Which of the following statements are correct about the electrical resistance and resistivity of a wire? 1. Both quantities depend on the area of cross-section of the wire 2. Both depend on the temperature 3. Resistance of the wire is directly proportional to the resistivity of the wire 4. Resistivity of the wire is directly proportional to the length of the Select the correct answer using the code given below:
wire
2 and 3
Understanding Electrical Resistance and Resistivity
Let's analyze the given statements about the electrical resistance and resistivity of a wire. Electrical resistance ($R$) is a measure of how much a material opposes the flow of electric current. Resistivity ($\rho$) is an intrinsic property of the material itself, indicating its fundamental opposition to current flow, independent of its dimensions.
The relationship between resistance, resistivity, length ($L$), and area of cross-section ($A$) of a uniform wire is given by the formula:
\( R = \rho \frac{L}{A} \)
Now let's evaluate each statement:
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Statement 1: Both quantities depend on the area of cross-section of the wire.
Resistance ($R$) depends on the area of cross-section ($A$) as shown in the formula \( R = \rho \frac{L}{A} \). Specifically, resistance is inversely proportional to the area of cross-section. However, resistivity ($\rho$) is a material property and does not depend on the dimensions (length or area) of the wire. Therefore, this statement is incorrect.
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Statement 2: Both depend on the temperature.
Both resistance ($R$) and resistivity ($\rho$) of most materials, especially conductors, are dependent on temperature. As temperature increases, the thermal vibrations of atoms in the material increase, which hinders the flow of electrons, thus increasing both resistivity and resistance. For semiconductors and insulators, the dependence can be different. However, generally speaking for metallic wires, both quantities depend on temperature. Therefore, this statement is correct.
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Statement 3: Resistance of the wire is directly proportional to the resistivity of the wire.
From the formula \( R = \rho \frac{L}{A} \), if the length ($L$) and area of cross-section ($A$) of a specific wire are kept constant, the resistance ($R$) is directly proportional to the resistivity ($\rho$) of the material it is made from. Therefore, this statement is correct.
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Statement 4: Resistivity of the wire is directly proportional to the length of the wire.
Resistivity ($\rho$) is an intrinsic property of the material and does not depend on the dimensions of the wire, including its length ($L$). Changing the length of a wire made of a specific material changes its resistance, but not its resistivity. Therefore, this statement is incorrect.
Based on the analysis of each statement:
- Statement 1 is incorrect.
- Statement 2 is correct.
- Statement 3 is correct.
- Statement 4 is incorrect.
The correct statements are 2 and 3.
| Statement | Analysis | Correctness |
|---|---|---|
| 1. Both depend on area of cross-section. | Resistance depends on Area, but Resistivity does not. | Incorrect |
| 2. Both depend on temperature. | Both Resistance and Resistivity generally depend on temperature. | Correct |
| 3. Resistance is directly proportional to resistivity. | \(R = \rho \frac{L}{A}\). For fixed L and A, R is directly proportional to \(\rho\). | Correct |
| 4. Resistivity is directly proportional to length. | Resistivity is a material property, independent of length. | Incorrect |
Revision Table: Electrical Properties
| Property | Symbol | Depends On | Independent Of | Formula (for uniform wire) | Unit |
|---|---|---|---|---|---|
| Resistance | \(R\) | Resistivity, Length, Area of cross-section, Temperature | Current, Voltage (for Ohmic materials) | \( R = \rho \frac{L}{A} \) | Ohm (\(\Omega\)) |
| Resistivity | \(\rho\) | Material type, Temperature | Length, Area of cross-section | Derived from \( R = \rho \frac{L}{A} \), so \( \rho = \frac{R A}{L} \) | Ohm-meter (\(\Omega \cdot m\)) |
Additional Information on Resistance and Resistivity
What is Resistance?
Resistance is the opposition offered by a substance to the flow of electric current. It is measured in Ohms (\(\Omega\)). When a voltage ($V$) is applied across a resistor, a current ($I$) flows through it, and the relationship is given by Ohm's Law: \( V = I R \).
What is Resistivity?
Resistivity is an intrinsic property of a material that quantifies how strongly it resists electrical current. It is a fundamental characteristic of the material itself, similar to density or specific heat capacity. It is measured in Ohm-meters (\(\Omega \cdot m\)). Low resistivity indicates that a material allows current to flow easily (like copper or aluminum), while high resistivity means it opposes current flow strongly (like rubber or glass).
Temperature Dependence:
For most metals, resistivity increases linearly with temperature over a certain range. This is because as temperature rises, atoms vibrate more vigorously, scattering the charge carriers (electrons) more frequently, thus increasing resistance. This temperature dependence is often described by the formula:
\( \rho_T = \rho_0 [1 + \alpha (T - T_0)] \)
where \(\rho_T\) is the resistivity at temperature \(T\), \(\rho_0\) is the resistivity at a reference temperature \(T_0\), and \(\alpha\) is the temperature coefficient of resistivity.
Since \( R = \rho \frac{L}{A} \), if \(L\) and \(A\) are constant, resistance will also show a similar temperature dependence as resistivity.
Important Questions from Resistance and Resistivity
A circular coil of single turn has a resistance of 20 Ω. Which one of the following is the correct value for resistance between the ends of any diameter of the coil?
Which one of the following physical quantities does NOT affect the resistance of a cylindrical resistor?
Let us consider a copper wire having radius r and length l. Let its resistance be R. If the radius of another copper wire is 2r and the length is l/2 then the resistance of this wire will be
A fuse wire must be
The product of conductivity and resistivity of a conductor
