A current is flowing through a metallic wire. If the wire is heated, which quantities change?
All of the above
Understanding the Effect of Heating a Metallic Wire
When a current flows through a metallic wire, and the wire is heated, its physical properties change. This change in properties affects how easily the current can flow through the wire. Let's analyze how heating affects the key quantities mentioned in the options: drift speed, resistivity, and resistance of the metallic wire.
How Heating Affects Resistivity of a Metallic Wire
Resistivity is an intrinsic property of the material that determines how strongly it resists the flow of electric current. For metals, resistivity is primarily due to the scattering of free electrons by the vibrating atoms (ions) of the metal lattice. When a metallic wire is heated, the thermal energy of the atoms increases, causing them to vibrate more vigorously. These increased vibrations lead to more frequent collisions between the free electrons and the lattice atoms. These increased collisions impede the directed motion of the electrons, thereby increasing the resistivity of the material.
The relationship between resistivity and temperature for metals is approximately linear over a certain temperature range, given by:
\(\rho = \rho_0 [1 + \alpha (T - T_0)]\)
where:
- \(\rho\) is the resistivity at temperature \(T\).
- \(\rho_0\) is the resistivity at reference temperature \(T_0\).
- \(\alpha\) is the temperature coefficient of resistivity (positive for metals).
Since \(\alpha\) is positive for metallic conductors, increasing the temperature \(T\) above \(T_0\) increases the resistivity \(\rho\).
Changes in Resistance When Heating a Metallic Wire
Resistance (\(R\)) of a wire is related to its resistivity (\(\rho\)), length (\(L\)), and cross-sectional area (\(A\)) by the formula:
\(R = \rho \frac{L}{A}\)
Assuming the length and area of the wire do not change significantly due to heating (thermal expansion is usually small compared to the change in resistivity's effect), any change in resistivity directly affects the resistance. Since heating a metallic wire increases its resistivity, it also increases its resistance. This means the resistance of the metallic wire changes when it is heated.
Impact on Drift Speed in a Heated Metallic Wire
Drift speed (\(v_d\)) is the average velocity attained by charge carriers (electrons in metals) in a material due to an electric field. The current (\(I\)) flowing through a conductor is related to the drift speed by the equation:
\(I = n e A v_d\)
where \(n\) is the number density of charge carriers, \(e\) is the charge of each carrier, and \(A\) is the cross-sectional area. If the potential difference across the wire is kept constant, the current decreases as the resistance increases (due to heating). Since \(I = neAv_d\) and \(n\), \(e\), \(A\) are constant for the given metallic wire, a decrease in current \(I\) implies a decrease in drift speed \(v_d\).
Alternatively, even if the current is kept constant (e.g., by a constant current source), the electric field inside the wire will increase to maintain the current against the increased resistance. However, the increased scattering of electrons at higher temperatures reduces the time between collisions and the average distance travelled between collisions. For a given electric field, the average drift speed is proportional to the mean free time between collisions. Since the mean free time decreases with increased thermal vibrations, the drift speed tends to decrease for a given electric field or current density.
In summary, heating the metallic wire leads to increased resistance and either decreased current (for constant voltage) or altered electric field and reduced mean free time, all of which result in a change (specifically, a decrease) in the drift speed of electrons.
Conclusion: Quantities that Change Upon Heating
Based on the analysis:
- Resistivity of the metallic wire increases with temperature.
- Resistance of the metallic wire increases with resistivity.
- Drift speed of electrons in the metallic wire changes (decreases) due to increased scattering and/or changes in current/electric field.
Therefore, all three quantities — drift speed, resistivity, and resistance — change when a metallic wire is heated.
| Quantity | Change upon Heating Metallic Wire | Reason |
|---|---|---|
| Resistivity | Increases | Increased thermal vibrations scatter electrons more. |
| Resistance | Increases | Directly proportional to resistivity (\(R \propto \rho\)). |
| Drift Speed | Changes (Decreases) | Increased scattering reduces mean free time; current may decrease. |
Thus, when a current is flowing through a metallic wire and it is heated, all of the mentioned quantities change. This covers drift speed, resistivity, and resistance.
Important Questions from Resistance and Resistivity
Which of the following metals has the lowest electrical resistivity?
When electric current is passed through a wire, the amount of heat produced in a wire depends upon _______.
I. Length
II. Thickness
A uniform wire of resistance 9Ω is bent in the form of an equilateral triangle. Find the effective resistance across a side of the triangle.
The value of carbon resistance is 54 × 103 Ω. The percentage tolerance is 5%. What is the colour code sequence of carbon resistance?
Which of the following relations are wrong?
I. The specific conductance is given by the relation \(k = \frac{1}{R}\left( {l/A} \right)\)
II. The equivalent conducting is given by the relation \(\lambda = \frac{{100\;K}}{C}\)
III. The specific resistance is given by the relation \(\rho = \frac{{Rl}}{A}\)
