Relationship between the Resistance and Dimensions of a Conductor

With a uniform wire/conductor of a given material and a given length, the resistance obtained by dividing Potential difference between any 2 points by Current is directly proportional to the distance between them. That is to say if the distance between the 2 points (on the conductor) increases the resistance also increases.

If 2 resistors each having resistance R ohms are connected in parallel the equivalent resistance Re is given by Product / Sum (Product over Sum)

Re = R*R/R+R
Re = R*R/2R
Re = R/2
Re = (1/2)*R
Each resistor will offer half of its resistance.

See the working in the slide. Click the previous button then after click on the slide screen.

Therefore, if two wires of the same material, having the same length and diameter are connected in parallel their resistance is half that of one wire. The effect of connecting two wires in parallel is similar to doubling the area of conductor. Connecting 7 wires in parallel is the same as increasing the cross section area of a wire 7 times and this reduces the resistance to a 1/7 (seventh) of one wire. Generally, the resistance of a given length of a conductor is inversely proportional to the cross section area. 


Other factors that influence resistance are; nature of the material (different materials have different resistances) and temperature (the resistance of some materials increase with an increase in temperature). So everything we have discussed above is true if temperature is constant.

Resistance of a wire = (Length of wire (L)/Cross Sectional Area (A))*a constant for a given material (ρ)

See the equation in the slide.

The constant of a material is called Resistivity of a material and it is represented by a Greek symbol rho (ρ). Resistivity is measured in ohm meter (Ωm)

Definition for Resistivity

This is the resistance of the specimen having one meter long and one meter square of cross sectional area.

Calculating Current in Two Parallel Resistances

Like I have already stated in one of the previous posts, voltage is the same in parallel resistor connection.
Voltage for resistor 1 = Voltage for resistor 2

That is to say (Voltage supplied), V = (I1R1 = I2R2).
Not forgetting that V = IR
Also Is = I1 + I2. (Is is the current supplied)
I-I1 = I1-I1+I2
I2 = I-I1
Substitute I2 in the equation V = (I1R1 = I2R2)
I1R1 = R2(I-I1)
I1R1 = IR2 - I1R2
I1R1 + I1R2 = IR2
I1(R1+R2) = I*R2
I1 = I*R2 / R1+R2
Similarly
I2 = I*R1 / R1+R2

OHM'S LAW

Ohm's law states that for a given current, the Potential difference (P.d) between two points is directly proportional to the resistance of the circuit between those points.

V = IR

Calculating Resistances in a Circuit

If current is to flow through a circuit, it must go against resistance with majority of it taking paths of least resistance and if resistance is great (in a certain path), current will not flow.
Circuits are designed with resistances both in parallel and in series.

Series Connection of Resistances
The figure below shows three resistors having resistances of R1, R2, R3, connected in series across the supply voltage labelled Vs.

Note. When resistors are connected in series the total current flowing through the circuit is the same (current supplied (Is)) while voltage drop depends on the value of the resistance of the individual resistor and total voltage of the circuit is the sum of the voltage drops of the individual resistors.
Vs = V1+V2+V3
From ohm's law
V=IR
Substitute V for IR in the equation Vs = V1+V2+V3
IsRT=IsR1+IsR2+IsR3 (Is being current supplied and RT being total resistance)
Divide both sides by Is
IsRT/Is=Is(R1+R2+R3)/Is (Is dies on both sides)
RT=R1+R2+R3

Parallel Connection of Resistances
For resistors in parallel connection, the supply voltage is the same while current flowing depends on the value of the resistance of the individual resistor. The total supply current is the sum of all the currents flowing  through the circuit.

In the figure above three resistors having resistances of R1, R2 and R3 respectively, are connected across supply voltage Vs.
From total current
Is = I1+I2+I3
And from Ohm's law
I = V/R
Substituting I for V/R in the equation Is = I1+I2+I3
Vs/RT = Vs/R1+Vs/R2+Vs/R3 (Vs being supply voltage and RT being total resistance)
Vs/RT *Vs = Vs(1/R1+1/R2+1/R3)/Vs
1/RT = 1/R1+1/R2+1/R3
If there are two resistors connected in parallel, the total resistance RT is given by product/sum (Product over Sum)
1/RT= 1/R1+1/R2
1/RT = R1+R2/R1*R2
RT = R1*R2 /R1+R2