Georg Simon Ohm (1787-1854) found that the ratio Potential Difference (in volts)/ Current (in ampere) [V/I] was constant for a specific resistor. This means that even if more energy sources were added on to the circuit, the quotient of the potential difference divided by the current would be the same. This relationship is known as being directly proportional which is represented by the sign "α" (lowercase alpha).
V α I
What does directly proportional mean? It basically means that when one variable goes up, the other variable goes up as well, and when one variable goes down, the other variable goes down as well. In our case, the two variables are POTENTIAL DIFFERENCE (V) and CURRENT (I).
Furthermore, current is inversely proportional with resistance which means that when the current goes up, the resistance goes down, and when the current goes down, the resistance goes up. It is represented by the following:
I α 1/R
Ohm's law can be represented by the mathematical triangle:
(*note: when looking for the current, cover I and you are left with V/R, when you are looking for the resistance, cover R and you are left with V/I, and when you are looking for the potential difference, cover V and you are left with IR [multiply])
Kirchhoff's Law
Gustav Robert Kirchhoff (1824-1887) came up with the two laws about current and voltage.
Current: "The total amount of current into a junction point of a circuit equals the total current that flows out of that same junction."
Voltage: "The total of all electrical potential decreases in any complete circuit loop is equal to any potential increases in that circuit loop."
CURRENT (I)
SERIES
IT = I1 = I2 = I3 = … = In
PARALLEL
IT = I1 + I2 + I3 + … + In
VOLTAGE/POTENTIAL DIFFERENCE (V)
SERIES
VT = V1 + V2 + V3 + … + Vn
PARALLEL
VT = V1 = V2 = V3 = … = Vn
WITH OHM'S LAW AND KIRCHHOFF'S LAW IN MIND, Resistance can be solved mathetmatically using both laws.
RESISTANCE (R)
SERIES
VT = V1 + V2 + V3 + … + Vn
(using Ohm’s Law we know V= IR)
ITRT = I1R1 + I2R2 + I3R3 + … + InRn
(which can be written as)
ITRT = ITR1 + ITR2 + ITR3 + … + ITRn
ITRT = IT(R1 + R2 + R3 + … + Rn)
(IT can be crossed out, which is represented by the red)
ITRT = IT(R1 + R2 + R3 + … + Rn)
RT = (R1 + R2 + R3 + … + Rn
PARALLEL
IT = I1 + I2 + I3 + … + In
(using Ohm’s Law we know I= V/R)
VT /RT = V1/R1 + V2/R2 + V3/R3 + … + Vn/Rn
(which can be written as)
VT /RT = VT/R1 + VT/R2 + VT/R3 + … + VT/Rn
(Multiply by 1/VT in order to isolate R)
(1/VT) (VT/RT) = (1/VT) (VT/R1 + VT/R2 + VT/R3 + … + VT/Rn)
1/RT = 1/R1 + 1/R2 + 1/R3 + … + 1/Rn
VT = V1 + V2 + V3 + … + Vn
(using Ohm’s Law we know V= IR)
ITRT = I1R1 + I2R2 + I3R3 + … + InRn
(which can be written as)
ITRT = ITR1 + ITR2 + ITR3 + … + ITRn
ITRT = IT(R1 + R2 + R3 + … + Rn)
(IT can be crossed out, which is represented by the red)
ITRT = IT(R1 + R2 + R3 + … + Rn)
RT = (R1 + R2 + R3 + … + Rn
PARALLEL
IT = I1 + I2 + I3 + … + In
(using Ohm’s Law we know I= V/R)
VT /RT = V1/R1 + V2/R2 + V3/R3 + … + Vn/Rn
(which can be written as)
VT /RT = VT/R1 + VT/R2 + VT/R3 + … + VT/Rn
(Multiply by 1/VT in order to isolate R)
(1/VT) (VT/RT) = (1/VT) (VT/R1 + VT/R2 + VT/R3 + … + VT/Rn)
1/RT = 1/R1 + 1/R2 + 1/R3 + … + 1/Rn
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