Physics conservation of charge and conservation of energy resistors proof❗️

Hello, I have come across the question attached and honestly do not know where to begin. I would be very grateful if someone was able to go into detail explaining the question and a suitable method to approach finding a solution. I truly appreciate any help 😁
I have attempted to resolve a solution but feel that may answers thus far are rather unrelated.

A student is using the laws of conservation of charge and conservation of energy to try and prove that R = R1 + R2 for two resistors in series and that 1 / R = 1 / R1 + 1 / R2 for two resistors in parallel.
a. Write the law of conservation of charge in terms of current.

a. Is this question essentially referring to the electric current rule, which states that the algebraic sum of the currents entering a junction is equal to zero? Since in order to conserve charge the sum of all the currents arriving at a junction in a circuit is equal to the sum of all the currents leaving that point.
i.e. sigma I = 0
Thus, I in + I out = 0
I think this is shown by the continuity equation of the law of charge conservation as:
Since I = Q/t
∂Q/∂t = Q in (t) + Q out (t)
where ∂Q/∂t is the electric charge accumulation rate at time t, Q in (t) is the amount of charge flowing in and Q out (t) is the amount of charge flowing out .
I know that the total amount of charge with a circuit cannot increase or decrease when the circuit is functioning. When something changes its charge it doesn't create charge but transfers it.

However, I do not think that I have arrived at the correct answer and feel that I may have deviated from the question.

b. State the law of conservation of energy and show how this can be simplified to give EMF = V1 + V2 for two components in series.
b. The law of conservation of energy states that energy can neither be created nor destroyed - only converted from one form of energy to another. This means that the total energy of an isolated system remains constant. I know that any group of emfs that follow in series in a circuit will have a total emf that is the sum of their individual values.
Therefore, V total = V1+V2+V3...

I do not know how I can apply this knowledge to prove that it can be simplified to EMF = V1+V2 for two components in series?

c. To complete the student's proof I have attempted as below;
Resistors in parallel have a total current through them that is the sum of their individual currents.
The pd across them will be the same in each branch.
Therefore;
I total = I 1 + I 2
and since I=V/R
V/Rtotal = V1/R1+V2/R2
1/Rtotal = 1/R1+1/R2

I do not think that I have completed the proof though and feel that I have just stated the general rule.

I am rather confused by this question and would greatly appreciate any help. Thank you very much 😁

Part a is only asking you to state the conservation of charge in terms of current NOT asking you derive it. So just state it.

While for part b, IMO you are “stitching” concepts without really explaining how are they are connected.

For example, you state the conservation of energy as “energy can neither be created nor destroyed - only converted from one form of energy to another. This means that the total energy of an isolated system remains constant.”

What is your system for this question?
What is the conversion of energy?
How do you translate the above answer into equation?




is, in fact, Kirchhoff voltage rule and you can actually find the “derivation” in standard physics textbooks.


A “simple” way of using the conservation of energy for a circuit that has a battery of emf E connected to two different resistors (R₁ and R₂) in series is



The ... is left for you to fill in the blanks.

For part c, it seems that you have missed out the “proof” for resistors in series.

The proof for resistors in parallel seems to be good. Good work.

Thank you for your reply and your detailed post.

a. I am not sure how to state the conservation of charge in terms of current. Would this be sigma I in = sigma I out
or current into the junction is equal to the amount of current flowing out? ...

......
b. I think the system is the circuit.

Energy delivered by EMF = Energy transferred to R1 + Energy transferred to R2
Kirchoff's Voltage Law states that the algebraic sum of all the potential differences around the loop must be equal to zero as: ΣV = 0.

Since the two resistors, R1 and R2 are wired together in a series, they are both part of the same loop so the same current must flow through each resistor. Thus the voltage drop across resistor, R1 = I*R1 and the voltage drop across resistor, R2 = I*R2;

V=(-RI1)+(-RI2)=0
V=RI1+RI2
V=I(R1+R2)
V=IRT
(RT=R1+R2)
I=V/RT=V/R1+R2

VR1=IR1=V(R1/R1+R2)
VR2=IR2=V(R2/R1+R2
Therefore, EMF=V1+V2

I do not think that I have fully shown how this can be simplified. Could you offer any help?
...

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c. To complete the student's proof I have attempted as below;
A group of components in series have a total current through them that is the same throughout each one.
The pd across the resistors is equal to the sum of their individual values (accounting for the direction of their positive and negative sides).
Therefore;
potential difference: Vtotal= V1 + V 2
and since V/1/R
IR total = IR1+IR2
Rtotal = R1+R2

Can I expand upon any areas here in completing the proofs or simplifying? 😁

Yes. You can state using maths or words.




I thought fill in the blanks is an easy task. Learn to exercise some logical think. I don’t think you ever come across a phrase like “energy delivered by emf” in your school notes or any physics texts.



The question states it clearly that you need to work from conservation of energy to arrive


But you are doing something else.

I mention that the equation that you are deriving is Kirchhoff’s voltage rule but that does not mean you need to reiterate the rule.

Work from conservation of energy, no more no less, to derive the following:





Not sure what you mean by “since V/1/R” but I believe is a typo error.

The working seems ok.

The missing parts for this answer and the answer in the original post would be there is no mentioning of the application of conservation of charge and conservation of energy in a series and parallel set-up. The 2 conservation laws are “implicitly” used, so it is a matter of fact whether you know it and whether your teacher is convinced that you know it.

Thank you for your reply. Sorry I am still a little stuck on question b.

b. I do not know the formula for the conservation of energy, would it be that Kirchoff’s Second Law states that the sum of the potential differences across components in any complete loop around the circuit must equal the sum of the electromotive forces supplying it.
In other words ΣE = IR?
Then rearrange this to EMF = V1 + V2.

So ΣE = IR for two reistors in series:
ΣE= I1R1 +I2R2
I1R1=V1
I2R2=V2
ΣE=V1+V2

c. Shoudl i include information of the conservation of charge and energy then in the proofs? Sorry I was just a little confused by what you mean.

b. I do not know the formula for the conservation of energy,

The question states that the student is trying to use conservation of charge and conservation of energy to prove ….. and part c is asking you to complete the proof.

The proof provided by you states the following:

Resistors in series
Resistors in parallel have a total current through them that is the sum of their individual currents.
The pd across them will be the same in each branch.

Resistors in parallel
A group of components in series have a total current through them that is the same throughout each one.
The pd across the resistors is equal to the sum of their individual values (accounting for the direction of their positive and negative sides).

You just state them without linking them to the conservation law and did not show how do you arrive these results, so you are just stating results without explaining how you arrive the result.
The lenient teachers would give the benefit of doubts for such workings.

However, I doubt your understanding as can be seen from part b.


For (b), it seems that you are adamants to change your thinking and use the given hints.

IMO, you keep doing the following:

You are asked to show A = B from a conservation law,
Then you state
A = B (by inferring from another rule that is derived from the conservation law.)
B = C
C = A
B = A
So A = B.


In the past posts, you are just doing a different version and expect it to be correct.
The question is clear in asking you to derive
EMF = V₁ + V₂
but you are using
EMF = V₁ + V₂
to derive
EMF = V₁ + V₂

If this is done once, I am ok but this is done thrice. This is insane!


Didn’t I state the following in post #4:


This is the conservation of energy for the circuit that has a battery of emf EMF connected to two different resistors (R₁ and R₂) in series

Your job is to do the following:


….
….




The missing part(s) are your jobs.

Hints for the missing parts:
Make use of the definition of emf and potential difference across a resistor.
OR
Imagine a unit of charge Q passes through the battery, what is work done by the battery? (See the definition of emf)

Next, imagine the same unit charge Q passes through resistor R₁ of resistance R₁, what is the work done on the unit charge? (See the definition of potential difference)
The word “imagine” is important.


PS: A lot of students just memorise and state definition emf and potential difference without really thinking about the implication and meaning.

Thank you for your reply. When I stated that I did not know the conservation law I did realise that you had shown
Energy delivered by the battery = Energy transferred to resistor R1 + Energy transferred to resistor R2,

I rather meant that I did not know if there is an alternative mathematical formula.

"If this is done once, I am ok but this is done thrice. This is insane!" I agree, I am just very confused and keep circling the same idea.

Energy delivered by the battery = Energy transferred to resistor R1 + Energy transferred to resistor R2

potential difference across one resistor=IR

Are you referring to EMF=E/Q ?
For the work done by the batterry do you mean;

W=V*Q
W=V*It
W=IR*It
W=I^2Rt

V=IR
Q=It

Energy = power x time
Power = voltage x current
Energy=V*I*t
Work done = energy
I^2Rt=VIt

EMF=E/Q
EMF=I^2Rt/Q
EMF=I^2Rt/It
EMF=IR
EMF=V

For two components this becomes;

EMF=V1+V2?

Sorry I am still confused but I am trying.

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Sorry I am still confused but I am trying.

T.....
c. I apologise but I am not sure how to link the completed proofs to the conservation law, could you perhaps get me started? Sorry to ask.

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I know that you are confused and can see that you are trying but it does not mean you should just “throw” out all the equations or formulas to expect people to tell you what is good and bad. IMO, doing what you are doing is “adding oil to fire”

I believe I quote the following:
“PS: A lot of students just memorise and state definition emf and potential difference without really thinking about the implication and meaning.”
And you still choose to do ….


Physics is NOT maths and the way you are doing is “plug it in and pray it is right” without thinking the meaning or physical interpretation.

Say what you “understand” by emf and potential difference. Don’t give me maths formula or equations.

For part c:

Resistors in series:
Conservation of charge
Method 1 (simple and maybe tedious for some)
1. Draw a circuit diagram that has a battery that connected with 3 resistors in series.
2. Place a junction (a dot) between the battery and resistor R1 and label the 2 currents that enter and exit the junction as I and I₁, respectively.

3. Do step 2 for 2 more positions (between R1 and R2, between R2 and battery).
4. Conclude that the current is the same throughout the circuit due to conservation of charge.

Method 2 (Simple but require one to think)
Write a concise explanation of method 1.
As the battery and resistors are connected by a single loop and the charge do not build up or disappear at particular locations in the circuit, so by the conservation of charge, the current must be the same throughout the whole circuit.

Conservation of energy:
If you have done part b properly, you can infer from (b) and state
From the conservation of energy, we know

for the circuit that is drawn.
Combine the results to show the

I have tried to follow method 1 which you have suggested and drawn a circuit diagram which I have attached. Is this what you meant, I think I might have put the junctions in the wrong places?

B) Thank you for your reply, I have attempted to take your advice below.
Ok , I understand that EMF is a supply voltage, equal to the terminal potential difference when no current flows. EMF is the energy transferred per unit charge when one type of energy is converted into electrical energy, ie. in this situation the amount of chemical energy in the battery that is provided to each coulomb of charge passing through. The is a flow of charge around the circuit, as it does so it gains and loses energy, the resistor would gain thermal energy where the battery becomes depleted of electrical energy in providing this.

Potential difference on the other hand is the difference in electric potential between two points; and is the energy transferred per unit charge when electrical energy is converted into another form of energy, ie. in a resistor electrical energy is transformed into thermal energy.

Energy delivered by the battery = Energy transferred to resistor R1 + Energy transferred to resistor R2

The "energy delivered by the battery" is equal to the EMF of the circuit and the "energy transferred to one resistor" is equal to the p.d across resistor.
The potential difference can be calculated using Ohm's Law and is proportional to the resistance.
V=IR
When resistors are connected in series, the total of all the potential differences in the circuit is equal to the the potential difference of the supply:
V=V1+V2+V3...
This relationship expresses the law of conservation of energy.

The p.d. is a measure of the energy supplied to each electron in the circuit, the potential difference across each component is the energy converted by each component. Therefore, the energy supplied by the EMF equals the energy converted, proving that energy is conserved since it cannot created nor destroyed in the circuit.

😊

"energy delivered by the battery" is equal to the EMF of the circuit

I really hope you can understand “simple” instruction(s). In the previous post, I only ask for
“Say what you “understand” by emf and potential difference. Don’t give me maths formula or equations. “

But ….

It is hard to guide when you keep overdoing of things, especially linking unnecessary concepts and worst still making wrong connection or equation:


Check the SI unit of LHS and RHS quantities of the quoted equation should easily tell you that it cannot be right.

I only want to know what you know about emf and potential difference. Don’t give me maths formula or equations.
Just write 2 to 3 statements in explaining emf and potential difference respectively. So in your reply I should not be seeing more than 6 sentences.Make them concise straight to the point instead of describing unnecessary stuff.

Electromotive force is a supply voltage, the energy transferred per unit charge when one type of energy is converted into electrical energy. However, EMF is not actually a force. It is usually measured in units of volts, equivalent to one joule per coulomb of electric charge.

Potential difference is is the energy transferred per unit charge when electrical energy is converted into another form of energy. A potential difference is measured in volts, a measure of one volt is equal to one joule of energy being used by one coulomb of charge when it flows between two points in a circuit.

How do you translate the explanation into an equation that links emf and energy conversion?



How do you translate the explanation into an equation that links potential difference and energy conversion?

Thank you for your reply.

The energy delivered by the emf is equal to the energy transferred to the resistor. The resistor then transfers this electrical energy to thermal energy, so it is a potenital difference.
Would this be as you have earlier stated:
Energy supplied by the EMF=energy transferred to resistor 1 +energy transferred to resistor 2

Look at how I ask the questions. I did not ask you to make any connection and again ....

Oh you meant to consider the equations of emf and pd individually. I do not think what I have written is correct but is more along the lines of what you meant;

…

Energy delivered by EMF source = energy transferred to electrical energy across a component

Energy transferred by Potential Difference = electrical energy across a component converted to another form of energy