Introduction to Resistivity
Electrical resistivity is the resistance of a material's movement of current from one end to the other. It is a straightforward 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; \[ \rho = \frac{RA}{I} \]. 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-metre. If distances are measured in centimetres, resistivity can be expressed in ohm-centimetre.
At 200C (680 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 1012 to 1012 ohm-metres.
The value of resistivity is often affected by the temperature of the material; resistivity tables usually list values at 200C. 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 -
\[ \rho = \frac{RA}{I} \]
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 metre)
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:
\[ I = \frac{V}{R} \]
\[ R = \frac{V}{I} \]
What is the Difference between Resistance and Resistivity?
The resistance of a material refers to the opposition of the flow of electrons in a material whereas resistivity occurs in a material when resistance is offered. The resistance is calculated as the ratio of voltage (V) to the current (I) applied on a material and the resistivity is the ratio of the electric field (E) to the current density (J). The resistance is measured in terms of ohms whereas resistivity is measured in terms of ohms metre. The symbol for representing resistance is R while the symbol for the representation of resistivity is ρ. The resistance of a material is dependent on factors like length of the conductor, cross-sectional area of the conductor, and the temperature of the conductor while the resistivity of a conductor is dependent only on the temperature of the conductor.
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,R2 and 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
\[\rho = \frac{RA}{A}\]
Put the value of R, A and l in the above formula
\[\rho = \frac {2\times 0.25}{0.15} Ω \]
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
\[ \rho = \frac{RA}{I} \]
Put the value of R, A and l in the above formula
\[ \rho = \frac{3 \times 0.5}{0.2} = 7.5 Ω \]
Hence resistivity value is 7.5 Ω
Questions for Practising and Self-Assessment
Here are some questions for you to practice and test your knowledge on the topic.
Question 1: Calculate the resistance between two opposite faces if there is a thin square sheet with side L and thickness t.
Question 2: wWhat will you observe if pieces of copper and germanium are cooled to 80K from room temperature?
Question 3: What will be the resistance of a wire at 0 degrees celsius if its resistance at 50 degrees celsius is 5 ohm and at 100 degrees Celsius is 6 ohm?
Question 4: What would be the resistivity of a material if it has a resistance of 2 ohms, area of cross-section of 25 cm square, and length of 15 cm?
Question 5: What will be the resistivity of a metal wire which has a length of 2 m and diameter of 0.6 mm if it has a resistance of 50 ohms?
(Tip: keep the SI unit of length and diameter same)
Question 6: Calculate the resistivity of a wire having length 0.2 m, area 0.5 metre square, and resistance of 3 ohms.
Question 7: What factor can reduce the resistance of material through an electric cord?
Question 8: What will be the change in resistivity of material if its cross-sectional area is doubled but resistance and length are constant?
Question 9: What should we do if we want to reduce the resistance of a wire by a factor of 2?
Question 10: How will the resistance of an insulator change with a rise in temperature?
Question 11: Which material has a resistance equal to zero?
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 formulas.
FAQs on Resistivity Formula
1. What are the factors affecting electrical resistance?
The electrical resistance in any circuit represents the ratio of the voltage in the circuit and the current flowing through the circuit. 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
The electrical resistance of a circuit increases with the increase in the length of the conductor and decrease in the cross-sectional area of the conductor.
2. What happens to the resistance of pure metals with an increase in temperature?
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. With the increase in temperature, the thermal motion among the atoms of the lattice also increases due to which the charge carriers bump more into the lattice vibrations which increases the resistivity of the pure metal. In the case of semiconductors, this theory is not true, and therefore, there are semimetals also in which the two effects compete with each other, and trends in temperature range can be observed.
3. Why is resistance less in parallel?
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. In parallel circuits, the net resistance acting on the circuit decreases because more components are added in the circuit because of which current has more paths to pass through. The potential difference across the two resistors is the same but the current flowing through them depends on their resistances.
4. What is the resistivity of different materials?
The resistivity of different materials is different. The resistance and resistivity of pure metal increase with rising in the absolute temperature of the pure metal, if the temperature coefficient of the metal is (1/273) degrees celsius inverse. The resistivity of alloys is similar to pure metal, that is, it also increases with the rise in temperature, though the increase in the case of alloys is much smaller as compared to pure metals. The resistivity of semiconductors decreases exponentially as the temperature of the material rises. The resistivity of electrolytes also decreases when the temperature of the material rises.
5. Briefly discuss the important points of electrical resistance and resistivity.
The electrical resistance of a material increases or decreases with a rise and fall in temperature. The resistance of materials like insulator, semiconductor, and electrolyte decreases as the temperature increases. Due to an increase in temperature, some of the electrons present in the material acquire energy and therefore, become free to conduction. Therefore, in short, as the temperature increases, the conductivity and the resistance of the material increase and decrease respectively. The resistivity is a simple and accessible quantity of a material that refers to the resistance of current from one end to the other end.
