Resistivity Formula

Introduction to Resistivity

Electrical resistivity is the resistance of a material's movement of current from one end to the other. It is a straight forward and insightful metric for describing a material. It's the electrical conductivity's inverse. The resistivity is denoted by ρ and is proportional to both the material resistance and the volume. The region of a cross-section of a given material is inversely proportional to its resistivity.


The resistance R of a specimen, such as a wire, multiplied by its cross-sectional area A and divided by its length l equals resistivity, which is usually symbolised by the Greek letter rho; ρ = \[\frac{RA}{I}\]. The ohm is the unit of resistance. The ratio of the area in square metres to length in metres is simplified to just metres in the metre-kilogram-second (mks) scheme. The unit of resistivity in the metre-kilogram-second system is ohm-meter. If distances are measured in centimetres, resistivity can be expressed in ohm-centimetre.


At 20° C (68° F), the resistivity of a very strong electrical conductor like hard-drawn copper is 1.77 x 10\[^{-8}\] ohm-metre or 1.77 x 10\[^{-6}\] ohm-centimetre. Electrical insulators, on the other hand, have resistivities ranging from 10\[^{12}\] to 10\[^{12}\] ohm-metres.


The value of resistivity is often affected by the temperature of the material; resistivity tables usually list values at 20° C. The resistivity of metallic conductors increases as the temperature rises, while the resistivity of semiconductors such as carbon and silicon decreases as the temperature rises.


Formula of Resistivity

The resistivity formula is expressed as -

ρ = \[\frac{RA}{I}\]

Where ρ is the resistivity, R is the resistance, l is the material's thickness, and A is the cross-sectional area.


Resistance Formula

Electrical resistance is proportional to the conductor's length (L) and inversely proportional to its cross-sectional area (A). The following relationship gives the resistance formula.

R = \[\frac{\rho L}{A}\]

where ρ is the resistivity of the material (measured in Ωm, ohm meter)


Ohm's Law Formula

The relationship between an electric current and a potential difference is defined by Ohm's law.

If all physical conditions and temperature remain constant, Ohm's law states that the voltage across a conductor is directly proportional to the current flowing through it.

Mathematically, ohm's law formula can be written as,

V = IR

Resistance is the constant of proportionality in the equation, with units of ohms and the symbol R.

The current and resistance can be calculated using the same formula by rewriting it as follows:

I = \[\frac{V}{R}\]

R = \[\frac{V}{I}\]


Resistors in Parallel Formula

When both terminals of a resistor are connected to each terminal of the other resistor or resistors, they are said to be connected in parallel.

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Since the supply current will flow in different directions, the current may not be the same across all of the parallel network's branches. In a parallel resistive network, however, the voltage drop over all of the resistors is the same. Then, all parallel-connected elements have a common voltage across them, and this is valid for all resistors in parallel.

In the above diagram, three resistors are connected in parallel. Let R1,R2and R3 be individual resistance.

Resistors in parallel formula are given below

\[\frac{1}{R_{T}}\] = \[\frac{1}{R_{1}}\] + \[\frac{1}{R_{2}}\] + \[\frac{1}{R_{3}}\]


Solved Examples:

1. Calculate the Resistivity of a Material with a Resistance of 2 and a Cross-Sectional Area and Length of 25 Cm² and 15 Cm, Respectively.

Sol: Given

R = 2 Ω

l = 15 cm = 0.15 m

A = 25 cm² = 0.25 m²

We know the Resistivity formula is

ρ = \[\frac{RA}{I}\]

Put the value of R, A and l in the above formula

ρ = \[\frac{2 \times 0.25}{0.15}\]Ω

Hence resistivity = 3.333 Ω


2. The Wire's Length and Area are 0.2 M and 0.5m², Respectively. Calculate the Resistivity of That Wire Whose Resistance Is 3 Ω. 

Ans: Given

R = 3 Ω

l = 0.2 m and

A = 0.5 m²

Resistivity formula is

ρ = \[\frac{RA}{I}\]

Put the value of R, A and l in the above formula

ρ = \[\frac{3 \times 0.5}{0.2}\] = 7.5 Ω

Hence resistivity value is 7.5 Ω


Conclusion

In the above article, we have discussed resistivity and formulas to calculate resistivity. A conductor's resistance (R) is proportional to its length (L) as R ∝ L. We have seen what factors resistivity depends on. We have also discussed ohm’s law and resistors in parallel formula. 

FAQs (Frequently Asked Questions)

1. What are the Factors Affecting Electrical Resistance?

Ans: The electrical resistance of a conductor is dependent on the following factors:

  • The cross-sectional area of the conductor

  • Length of the conductor

  • The material of the conductor

  • The temperature of the conducting material

2. What Happens to the Resistance of Pure Metals With an Increase in Temperature?

Ans: The resistance of pure metals increases as the temperature rises. The explanation for this is that the number of electrons in the conduction band increases, reducing mobility and thereby increasing resistance.

3. Why is Resistance Less in Parallel?

Ans: When resistors are connected in parallel, more current flows from the source than would flow if they were connected separately, resulting in a lower total resistance.