The temperature of a tungsten sample is raised while a copper zample is maintained at 20℃. At what temperature will the resistivity of the tungsten sample be four times that of the copper sample?
Solving the question
To find the temperature at which the resistivity of the tungsten sample is four times that of the copper sample, we need to consider the temperature coefficient of resistivity for both materials.
The temperature coefficient of resistivity (alpha) represents the change in resistivity per degree Celsius change in temperature. It is typically given in units of per degree Celsius (℃^-1) or per Kelvin (K^-1).
Let's denote the temperature of the tungsten sample as T (in degrees Celsius) and the resistivity of the tungsten sample as ρ_w. Similarly, let's denote the resistivity of the copper sample as ρ_c.
The resistivity of a material can be related to its resistivity at a reference temperature (usually 20℃ or 25℃) using the equation:
ρ(T) = ρ_ref * (1 + α*(T - T_ref))
Where:
- ρ(T) is the resistivity of the material at temperature T
- ρ_ref is the resistivity of the material at the reference temperature
- α is the temperature coefficient of resistivity
- T_ref is the reference temperature
Based on the problem statement, we want the resistivity of the tungsten sample to be four times that of the copper sample. Mathematically, we can express this as:
ρ_w = 4 * ρ_c
Substituting the resistivity equation for both materials, we have:
ρ_ref_w * (1 + α_w*(T - T_ref)) = 4 * (ρ_ref_c * (1 + α_c*(20 - T_ref)))
Simplifying the equation:
(1 + α_w*(T - T_ref)) = 4 * (1 + α_c*(20 - T_ref))
Expanding and rearranging:
α_w*T - α_w*T_ref + 1 = 4*α_c*20 - 4*α_c*T_ref
α_w*T - 4*α_c*T_ref = 4*α_c*20 - 1
α_w*T = 4*α_c*(20 - T_ref) + 1 + 4*α_c*T_ref
T = (4*α_c*(20 - T_ref) + 1 + 4*α_c*T_ref) / α_w
Here, T_ref represents the reference temperature, which is typically given as 20℃.
To find the specific temperature T, we need to know the temperature coefficients of resistivity (alpha) for both tungsten and copper.
Once we have those values, we can plug them into the equation to determine the temperature at which the resistivity of the tungsten sample is four times that of the copper sample.