Mood Brightener: ...more from Stay Homas. (Confination XI)

 

Circuits: Doing the Math!

The intent of spending time creating conceptual models for the movement of energy and charge in circuits is that conceptual knowledge is typically longer lasting and more accessible than mathematical knowledge. None the less, now that we have a conceptual model, we are going to translate that model into mathematics both to see how that process works and to explore some safety issues in your home and life that are hard to address clearly without the math.

Kirchhoff's Rules:

There are two rules that encapsulate most of what we have explored in our conceptual model. They are Kirchhoff's voltage (energy) and current (flow rate) rules.

Voltage Rule:

Our conceptual rule was that the carriers had to deliver all their energy before returning to the factory. If I follow the path of an individual carrier completely through a circuit I should be able to see where that energy was 'loaded on' and 'delivered'.

What this means is if I add up all the energy losses and gains around any complete loop (following an individual carrier) that it all balances out. What the factory gives the loads take away.

Junction Rule:

Our conceptual rule was that the overall number of carriers doesn't change. There may be forks in the road where the traffic divides or rejoins but overall what flows out of the factory must flow back. This rule is applied at any location where the current might branch or divide.

Ohm's Law:

The last mathematical tool we need is something that expresses the observation that the more energy a charge arrives with the easier it is to move through a load. Another way of saying this is that greater energy delivered leads to higher current (flow). This is captured in an expression that is known as Ohm's Law. In this expression R is a characteristic of the load known as the resistance. Think of it as a measure of how 'rough the road' is.

With these three tools and a clear sense of our process we can figure things out mathematically that are too much to keep track of conceptually. We will not push it very hard but it is possble to analyze remarkably complex circuits with just these simple tools. We are not going to push that hard but we will explore our basic circuits using this tool.

Here is the process:

i: Label all possible currents -- each wire or portion of a wire needs a label

ii: Label each location where some energy is added or removed (Δ V)

iii: Write down Kirchoff's Junction Rule for all junctions - drop the last one because it's redundant!

iv: Write down Kirchoff's Loop rule for all loops.

v: Write Ohm's Law for each Δ V in the previous Loop equations.

vi: Do the math.

vii: Calculate the power (brightness) of each object using P = I * V

A seven step mathematical process can certainly be intimidating but follow the example of our reference circuit in this video to see the steps being applied.

Circuit I:

 

Circuit II: Series

 

Circuit III: Parallel

 

Thoughts about Ohm's Law:

Turns out if you measure the resistance of a human being it's very different if you're including the dead skin on our outside or starting with our juicy insides. We're mostly a bag of salt water which has pretty low resistance. Our skin is sort of like insulation on a wire. Most people find that their personal resistance is around 1 MΩ. If we touch a 120V live wire in our house that would mean, roughly, that the current through our body is I = 120V/1 MΩ = .12 mA. This is not nothing but even if it passes through your heart it's not enough to notice in a healthy person. It usually take about 5 mA to make your heart unhappy. Some people have more conductive skin for a variety of reasons and their resistance can get down to 80 kΩ. If they handle a live wire they might get 2 mA through their heart which isn't good though it won't kill.

Your car battery:

You can handle your car battery with safety because under normal circumstances the probably current through your body is given by I = 12V/1 MΩ which gives I = 12 μA. Actually hard to notice. But what if you're working on your car in the winter because you need to jump your battery and your hands are wet with maybe some road salt mixed in. Now you need to be much more careful because your skin is not nearly as good an insulator under these circumstances. It is quite rare for someone to get killed this way but people have gotten some nasty shocks working on car batteries with wet hands.

Electric Vehicles:

Hopefully now you can appreciate the potential danger of the battery in an electric vehicle which range from 250V to 400V roughly. These batteries can move dangerous amounts of charge through your body under many circumstances. I only bring this up so you will be cautious when you help in an accident where one of the vehicles is an electric or hybrid vehicle. We're not used to being careful about this just yet.

Household Appliances:

For a homework problem recently you were asked to find the data on the back of an appliance in your house. Here's how that's relevant. In your house the 'battery' delivers 120 V of energy (10x your car battery) to each charge. I know - in your house it's AC and not DC like a battery but the math is essentially the same. We can talk about the difference between AC and DC in class if you want. Depending on your particular appliance it sometimes tells you a power consumption (in Watts) or the current draw (in Amps) or perhaps both. If you have a 1200 W microwave (pretty powerful for a microwave) then using P = I*V and rearranging...

I = P/V = 1200 W / 120 V = 10 A

This means your microwave uses 10 A on full power. Hard to know what to make of that but when you realize that each circuit in your house can only handle 15 A or 20 A depending on the specific breaker this is half of the maximum capacity of that circuit. It is not uncommon to have the microwave on it's own circuit breaker just for safety.

A space heater is often rated at 1000 W to 1500 W. Like a microwave a space heater uses up most if not all of the capacity of a given circuit. If you plug two space heaters into the same circuit to stay warm in the winter you will often discover that the circuit breaker pops. This is why.

A 100 W light bulb uses I = 100W/120V or 0.9 A. That means you can have about 15 light bulbs on a given circuit if you really needed to. That would be a lot but it gives you a sense of scale.

Appliances like stoves and ovens use so much energy that they have their own special breakers to protect them and they use much larger wire to keep the wire from overheating.

Questions from Class:

[set aside for questions that come up in class]

Assignment Breadcrumb Reading: Bb Quiz

Why Turn It In To Math?

Why do those physicists have to turn everything into math? Why does writing down our conceptual understanding in mathematical terms add value?

Before Next Class:

Assignment HW: Bb Assignment

Two Bulbs in Series:

Apply our circuit analysis process to Circuit II of our three basic circuits and show that the mathematical results are the same as our conceptual analysis. This means you compare your calculated results with the current and brightness of bulb A in the Circuit I.

Assignment HW: Bb Assignment

Two Bulbs in Parallel:

Apply our circuit analysis process to Circuit III of our three basic circuits and show that the mathematical results are the same as our conceptual analysis. This means you compare your calculated results with the current and brightness of bulb A in the Circuit I.

Looking Ahead:

Look ahead to the next Breadcrumb: Fields I

Assignment Breadcrumb Reading: Bb Quiz

How Many Fields:

How many physical fields do physicists have reproducible evidence for at this time?