To what temperature must you raise a copper wire, originally at 20.0°C, to double its resistance, neglecting any changes in dimensions? (Does this happen in household wiring under ordinary circumstances?)
To find the temperature at which the resistance of a copper wire doubles, we need to use the temperature coefficient of resistance for copper, which is given as 0.00393 per degree Celsius.
The formula we can use is:
R2 = R1 * (1 + α * (T2 - T1))
Where:
R1 is the initial resistance of the wire,
R2 is the doubled resistance we want to achieve,
α is the temperature coefficient of resistance, and
T1 and T2 are the initial and final temperatures, respectively.
In this case:
R1 is the initial resistance of the wire,
R2 is 2 * R1,
α is 0.00393 per degree Celsius, and
T1 is 20.0°C.
We can rearrange the formula as:
T2 = ((R2 / R1) - 1) / α + T1
Substituting the known values:
T2 = ((2 * R1 / R1) - 1) / 0.00393 + 20.0
T2 = 507.93 + 20.0
T2 ≈ 527.93°C
Therefore, to double the resistance of the copper wire, you need to raise its temperature to approximately 527.93°C.
This phenomenon does not occur in household wiring under ordinary circumstances. The temperature rise required to double the resistance of a copper wire is far beyond the operating temperatures of household appliances and electrical systems. Household wiring operates at much lower temperatures, so we don't typically see a doubling of resistance due to temperature effects.
To determine the temperature to which you need to raise the copper wire to double its resistance, you can use the Temperature Coefficient of Resistance (α) for copper.
The general formula to calculate the change in resistance with temperature is:
ΔR = R₀ * α * ΔT
Where:
ΔR is the change in resistance
R₀ is the initial resistance
α is the temperature coefficient of resistance
ΔT is the change in temperature
For copper, the temperature coefficient of resistance is approximately 0.0039 (°C)^-1.
To double the resistance, the change in resistance (ΔR) is equal to the initial resistance (R₀).
Therefore, we can rewrite the formula as follows:
R₀ = R₀ * α * ΔT
Since we want to double the resistance, let's let R₁ be the new resistance, so:
R₁ = 2 * R₀
Now, substituting R₁ into the formula:
2 * R₀ = R₀ * α * ΔT
Simplifying:
2 = α * ΔT
Now, let's solve for ΔT:
ΔT = 2 / α
Substituting the value for α (0.0039):
ΔT = 2 / 0.0039
ΔT ≈ 512.82°C
In order to double the resistance of the copper wire, you would need to raise its temperature to approximately 512.82°C.
Regarding household wiring under ordinary circumstances, temperatures of this magnitude are not typically encountered. Normal household wiring operates within a safe temperature range to avoid hazards such as overheating, fire, or damage to insulation.