A platinum resistance thermometer has a resistance of 10.59ohms at 0 degree celcius and 15.59ohms at 100 degree celcius. Assuming that the resistance changes uniformly with temperature, what is: (i) the temperature when the resistance is 10.59ohms? (ii) the resistance of the thermometer when the temperature is 55degree celcius?
i) is easy as that is in the question and is a prompt for the next part:
As we are told that response is linear the equation for the resistance will be of the form
y=mx+c or
R=mT+c if you like
we are told c in the question (c=10.59 ohms)
to calculate m
m=(15.59-10.59)ohms/100 C
= 0.0500 ohms/C so the equation is
R=0.0500T+10.59
Now substitute 55 deg C and calculate R
To calculate the temperature when the resistance is 10.59 ohms, we can use the concept of linear interpolation.
We know that the resistance changes uniformly with temperature, which means it follows a straight line relationship.
Let's first establish the equation of the straight line using the given data points: (0, 10.59) and (100, 15.59).
The equation of a straight line passing through two points can be written as:
y - y1 = (y2 - y1) * (x - x1) / (x2 - x1),
where (x1, y1) and (x2, y2) are the coordinates of the two points.
Using the values (0, 10.59) and (100, 15.59) in the equation, we can find the equation of the line.
(y - 10.59) = (15.59 - 10.59) * (x - 0) / (100 - 0).
Simplifying this equation gives:
(y - 10.59) = 5 / 100 * x.
Now, we can substitute the resistance value of 10.59 ohms into the equation and solve for temperature (x):
(10.59 - 10.59) = 5 / 100 * x.
0 = 5 / 100 * x.
x = 0.
Therefore, when the resistance is 10.59 ohms, the temperature is 0 degrees Celsius.
To find the resistance of the thermometer when the temperature is 55 degrees Celsius, we can use the equation of the straight line we established earlier:
y = 5 / 100 * x + 10.59.
Substituting x = 55 into the equation:
y = 5 / 100 * 55 + 10.59.
Simplifying this equation gives:
y = 2.75 + 10.59.
y = 13.34.
Therefore, when the temperature is 55 degrees Celsius, the resistance is approximately 13.34 ohms.