# Applying the conservation of energy law to electrical circuits.

1. Apr 27, 2013

### CraigH

Hi guys, I've realised that to truly, deeply, and fully understand something it often helps if you think about the conservation of energy, and then just build up from that.
So here I have tried to explain why a battery has a voltage, and why there are voltage drops across resistances, capacitances, and inductances, using the law of conservation of energy. This explanation should also help see the reasoning behind kirchhoff's voltage law.

However I'm very unsure if everything I have said is true, so it would help if someone could read it, and say if everything I have said is correct. Please be very critical, I want to fully understand this seemingly but not actually simple subject. One of the main uncertainties I have is when I say an electron has a kinetic energy. An electron does have a kinetic energy but i'm not sure if this is how the energy is transmitted along the circuit. Anyway, here is my explanation.

Applying the conservation of energy law to electrical circuits

A voltage is just a difference in electrical potential energy (EPE) between two places, per unit charge, where the electrical potential energy arises because of the forces between charges. If there is more electrical potential energy at one place than there is another, then a current will flow between these places until it causes the difference in electrical potential energy to equal zero. e.g in a wire with the electrons all bunched up at one end, the EPE here will be higher than at the other end, so electrons will flow until the difference in EPE at all places along the wire equals zero

In an electrical circuit a battery ensures that there is always more electrical potential energy at one side than there is the other (by using a chemical reaction to send electrons to the cathode), so a current constantly flows.

A voltage drop across a resistance occurs because some of the kinetic energy that the moving electrons have is transferred to the molecules inside the resistance (hence ohmic heating). For this situation, if we again consider the wire with a different EPE at each side, but this time with a resistor in the middle, then it will take longer for the flowing current to cause the difference in EPE to equal zero, as now each electron has less energy.(because some was lost due to ohmic heating)

A voltage drop across a capacitance is caused by the electrons building up on one side of the capacitor, and leaving the other side of the capacitor, causing an electrical potential energy difference between the two plates of the capacitor. In this case the energy of the electrons is being transferred back into electrical potential energy. The electrons will flow until the EPE per unit charge across the capacitor equals the EPE per unit charge from the source. In this case the energy is not transferred out of the circuit like it is with the resistance, it is just stored in the electrical field between the capacitor plates. If the source is removed, the EPE will be converted back into the kinetic energies of the electrons as they flow from the charged capacitor plate, and distribute themselves along the wire so that the electric fields created by each cancel each other out. If there was no resistance in the wire the kinetic energies of the charges could not be transferred, so the charges would forever oscillate between having maximum potential energy (when the capacitor plate is full) to having maximum kinetic energy (when the rate of discharge/charge is at its maximum). However this is not the case as a wire has a resistance, and the energies will be transferred out of the circuit as heat.

A voltage drop across an inductance is caused by a changing current, which creates a changing magnetic field, which induces an opposing voltage. In this case the energy from the electrons is being taken and stored in the magnetic field around the inductance. More specifically, the kinetic energies of the electrons are being transferred to magnetic potential energy, as the magnetic field opposes the acceleration of the electrons, and this causes them to have less kinetic energy, but energy is conserved because it is stored in the magnetic field around the inductance. (I'm very unsure about this one, it still doesn't make full sense to me)

Last edited: Apr 27, 2013
2. Apr 27, 2013

### Simon Bridge

Nice essay - clearly written, and you have avoided excessive jargon.
What level is this to be assessed at?

Working out when you've made a mistake is the thing though isn't it? How can you do that without having to ask someone else?

What was the wording of the assignment? For the sake of discussion I'll go by:
... the trick is to restate the points you have to cover as questions, then answer them.

i.e.
Point: explain why a battery has a voltage
Question: "How does a battery get it's voltage?" [*]
You wrote:
In an electrical circuit a battery ensures that there is always more electrical potential energy at one side than there is the other (by using a chemical reaction to send electrons to the cathode), so a current constantly flows.

The part of that sentence that answers the question is the bit in parentheses.
The bit before is just the question written as a statement - so it is "begging the question".
The bit just after relates the result of the voltage.

So you have put the actual answer to the question as an incidental "aside" - when it should be the main focus. Put it at the front, vis:

The battery uses a chemical reaction to send electrons to the cathode. This ensures that there is always more electrical potential energy at one side than there is the other, so a current constantly flows.

(In general, you should avoid parenthetical statements ;) )
Now you've gone from a C+/B- to an A+.

Of course I'm only guessing that's an assessment point.
But this is how you go about checking essays - work out what the points are that have to be covered, reword them as questions, check to see that the questions are answered prominently.

In the bits about "voltage drop", its not terribly clear what is meant by "drop". You basically say that if there is a potential difference then there is a voltage drop. You don't say if this is the DC or AC case (or both), but you only mention the changing current for the inductor.

But as I say, it is competent - you'll pass.
Your main concern is that you've got your facts right - and, I suspect, if you go through the assignment like I said, getting the questions, looking for the answers, you'll feel more confident about it.

--------------------------

[*] notice the deliberate shift from "why" to "how".
This is "interpreting the question" - it can make sense to ask/answer "why" questions in science, but it is good mental discipline to avoid the wording. I like to say, Science does not do "why" questions.

Last edited: Apr 27, 2013
3. Apr 27, 2013

### CraigH

Ah yes I guess I should have been more specific, and said that the EPE per unit charge at the cathode is higher than it is at the anode.

This isn't actually assessed, I just wrote it to test and improve my understanding of the subject. If I can fully understand this it will really help me in my course. I'm struggling with the Electromagnetism and RF engineering module at the moment, so I thought that I would start from basics, and try to understand fully how energy is transmitted electrically.

4. Apr 27, 2013

### jeffrey c mc.

Energy/inductance(resistance) and Power/inductance(energy); E over I(R); P over I(E) Ohms Law; your discussion should have accounted for these two also. Your discussion on battery cells could be cleaned up some by going into the chemistry of the electrolyte and the cathode plate. Inductance is a product of resistance, but is also related to the Energy Potential in the circuit. Power; in electricity, the watt; is a product of the EP and the total inductance in the circuit.

5. Apr 27, 2013

### CraigH

Thanks for your answers guys, but this is not actually an assessed essay. I'm trying to fully understand how energy is transmitted through electricity. My previous understanding was that in DC a battery "pumps" electrons around a circuit, and power equals IV, and in AC the alternating currents cause an alternating power, but there is rms power...etc. However this understanding no longer cuts its for me. When studying RF engineering I need to know about signal velocity, the velocity of the wavefront, reflected power, harmonics, and standing waves/partially standing waves in electrical systems. To understand these I thought I would first start from basics, and make sure I fully understand how energy is transmitted through electricity. The main uncertainty I have is if it is the kinetic energies of the electrons that are the intermediate energy transmission between the EPE caused by the source and the energy transferred out of the system, e.g the thermal energy of a resistor.

I wrote an essay like explanation of how I currently understand how electrical energy is transmitted, so if I am going wrong somewhere, people can point out exactly where

Last edited: Apr 27, 2013
6. Apr 27, 2013

### Simon Bridge

Technically, the essay is assessed - but, now, it is being assessed by us ;)
... and anybody else you show it to.
That makes your concerns easier to address - we don't have to worry about some faceless marker, it's just us.

It is useful to treat all your work as if it were being graded by someone, if you want to do well.
The approach of breaking down the points you have to cover into questions is still valid.

This essay has covered the bare basics of electric circuits up to senior high-school level.

If you have just left High School, and have started College, then your understanding is about what I'd expect for the bulk of students starting engineering or physical sciences.

jeff's concerns, though useful, actually fall outside the assignment you have set yourself, described in post #1 ;) OTOH: the essay does not address things like power transmission and electrical signals. You have not directly addressed the difference between the AC and DC cases either, although the "battery" implies DC. You have only covered simple reactive circuits. I am guessing this is just laying a foundation for yourself at this stage?

7. Apr 28, 2013

### BruceW

I agree that ohmic heating occurs due to the electrons 'bouncing around' inside the resistor. But the electrons are not necessarily losing kinetic energy overall. They will lose kinetic energy to ohmic heating, but also will gain kinetic energy from the electric field which is driving them.

Not sure what you mean here. For this oscillation to occur, you would need some inductance in the circuit. The energy can't just disappear and turn up again later.

Yes, this is pretty much right. There is also something causing the acceleration of the electrons, for example an electric field in the circuit, so this uses up some of the energy from the battery to increase the KE of the electrons, but then some of this KE is taken by the inductor and put into the magnetic field. So in essence, some of the energy of the battery goes into the magnetic field, and as you say, the KE of the electrons is an 'intermediate' form of energy.

I think on the whole, the stuff you wrote is pretty much correct.