Resistivity and Resistance Formulas, MCQs, Assertion Reason Based Q&A, Solved Examples | Class 10 Science Electricity Notes

ANAND CLASSES Study Material and Notes to learn the relationship between resistance, length, and area of a conductor. Explore resistivity, its formula, units, examples, and practical uses in electric circuits. It includes FAQs, conceptual Q&A, MCQs with explanations, a “Do You Know?” section, a worksheet, a test paper with marks distribution, and key points for quick revision.

🔍 Q&A Format: Resistivity and Resistance

Q1. How is the resistance of a conductor related to its length and area of cross-section?

A.

Resistance (R) is directly proportional to the length (l) of the conductor:
$$R \propto l \tag{1}$$

Resistance is inversely proportional to the area of cross-section (A):
$$R \propto \frac{1}{A} \tag{2}$$

Q2. How do we combine the two relationships of resistance to get a general formula?

A.
By combining equations (1) and (2) in Q1, we get: $$R \propto \frac{l}{A} $$ $$ R = \rho \frac{l}{A} \tag{3}$$

Where ρ (rho) is a constant called resistivity or specific resistance of the material.

Q3. What does equation (3) imply about the relationship between resistance and resistivity?

A.
For a conductor of fixed length and area, the resistance is directly proportional to its resistivity.

• If the resistivity doubles, resistance also doubles.
• If the resistivity triples, resistance also triples.

Q4. What is the formula to calculate resistivity?

A.
Rearranging equation (3): $$\rho = R \frac{A}{l} \tag{4}$$

Where:

• R = Resistance
• A = Area of cross-section
• l = Length of the conductor

Q5. What is the definition of resistivity?

A.
Resistivity is defined as the resistance of a conductor of unit length (1 m) and unit cross-sectional area (1 m²).
So, $$\rho = R$$

Hence, resistivity is the resistance of a cube of material having sides of 1 metre.

Q6. What is the SI unit of resistivity?

A.
Using the formula: $$\rho = R \frac{A}{l}$$

Putting units:

• R = ohm (Ω)
• A = m²
• l = m

Unit of ρ : $$\text{Unit of } \rho = \frac{\Omega \cdot m^2}{m} = \Omega \cdot m$$

So, the SI unit of resistivity is ohm-metre (Ω·m).

Q7. On what factors does the resistivity of a substance depend?

A.

• Does not depend on length or thickness.
• Depends on:
  • Nature of the material
  • Temperature
    Resistivity is a characteristic property of the material.

Q8. What does low or high resistivity indicate?

A.

• Low resistivity: Good conductor (e.g., Silver, Copper, Aluminium)
• High resistivity: Poor conductor or insulator (e.g., Glass, Rubber, Ebonite)

Q9. What is the resistivity of copper? What does it mean?

A.
Resistivity of copper = $1.69 \times 10^{-8} \ \Omega \cdot m$
It means a copper rod of 1 m length and 1 m² area has resistance of $1.69 \times 10^{-8} \ \Omega$.

Q10. Why don’t we use silver for electrical wiring despite it being the best conductor?

A.
Silver has the lowest resistivity, making it the best conductor, but it is very expensive.
Hence, copper and aluminium are commonly used due to low cost and low resistivity.

Q11. Why are alloys used for making resistors and heating elements?

A.
Because:

1. High resistivity — provides high resistance.
2. Do not oxidize easily at high temperatures.

Example:

• Nichrome (Ni-Cr-Mn-Fe) is used in electric irons, toasters, etc.

The heating elements (or heating coils) of electrical heating appliances such as electric iron and toaster, etc., are made of an alloy rather than a pure metal because

(i) the resistivity of an alloy is much higher than that of pure metal, and

(ii) an alloy does not undergo oxidation (or burn) easily even at high temperature, when it is red hot.

For example, nichrome alloy is used for making the heating elements of electrical appliances such as electric iron, toaster, electric kettle, room heaters, water heaters (geysers), and hair dryers, etc., because :
(i) nichrome has very high resistivity (due to which the heating element made of nichrome has a high resistance and produces a lot of heat on passing current).
(ii) nichrome does not undergo oxidation (or burn) easily even at high temperature. Due to this nichrome wire can be kept red-hot without burning or breaking in air.

Q12. How does temperature affect resistivity?

A.

• Metals (Conductors): Resistivity increases with temperature.
• Insulators: Resistivity remains nearly unchanged.
• Semiconductors: Resistivity decreases with increase in temperature.

Q13. What are semiconductors and how are they useful?

A.
Semiconductors like silicon and germanium have resistivity between that of conductors and insulators.
Their resistivity changes with temperature and impurities, making them ideal for:

• Solar cells
• Transistors

Q14. What are the resistivities of some common substances at 20°C?

A.

Q15. Sample Problem: Calculate the resistivity of a copper wire of length 2 m, area $1.7 \times 10^{-6} m²$, and resistance $2 \times 10^{-2} \ \Omega.$

A.
Using: $\rho = \frac{RA}{l}$ $$\rho = \frac{2 \times 10^{-2} \times 1.7 \times 10^{-6}}{2} = 1.7 \times 10^{-8} \ \Omega\cdot m$$

Q16. Sample Problem: A copper wire has diameter 0.5 mm and resistivity $1.6 \times 10^{-8} \ \Omega\cdot m.$ Find the length for resistance of 10 Ω.

A.
Radius = 0.25 mm = $0.25 \times 10^{-3} \ m$
Area, $A = \pi r^2 = 3.14 \times (0.25 \times 10^{-3})^2 = 0.1964 \times 10^{-6} \ m^2$

Using: $\rho = \frac{RA}{l} \Rightarrow l = \frac{RA}{\rho} $

$$l = \frac{10 \times 0.1964 \times 10^{-6}}{1.6 \times 10^{-8}} = 122.7 \ m$$

Q17. What happens to resistance if the diameter of a wire is doubled?

A.
Resistance $$R \propto \frac{1}{d^2}$$, so if diameter is doubled: $$R \rightarrow \frac{1}{4} R$$

i.e., resistance becomes one-fourth.

Q18. Sample Problem: A 6 Ω wire is folded to make it half-length. What is the new resistance?

A.
Length becomes $\frac{l}{2}$, area becomes 2A:

$$R = \frac{\rho \cdot \frac{l}{2}}{2A} = \frac{\rho \cdot l}{4A}$$

So new resistance = $\frac{1}{4} \times 6 = 1.5 \ \Omega$

✅ Frequently Asked Questions (FAQs)

Q1. What is resistance?

Ans: Resistance is the property of a conductor that opposes the flow of electric current through it. It is denoted by R and measured in ohms (Ω).

Q2. What factors affect the resistance of a conductor?

Ans: Resistance of a conductor depends on:

• Length (l) – Resistance ∝ length
• Area of cross-section (A) – Resistance ∝ 1/Area
• Nature of material – Different materials offer different resistance
• Temperature – Resistance usually increases with temperature in conductors.

Q3. Define resistivity.

Ans: Resistivity (ρ) is a material-specific constant that measures how strongly a material opposes the flow of electric current. It is given by: $$\rho = \frac{RA}{l}$$

Unit of resistivity is ohm-meter (Ω·m).

Q4. On what factors does resistivity depend?

Ans: Resistivity depends on:

• The nature of the material
• Temperature
  It does not depend on the length or thickness of the wire.

🔍 Conceptual Questions with Answers

Q1. Why does a long wire offer more resistance than a short wire?

Ans: Resistance is directly proportional to the length of the wire. A longer wire has more atoms to obstruct the flow of electrons, thus offering more resistance.

Q2. What happens to the resistance if the diameter of a wire is doubled?

Ans: Resistance is inversely proportional to the square of the diameter. If diameter is doubled, resistance becomes one-fourth.

Q3. Two wires of the same material and length have different thicknesses. Which has more resistance?

Ans: The thinner wire has more resistance because it has a smaller cross-sectional area.

Q4. Why are alloys used in heating appliances instead of pure metals?

Ans: Alloys like nichrome have:

• High resistivity (produce more heat)
• High melting point
• Do not oxidize easily when red hot.

5. Why is copper used more widely in household electrical wiring than silver?

Ans: Though silver is a better conductor, copper is used because it is:

• Much cheaper,
• Has low resistivity, and
• Is ductile and durable.

Q6. A wire has resistance R. What will be its new resistance if it is stretched to double its length without changing its volume?

Ans: If volume remains constant and length doubles then area of cross section of wire becomes half (only case of stretch), resistance becomes four times because: $$R’ = \rho \frac{(2l)}{A/2} = 4R$$

Q7. Why does resistance increase with temperature in metals?

Ans: As temperature increases, atoms in the metal vibrate more, causing more frequent collisions with electrons, increasing resistance.

Q8. How does resistivity of a material help us in identifying it?

Ans: Each material has a unique resistivity at a given temperature. Comparing the measured resistivity with standard values helps in identifying the material.

Q9. How does the resistance of a wire change when its cross-sectional area is halved?

Ans: Resistance is inversely proportional to the area. So, if area is halved, resistance becomes twice.

Q10. A wire of resistivity ρ has length l and area A. If both length and area are doubled, how does resistance change?

Ans: $$R = \rho \frac{l}{A} $$ $$\text{and new } R’ = \rho \frac{2l}{2A} = \rho \frac{l}{A} = R$$

So, resistance remains the same.

✅ Multiple Choice Questions (MCQs) with Explanation

Q1. The unit of resistivity is:
a) Ohm
b) Ohm-meter
c) Meter/Ohm
d) Ohm/meter
✔️Correct Answer: b) Ohm-meter
Explanation: Resistivity is measured using the formula ρ = RA/l, which gives the unit Ω·m.

Q2. Which of the following materials has the lowest resistivity?
a) Copper
b) Silver
c) Iron
d) Nichrome
✔️Correct Answer: b) Silver
Explanation: Silver has the lowest resistivity (1.60 × 10⁻⁸ Ω·m), making it the best conductor.

Q3. If the length of a wire is doubled, then the resistance becomes:
a) Half
b) Same
c) Double
d) Four times
✔️Correct Answer: c) Double
Explanation: Resistance ∝ length, so doubling length doubles the resistance.

Q4. Which physical quantity does not affect the resistivity of a material?
a) Temperature
b) Length
c) Nature of material
d) Atomic structure
✔️Correct Answer: b) Length
Explanation: Resistivity is independent of length and area; it only depends on the material and temperature.

Q5. Which of the following statements is true about resistivity?
a) It changes with shape of material
b) It changes with the size of material
c) It is the same for all metals
d) It is constant for a given material at a given temperature
✔️Correct Answer: d)
Explanation: Resistivity is a constant property of a material at a given temperature.

Q6. A material has high resistivity. It can be used for:
a) Connecting wires
b) Heating elements
c) Electric bulbs
d) Batteries
✔️Correct Answer: b)
Explanation: Materials with high resistivity are poor conductors and produce more heat, hence used in heaters.

Q7. If the area of cross-section of a wire is tripled, what happens to the resistance?
a) Tripled
b) Becomes one-third
c) Remains unchanged
d) Becomes nine times
✔️Correct Answer: b)
Explanation: $R \propto \frac{1}{A} \Rightarrow R \text{ becomes one-third}$

Q8. Which one is not a good conductor of electricity?
a) Silver
b) Copper
c) Aluminium
d) Nichrome
✔️Correct Answer: d)
Explanation: Nichrome has high resistivity, so it is not a good conductor compared to metals like copper or silver.

💡 Do You Know?

• Silver is the best conductor of electricity, but we use copper and aluminium because they are cheaper and still have low resistivity.
• Alloys like nichrome are preferred in heaters because they don’t oxidize easily and have high resistance.
• Semiconductors like silicon and germanium have resistivity between conductors and insulators, and their resistivity decreases with increase in temperature.

📄 Worksheet: Practice Questions

1. Fill in the blanks:
a) Resistance is measured in __________.
b) Resistivity depends on the __________ of the material.
c) The formula of resistivity is __________.

2. True or False:
a) Resistivity depends on the length of the conductor.
b) Nichrome is a poor conductor of electricity.
c) Alloys have higher resistivity than pure metals.

3. Match the following:

📝 Test Paper – Class 10 Science: Resistivity and Resistance

Total Marks: 20 | Time: 30 Minutes

Section A – 1 Mark Each (5 Marks)

1. Define resistance.
2. What is the SI unit of resistivity?
3. Write the formula for resistivity.
4. Mention one factor that affects resistivity.
5. State whether resistivity increases or decreases in semiconductors with temperature.

Section B – 2 Marks Each (6 Marks)
6. Explain why copper is used in electrical wires instead of silver.
7. What will be the resistance of a wire if its length is doubled and thickness is also doubled?
8. State two reasons why alloys are used in heating elements.

Section C – 3 Marks Each (9 Marks)
9. A wire of length 2 m and area of cross-section 1.7 × 10⁻⁶ m² has a resistance of 2 × 10⁻² Ω. Calculate the resistivity of the material.
10. A copper wire has a diameter of 0.5 mm and resistivity of 1.6 × 10⁻⁸ Ω·m. Find the length of the wire if its resistance is 10 Ω.
11. Explain how resistance changes when a wire is folded in half.

📌 Important Points for Quick Revision

• R ∝ l and R ∝ 1/A
• Resistivity (ρ) is a property of material: $\rho = \frac{RA}{l}$
• SI Unit of resistivity = Ohm-meter (Ω·m)
• Resistance depends on length, area, material, and temperature.
• Resistivity depends only on material and temperature.
• Best Conductors: Silver < Copper < Aluminium
• Used in heaters: Nichrome, due to high resistivity and oxidation resistance
• Semiconductors: Germanium and Silicon (resistivity decreases with temperature)