Resistance | SI Unit of Resistance
Resistance | SI Unit of Resistance :- G.S. Ohm in 1828, discovered a basic law regarding the flow of current(I) in a conductor and potential difference(V) across its terminals, called Ohm’s Law. According to Ohm’s Law :-
where the constant of proportionality R is called the resistance of the conductor.
S.I. unit of resistance
Volt/Ampere or Ohm(Ω)
Definition of 1 Ohm : – If one volt potential difference applied between the ends of a conductor results in one ampere current through it, then the resistance of the conductor is said to be one ohm.
1 Ω = 1 V/1 A
Factors on which resistance of a conductor depends (SI Unit of Resistance)
Resistance of a conductor depends on :-
- The material of the conductor(nature of material of conductor)
- Temperature &
- Dimensions of the conductor (i.e., length and breadth)
Let us understand the dependence of resistance on the dimensions of the conductor :-
Consider a conductor in the form of a slab of length l and cross sectional area A.
For a given voltage V across the slab, let I is the current through it.
Let us place two such identical slabs side by side, so that the length of the combination is 2l.
If V is the potential difference across the ends of the first slab, then V is also the potential difference across the ends of the second slab because the slabs are identical.
The potential difference across the ends of the combination = 2V
The current flowing through the combination will be the same as that flowing through either of the slabs = I
The resistance of the combination R’ = (2V)/I = 2R
Thus, doubling the length of a conductor doubles the resistance. Hence, resistance is directly proportional to the length of the conductor, i.e.,
R ∝ l …..(1)
Now let us, imagine dividing the slab into two parts by cutting it lengthwise.
The slab can be considered as a combination of two identical slabs of length l, but each having a cross sectional area of A/2.
As I is the current through the entire slab area A, then current flowing through each of the two half-slabs = I/2
Potential difference across the ends of the half-slabs = V
The resistance of the half slab R” = V/(I/2) = 2(V/I) = 2R
Thus halving the cross-section of a conductor doubles the resistance. Hence, the resistance R is inversely proportional to the cross-sectional area, i.e.,
R ∝ 1/A …..(2)
From equations (1) and (2),
where ρ is a constant of proportionality and it depends only on the material of the conductor and not on its dimensions. This constant is called resistivity.
V = IR
Where σ = (1/ρ) is called the conductivity.
Equation (5) is called microscopic form of Ohm’s Law.
Temperature dependence of the resistance of the conductor
Resistance of conductors depends on temperature. If the resistance of a conductor is R0 at reference temperature((i.e., room temperature 293K or 20ºC) then the resistance of a conductor Rt at T 0C is given by the following formula :-
Here, α = resistance coefficient of temperature and ΔT = rise in temperature.
The value of α is positive for metals and negative for semi-conductors and insulators.
The resistance of a conductor decreases linearly with decrease in temperature and becomes zero at a specific temperature. This temperature is called critical temperature. At this temperature, the conductor behaves like a superconductor.
- If a wire is stretched to n times of it’s original length, its new resistance will be n2 times.
- If a wire is stretched such that it’s radius is reduced to (1/n)th of it’s original values, then resistance will increases n4 times similarly resistance will decrease n4 time if radius is increased n times by contraction.
- Resistivity of alloys is greater than their metals.
- Temperature coefficient of alloys is lower than pure metals.
- Resistance of most of non metals decreases with increase in temperature. (e.g. carbon)
- The resistivity of an insulator (e.g. amber) is greater then the metal by a factor of 1022
- Temperature coefficient (α) of semi conductor including carbon (graphite), insulator and electrolytes is negative.
Figure shows a conductor of length L having a circular cross-section. The radius of cross-section variance linearly from a to b. The resistivity of the material is ρ. Find the resistance of the conductor.
Let us consider a small strip of thickness dx at a distance x from the left end of the conductor.
Resistance of small strip of thickness dx,
Differentiating w.r.t. y, we get
Using the value of dx in equation (1),
Integrating both sides,
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