Find the resistance at 100°C of a silver wire 1 mm in diameter. How much wire is needed?
To find the resistance of a silver wire at 100°C, we will use the formula for the resistance of a wire:
R = (ρ * L) / A
Where R is the resistance, ρ is the resistivity of silver, L is the length of the wire, and A is the cross-sectional area of the wire.
First, let's find the resistivity of silver at 100°C. The resistivity of silver at room temperature is 1.59 x 10^-8 Ω.m.
The temperature coefficient of resistivity for silver is 0.0038 per degree Celsius. To find the resistivity at 100°C, we can use the equation:
ρ100 = ρ0 * (1 + α * (T - T0))
Where ρ0 is the resistivity at room temperature, α is the temperature coefficient of resistivity, T is the temperature in Celsius, and T0 is the reference temperature (room temperature).
ρ100 = (1.59 x 10^-8 Ω.m) * (1 + 0.0038 * (100 - 20))
ρ100 = 1.59 x 10^-8 Ω.m * 1.076
ρ100 = 1.71 x 10^-8 Ω.m
Now, let's calculate the resistance at 100°C. We will assume the wire has a length of 1 meter.
R = (ρ * L) / A
R = (1.71 x 10^-8 Ω.m * 1 meter) / [(π * (0.001m / 2)^2)]
R = (1.71 x 10^-8 Ω.m * 1) / [π * (5 x 10^-4m)^2]
R = (1.71 x 10^-8 Ω.m) / [π * 2.5 x 10^-7m^2]
R = 2.17 x 10^-1 Ω
So, the resistance of the silver wire at 100°C is approximately 0.217 Ω.
To find out how much wire is needed, we need to consider the wire's resistance per unit length. The resistance per unit length can be calculated using the formula:
R/L = (ρ * A) / L
where R/L is the resistance per unit length, ρ is the resistivity, A is the cross-sectional area, and L is the length.
R/L = (1.71 x 10^-8 Ω.m * (π * (5 x 10^-4m)^2)) / 1 meter
R/L = (1.71 x 10^-8 Ω.m * 3.14 * 2.5 x 10^-7m^2) / 1 meter
R/L = (1.71 x 10^-8 Ω.m * 7.85 x 10^-7m^2) / 1 meter
R/L = 1.34 x 10^-14 Ω/m
So, the resistance per unit length of the silver wire is approximately 1.34 x 10^-14 Ω/m.
To determine how much wire is needed, we need to know the desired resistance or length.