Series and parallel circuit problem solving

Introduction
My research is based on how Ohm’s law is derived and how it is used to solve problems in series and parallel connection and the resistance of a material. Ohm's Law shows the relationship between the voltage (V), current (I) and resistance (R). It can be written in three ways:
V = I × R or I =V/R or R = V/I
The resistance (R) of a material depends on Its length, cross-sectional area,
The resistivity, and
Resistance also depends on temperature, usually increasing as the temperature increases. For reasonably small changes in temperature, the change in resistivity, and therefore the change in resistance, is proportional to the temperature change.
Circuits consisting of just one battery and one load resistance are very simple to examine, but they are not usually found in practical applications. We find circuits when two or more constituents are connected together.
We have two basic ways of connecting circuit components:
Series
parallel 1. Series Circuit – is an electric circuit having its parts connected serially (without branching). It has only one path for the charges to move along and the charges must move in “Series” which means first going to one resistor then the next. Here if one of the circuits is broken down, then no charge will move through the circuit because there is only one path. Which is if one bulb burned out the whole lights will go off. The lamps on a Christmas tree are connected in series. Normally we expect al the lamps go out if one blew.

The following rules apply to a series circuit:
1. The sum of the potential minimum equals to the potential rise of the source. VT = VR1 + VR2 + VR3 +..
2. The current in a series connection is the same everywhere. IT = I1 = I2 = I3 = …
3. The sum of the individual resistance is equal to the total resistance of the circuit.
RT = R1 + R2 + R3
Ohm’s law may be used in a series circuit as long as we remember, we can use the formula with either partial values or with total value, but we can not mix them.
Vtotal = Itotal Rtotal
V=I1R1
Combining series resistor and a voltage source
Then using KVL V- r1r2 = 0
By Ohm’s law
= V1=I1 R1 V2=I1 R2
=V- I1 R1 – I1 R2 = 0
= V- I1(R1+R2) = 0
Thus
I1 = V/R1 + R2
V= I1 (R1+ R2) Example of series circuits

We can see that when we want to solve for the equivalent resistance of a circuit is we will sum up the resistor value.
Req = R1 + R2 + R3 + …

= Rtotal= R1 + R2 = 5KΩ
= I1 = V/R1+ R2 = 5/2000+3000 = 1mA
Appling the formula I1 = V/R1+R2 = V1= I1 R1 = VR1/R1+R2 = 5(2000)/2000+3000 = 2V
For V2
= V1= I1R2 = VR/R1+R2 = 5(3000)/2000+3000 = 3v
2. Parallel Circuits When all the devices are connected using parallel connections, the circuit can be referred as a parallel circuit. In parallel circuit, each device has its own separate branch and a parallel circuit has more than one resistor. This makes the charge to move through several paths. Parallel circuits are mostly used in household electrical wiring. The voltage across each resistor parallel is the same.

The following rules apply to a parallel circuit
The potential drops of each branch equal to the potential rise of the source. VT = V1=V2=V3=…
The total current is equal to the sum the current in the branches.
IT = I1+I2+I3+…
The inverse of the total resistance of a circuit is equal to the sum of the inverse of the individual resistance.
1/RT = 1/R1+1/R2+1/R3+…
Here we have to remember that as the total resistance decreases, the total current increases. Case 1: Three 12 resistors are placed in parallel

1/Req = 1/R1 + 1/R2 + 1/R3
1/Req = 1/(12 ) + 1/(12 ) + 1/(12 )
1/Req = 0.25 -1
Req = 1 / (0.25 -1)
Req = 4.0
Case 2: A 5.0 , 7.0 , and 12 resistor are placed in parallel

1/Req = 1/R1 + 1/R2 + 1/R3
1/Req = 1/(5.0 ) + 1/(7.0 ) + 1/(12 )
1/Req = 0.42619 -1
Req = 1 / (0.42619 -1)
Req = 2.3 Ohm’s Law
States that the current through a conductor between two points is directly proportional to the potential difference across two points, and inversely proportional to the resistance between them, the mathematical equation describes this relationship as follow:
I = V/R
Where I is the current through the conductor. (in volts)
V is the potential difference across the conductor. (in ampere)
And R is the resistance of the conductor in units of Ohms (Ω). Increasing the resistance of the circuit will lower the current flow if the voltage is not changed. The formula can be reorganized so that the relationship can easily be seen for all of the three variables.
Resistor in series and parallel
A) Resistor in Series
Resistors can be connected in series, which is the currents flow through them one after another. Since all the current flowing through the first resistor has no other way to go it must also pass through the second resistor and the third and so on. Then, resistors in series have a Common Current flowing through them as the current that flows through one resistor must also flow through the others as it can only take one path. Then the amount of current that flows through a set of resistors in series is the same at all points in a series circuit.
The current through each of the resistor is the same on resistor in series.
1) I=I1=I2=I3
Voltage drops across the resistors must add up to the total voltage supplied by the battery.
2) VT = V1+V2+V3
Since V=IR Ohm’s law then,
3) Vtotal = I1R1+I2R2+I3R3
Ohm’s law must be satisfied for the complete circuit:
4) Vtotal = I.R equivalent
Combining equation [3] and [4]
5) I.R equivalent = I1R1+I2R2+I3R3
We know current through each resistor is the same (from [1]) it’s just I.
6) I.R equivalent = I (R1+R2+R3) = R equivalent = R1+R2+R3 =
B) Resistor in parallel One connected completely in parallel is known as parallel circuit.
At A the potential must be the same for each resistor. Similarly, at B the potential must also be the same for each resistor. So, between points A and B, the potential difference is the same. That is, each of the three resistors in the parallel circuit must have the same voltage.
[1]
Also, the current splits as it travels from A to B. So, the sum of the currents through the three branches is the same as the current at A and at B (where the currents from the branch reunite).
[2]
By Ohm's Law, equation [2] is equivalent to:
[3]
By equation [1], we see that all the voltages are equal. So the V's cancel out, and we are left with
[4]
This result can be generalized to any number of resistors connected in parallel.
[5]
Conclusion:
For a wide variety of materials and conditions, the electrical resistance R is constant for a given temperature; it does not depend on the amount of current through or the potential difference (voltage) across the object. Such materials are called Ohmic materials. For objects made of Ohmic materials the definition of the resistance, with R being a constant for that resistor, is known as Ohm's law. Parallel connection is used for in home the main reason for that is because; In parallel connections voltage will be same in all paths, only current will differ. That's why it’s widely used in home connections