EFFECT OF TEMPERATURE ON RESISTANCE: If the temperature increases, the resistance of the pure metal also increases as the resistivity of these materials depends on the temperature. So metals have positive temperature of co-efficient of resistance.
The increase of resistance is large and fairly regular for normal ranges of temperature, so temperature-resistance graph is straight line.
On the other hand, the resistance of electrolytes, insulators, and others non metallic substances normally decreases with increase of temperature.
In case of alloys, the resistance increases in small amount and irregularly with increase of temperature. High resistance alloys like Eureka(60% cupper & 40% nickel), manganin, the increase of resistance is negligible over an increase of temperature at considerable range.

TEMPERATURE CO-EFFICIENT OF RESISTANCE: Suppose in a metallic conductor, the measured resistance at 0°c is R0 and Rt at t°c. So the increase of resistance of that conductor is ΔR= Rt– R0.
Or, Rt-R0∞R X t
Or, Rt-R0 = αR0t
Or, Rt=R0(1+ αt)
α is a constant and known as temperature co-efficient of resistance of the conductor.
Or, α = Rt-R0/R0t.
If R0= 1 ohm and t=1°c then α= R0b>0.
So, temperature co-efficient of resistance may be defined as – with per °c temperature rise, the increase of the resistance of the conductor per ohm original resistance.

Points to be noted that if the temperature of a conductor decrease, its resistance also decreases. As per graph at -t°c temperature, the resistance of the conductor becomes zero, practically which is not possible, the curve departs from the straight line at very low temperature. The temperature is referred as inferred zero resistance temperature.
In case of copper the temperature is -234.5°c
The temperature co-efficient of resistance that is α also depends on the temperature. The value of α at 0°c is maximum value for any conductor.

Suppose an equation is Rt=R0(1+αt)
where Rt= Final resistance at t°c,
R0= Resistance at 0°c, and α0 is temperature co-efficient at 0°c.
If we decrease the temperature from t°c to 0°c, then
final resistance became R0.
so, R0=Rt(1+αt[-t])
So, αt=(Rt-R0)/Rtt.
AS, Rt=R0(1+α0t)
so, the equation is αt=[R0(1+α0t)-R0]/R0(1+α0t)t
= α0/(1+α0t).
So, αt= α0/(1+α0t).

RESISTANCE IN SERIES: In case of a circuit consisting some resistance are connected in series, then the total resistance is the algebraic sum of all individual resistance.
Let, R1, R2, R3 are connected in series in a circuit, R is the total equivalent resistance, then,
R = R1+R2+R3 or, IR = IR1+IR2+IR3
Or, V= IR1+IR2+IR3.
Here we see that, some current flows through all resistance, that means in series circuit current does not change. The voltage drops are individual according to their resistance but total voltage drops are additive and equal to the applied voltage. And of-course resistances are also additive.

VOLTAGE DIVIDER RULE: In series circuit, same current flows through each resistor, voltage drops varies with the resistance.
In this circuit voltage drop against R1 resistance is V1= V X R1/R = V X R1/(R1+R2+R3)
Similarly, V2= VXR2/R = V X R2/(R1+R2+R3)
V3= V X R3/R = V X R3/(R1+R2+R3).

RESISTANCE IN PARALLEL: When some resistances are connected parallel in a circuit such as
Then, 1/R= 1/R1+1/R2+1/R3
OR, V/R=V/R1+V/R2+V/R3
OR, I=I1+I2+I3
Here we see that, the current is not same of all branches, but the voltage is same across the individual resistor. The total current, I, is the algebraic sum of brunch current made by voltage divided by R1 or, R2 or R3 as I1,I2 &I3. As I1= V/R1, I2=V/R2, or, I3=V/R3.


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