Platinum resistance thermometer and Kirchoff's Laws

In summary, the conversation discusses a question about balancing resistors in a wire and converting log values. The answer given includes the use of loop equations and the relationship between resistance and length. The conversation also includes a calculation to find the resistivity of platinum based on the given information.
  • #1
penroseandpaper
21
0
Homework Statement
(b) The circuit shown in Figure 1 can be used to measure the resistance of a platinum resistance thermometer (PRT). AB is a uniform resistance wire of length 1.0 m and C is a sliding contact on this wire. A standard resistor R is included in the circuit. The position of C is adjusted until the voltmeter V reads zero.

(i) By applying Kirchhoff’s laws to loops ADCA and BCDB, deduce an expression for the resistance of the PRT in terms of l1, l and the value of the standard resistor.

(ii) The PRT consists of 9.0 m of wire of diameter 8.0 × 10−2 mm. The voltmeter reads 0 V when l1 = 0.44 m. If the standard resistor, R, has a resistance of 224 Ω, what is the resistivity of platinum?
Relevant Equations
Kirchoff's Laws
Hi,

I found this question/s online and I was wondering if somebody was able to explain the answer given, specifically the first bit that says R1/(R1+R2) = R4/(R3+R4) (is this something to do with an equivalent resistor to replace the two in series on each side? But why are R1 and R4 the numerators?)

Also, is it easy to convert the log value at the end with e in it to one with just 10 to the power of something?

The answer given was:

Let R1 be the resistance of the left part of the wire, R2 the right part, R3 the "R" and R4 the PRT. L1 and L2 the lengths of the left and right part of the wire, L the total lengths.

Your use of "I" for the lengths is too confusing as I usually means current. Don't need loop equations.

The current is zero when the voltage at the junction of R1,2 is equal to that at the junction of R3,4.

Therefore it is balanced when R1/(R1+R2) = R4/(R3+R4)

in terms of the wire, length is proportional to the resistance, so

L1/(L1+L2) = R4/(R3+R4)

L1/(L) = R4/(R3+R4)

(R3+R4)L1 = R4L

R3L1+R4L1 = R4L

R4(L–L1) = R3L1

R4 = R3L1 / (L–L1)"The PRT consists of 9.0 m of wire of diameter 8.0 × 10−2 mm. The voltmeter reads 0 V when l1 = 0.44 m. If the standard resistor, R, has a resistance of 224 Ω, what is the resistivity of platinum?"
8.0 × 10−2 mm ??

guessing that that is 0.08 mm. Please use correct terminology in the future.
R4 = R3L1 / (L–L1) = 224•0.44 / (1–0.44) = 176 Ω
R = ρL/A

r = 0.04 mm = 4e-5 m

A = πr² = 5.03e-9 m²

ρ = AR/L = 5.03e-9•176/9 = 9.83e-8 Ω-mResistance of a wire in Ω

R = ρL/A

ρ is resistivity of the material in Ω-m

L is length in meters

A is cross-sectional area in m²

A = πr², r is radius of wire in m

resistivity Pt 105e-9 Ω-m
 

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  • #2
If the voltmeter reads zero then the voltage drop along R1 and R4 (PRT) is the same. That voltage drop can be expressed as fraction of the source voltage, looking at each path individually: R1/(R1+R2) = R4/(R3+R4).
R1/R2=R4/R3 is true as well, you can derive this from the above expression or vice versa if you want.
8.0 × 10−2 mm ??
This should be 8.0 × 10−2 mm. As 10−2=0.01 this is 0.08 mm.
 

Related to Platinum resistance thermometer and Kirchoff's Laws

1. What is a platinum resistance thermometer?

A platinum resistance thermometer is a type of temperature sensor that uses the change in electrical resistance of a platinum wire to measure temperature. It is known for its high accuracy and stability over a wide temperature range.

2. How does a platinum resistance thermometer work?

According to Kirchoff's Laws, the electrical resistance of a conductor is directly proportional to its length and inversely proportional to its cross-sectional area and temperature. In a platinum resistance thermometer, the length and cross-sectional area of the platinum wire are fixed, so changes in temperature cause changes in the resistance, which can be measured and used to calculate the temperature.

3. What is Kirchoff's First Law?

Kirchoff's First Law, also known as the Law of Currents, states that the total current entering a junction in a circuit must equal the total current leaving the junction. This law is based on the principle of conservation of charge.

4. What is Kirchoff's Second Law?

Kirchoff's Second Law, also known as the Law of Voltages, states that the sum of the voltage drops around a closed loop in a circuit must equal the sum of the voltage sources in that loop. This law is based on the principle of conservation of energy.

5. How are Kirchoff's Laws used in a platinum resistance thermometer?

In a platinum resistance thermometer, Kirchoff's Laws are used to measure the change in electrical resistance of the platinum wire, which is directly related to the temperature. By applying Kirchoff's Laws and using a known reference temperature, the temperature of the sensor can be accurately calculated. Additionally, Kirchoff's Laws are used in the design and calibration of the thermometer to ensure accurate and reliable temperature measurements.

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