallel Circuits

$total} = E_{1} = E_{2}.... = E_{n$

$total} = I_{1} + I_{2}.... + I_{n$

$total} = \frac {1}{\frac{1}{R_{1}} + \frac{1}{R_{2}}.... + \frac{1}{R_{n}}$

$total} = P_{1} + P_{2}.... + P_{n$