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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 :-

$\displaystyle V\propto I$

$\displaystyle V=IR$

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),

$\displaystyle R\propto \frac{l}{A}$

$\displaystyle R=\rho \frac{l}{A}$ …..(3)

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.

Now

V = IR

$\displaystyle V=I\rho \frac{l}{A}$

$\displaystyle \Rightarrow \frac{V}{l}=\frac{I}{A}\rho$

And,

V/l = E (Electric field intensity) and I/A = J(Current density)

$\displaystyle \therefore E=J\rho$ …..(4)

$\displaystyle \Rightarrow J=\frac{1}{\rho }E$

$\displaystyle J=\sigma E$ …..(5)

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 R_t at T ⁰C is given by the following formula :-

$\displaystyle {{R}_{t}}={{R}_{0}}(1+\alpha \Delta T)$

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.

Note :-

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.

Example 1.

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.