IPhO 2019, Експеримент — Задача 2. Wiedemann-Franz Law

Part A: Electric conductivity of copper, aluminum and brass (1.5 points) (1,5 т.)

Theory

When a permanent magnet falls inside a hollow cylindrical conductive tube, it experiences a dissipative force due to induced eddy currents. Therefore, the magnet reaches a terminal velocity. For this geometry the terminal velocity can be expressed as:

Here $m$ is the mass of the magnet, $\sigma$ is the electrical conductivity of the material of the tube, $a$ is the inner radius of the tube, $r_m$ and $d$ are the radius and the height of the magnet, respectively, $M$ is the remanent magnetization of the magnet, $w$ is the thickness of the tube wall and $f\left(\frac{d}{a}\right)$ is a scaling function. In our case, $a \approx r_m$ , $d = 2r_m \approx 2a$ and $f(2) \approx 1.75$ . Therefore, the time it takes for the magnet to fall through the tube can be approximated by:

Here $L_0 = 0.2$ m is the length of the tube and we assume that the magnet reaches terminal velocity immediately upon its release.

The characteristics of the tube and the magnet which are needed for the calculation are:

$\mu_0 M = 0.65$ T, $w_{\text{Aluminum}} = w_{\text{Copper}} = 7.0\times 10^{-3}$ m, $w_{\text{Brass}} = 6.5\times 10^{-3}$ m, $m = 1.2\times 10^{-3}$ kg , $r_m = 3.0\times 10^{-3}$ m, $g = 9.8$ m/s$^2$

Experiment

A.1 Using the digital readout box in stopwatch mode, measure the time it takes the magnet to fall through the hollow tubes made of aluminum, copper and brass. Write down your measurements in table A1. [1 т.]

A.2 Using the equation above, find the electrical conductivities $\sigma_{\text{Aluminum}}, \sigma_{\text{Copper}}, \sigma_{\text{Brass}}$ for each of the three materials. [0,5 т.]

Part B: Thermal conductivity of copper (3.0 points) (3 т.)

The goal of this section is to measure the heat conductivity of copper close to the steady state.

Theory

The thermal conductivity $\kappa$ is defined by the equation $P(x) = -\kappa A\cdot \frac{\Delta T(x)}{\Delta x}$. This equation describes a linear relation between the local temperature gradient and the local power flowing through a cross section of the material. Here, $P(x)$ is the power flowing through a cross-section at location $x$, $A$ is the cross-sectional area of the rod and $\Delta T(x)/\Delta x$ is the temperature gradient at position $x$.

Experiment

Connect the digital readout box to the external outlet and calibrate Rod #1. Pour 4 liters (2 bottles) of water into the pot to fully immerse the heat exchanger and close the lid.

B.1 Write down the initial temperature of rod #1 when placed on the table. [0,1 т.]

Disconnect the readout cable from the rod. Remove the insulating cap and screw rod #1 onto the cover of the pot. Reconnect the cable to the readout box, as shown in figure 5. Be careful not to apply too much torque.

B.2 Draw a circuit that will allow you to supply power to the heater and to measure that power. Your circuit should contain the following: 9V power source, heater (already connected to the rod), the voltmeter, the ammeter and wires. You may use the wires as a switch to open and close the circuit. [0,5 т.]

The heat conductivity will be measured by applying heat power to one side of the rod while keeping the other side of the rod at the almost constant temperature of the water reservoir.

We aim to get close to a steady state for all thermometers. Connect the circuit from section B2 and apply power to the heater.

B.3 Perform appropriate measurements to compute the applied power $P$ to the heater, and write it down in the answer sheet. [0,1 т.]

Wait for 15 minutes while applying power (you can use this time to plan your experiments).

B.4 Write down in the supplied table the temperatures of all eight thermometers at approximate times: 15 min, 17.5 min, 20 min. [0,5 т.]

B.5 On one graph paper, draw three graphs of the temperature as a function of position for each of the measured times. These graphs will also be used in Part D. [1 т.]

B.6 Use the graph to extract the thermal conductivity of copper, $\kappa_0$, using your data from time approximately 17.5 min. Disregard any heat loss in this part. Estimate the average rate of temperature change of the rod, $\frac{\Delta T}{\Delta t}$, at time approximately 17.5 min. [0,5 т.]

B.7 Do you expect a higher / lower / the same value of $\kappa_0$, compared to the real value of $\kappa$? [0,3 т.]

Part C: Estimating the heat loss and the heat capacity of copper (4.0 points) (4 т.)

Theory

The heat capacity $C$ is defined by either of the following equations:

Here, $\Delta Q/\Delta t$ is the net heat rate transfer to the material and $\Delta T/\Delta t$ is the temperature change rate. The specific heat $c_p$ is the heat capacity per unit mass. The mass of the copper rod should be taken as $0.58$ kg.

Experiment

Turn off the heater power supply. Disconnect the circuit, unscrew and place rod #1 on the table. Place the insulating termination cap onto the rod, as you found it at the beginning of the experiment. Reconnect the heater circuit and reconnect the rod to the digital readout box.

WARNING: Do not leave the heater on in this part for long periods of time without monitoring the temperature.

Using a sequence of cooling, heating, and cooling again, we can extract both the heat loss and the heat capacity of the material. The heating step should change the average temperature by approximately $2.5^{\circ}$C. The necessary accuracy in this step can be achieved by a cooling-heating-cooling sequence of a total length of 10-15 minutes.

Here we aim to work near the average temperature measured close to the steady state of Part B.

In order to account for all the thermal energy stored in the rod, we would want to track its average temperature. The temperature at the center of the rod is a good approximate measure of the average temperature.

C.1 Perform a cooling-heating-cooling sequence and record your measurements in table C1, to obtain the average temperature. [1 т.]

C.2 Plot on a graph paper the average temperature versus time. [1 т.]

C.3 Using the graph, calculate the specific heat $c_p$ and the heat loss per unit time $P_{\text{loss}}$ around the average temperature of Part B. Describe your method using diagrams and equations. [1 т.]

There are two main mechanisms which should be taken into account in order to improve the accuracy of the heat conductivity extracted in part B.

• There is heat loss due to radial heat transfer through the insulation.
• The system did not reach steady-state at the time of the measurement

To first order approximation, you may assume that due to these mechanisms the change per unit length of power flow along the rod, $\Delta P(x)/\Delta x$, is constant.

C.4 Write down an equation correcting to first order the thermal conductivity found in part B while taking into account both mechanisms. Use $\kappa_0, P, c_p, m, P_{\text{loss}}, \frac{\Delta T}{\Delta t}$ from parts B,C to express the corrected value of the thermal conductivity, $\kappa_{\text{Copper}}$ and calculate its value. [1 т.]

Part D: Thermal conductivity of brass and aluminum (1.0 points) (1 т.)

Connect the insulated rod #2 to the digital readout box, and calibrate the thermometers of this rod as instructed at the beginning of Part B (while the red button is pressed, connect the insulated rod #2 to the readout box using the thermometers cable).

D.1 Write down the initial temperature of the rod when placed on the table. [0,1 т.]

Disconnect the cable and screw rod #2 onto the pot’s cover as shown in figure 4. Reconnect the cable to the readout box.

Repeat the procedure used in part B to get close to the steady state while heating.

Supply power to the heater for at least 15 minutes before making measurements.

You may assume, for the accuracy required in this part, that the rod is in steady state. In addition, you may assume that the heat loss per unit length is constant along the rod.

D.2 Write down the temperature readings from all eight thermometers of rod #2 and write down $\Delta T/\Delta x$ for each of its sections. [0,2 т.]

To first order approximation, you may use the same assumption as in task C.4, namely, that $\Delta P(x)/\Delta x$ is constant.

D.3 Express $\kappa_{\text{Brass}}$ and $\kappa_{\text{Aluminum}}$ using your previous measurements and evaluate their numerical values. [0,7 т.]

Part E: The Wiedemann-Franz law (0.5 points) (0,5 т.)

The Wiedemann-Franz law states that, in metals where heat transport is dominated by conduction electrons, the ratio of thermal and electrical conductivities depends linearly on the absolute temperature. Moreover, the law states that the slope $L = \frac{\kappa}{\sigma T}$ (known as the "Lorenz number") of this dependence is the same for most metals, and depends only on universal constants of nature. In reality, for metals at room temperature, this law holds with about $10\%$ accuracy.

E.1 Write down your findings for thermal and electrical conductivities ( $\kappa, \sigma$) in table E1. Calculate the value of $L$ for each material and display it in the same table E1, while assuming that the thermal conductivity does not depend on temperature to first order. [0,5 т.]