Thermoelectric Effects

(a) Seebeck effect

When the two junctions of a thermocouple (a loop of two dissimilar metals) are kept at different temperatures, an emf (thermo-emf) is produced and a current flows in the loop. This is the Seebeck effect. The Peltier and Thomson effects are about heat being absorbed or released when a current flows, and Joule heating is $I^2R$ heating.

(C) 20$^\circ$C

The neutral temperature lies midway between the cold-junction and inversion temperatures, $\theta_n = \dfrac{\theta_c + \theta_i}{2}$, so

$$\theta_c = 2\theta_n - \theta_i = 2(310) - 600 = 20^\circ\text{C}$$

(a) $40^\circ$C

The neutral temperature is the mean of the cold-junction and inversion temperatures:

$\theta_n = \frac{\theta_c + \theta_i}{2} \;\Rightarrow\; 320 = \frac{\theta_c + 600}{2} \;\Rightarrow\; \theta_c = 640 - 600 = 40^\circ\text{C}$

(a) Seebeck effect vs Peltier effect

The Peltier effect is reversible: reversing the current changes heating to cooling at the junction.

(b) Variation of thermo-emf with temperature

Keep the cold junction at $0^\circ$C and slowly raise the temperature $\theta$ of the hot junction. Experiments show that the thermo-emf varies as

$$E = a\theta + \frac{1}{2}b\theta^2$$

where $a$ and $b$ are constants for the pair of metals ($b$ is negative). The graph of $E$ against $\theta$ is a parabola.

   E
   ^
   |              .  -  -  .
   |          .'      :      '.
   |        /         :         \
   |      /           :           \
   |    /             :             \
   |  /               :               \
   +------------------+-----------------+-----> θ
  θc (0 °C)           θn                θi
                   (neutral)       (inversion)

1. Rise: As $\theta$ increases, $E$ increases, but less and less quickly.
2. Neutral temperature $\theta_n$: $E$ reaches its maximum. Here

$$\frac{dE}{d\theta} = a + b\theta_n = 0 \quad\Rightarrow\quad \theta_n = -\frac{a}{b}$$

$\theta_n$ depends only on the pair of metals, not on the temperature of the cold junction. 3. Fall and inversion temperature $\theta_i$: Beyond $\theta_n$ the emf decreases and becomes zero at $\theta_i$. Above $\theta_i$ the emf reverses its direction. 4. Because the parabola is symmetric about $\theta_n$,

$$\theta_n - \theta_c = \theta_i - \theta_n \quad\Rightarrow\quad \boxed{\theta_n = \frac{\theta_i + \theta_c}{2}}$$

For example, for a Cu–Fe thermocouple with the cold junction at $0^\circ$C, $\theta_n \approx 270^\circ$C and $\theta_i \approx 540^\circ$C.

(b) $-\dfrac{a}{b}$

At the neutral temperature the emf is maximum, so $\dfrac{dE}{d\theta} = 0$:

$$a + b\theta_n = 0 \quad\Rightarrow\quad \theta_n = -\frac{a}{b}$$

Keep the cold junction of a thermocouple (e.g. Cu–Fe) at a fixed temperature $\theta_c$ (say $0^\circ$C) and gradually heat the hot junction to temperature $\theta$. The thermo-emf is found to vary as

$$E = a\theta + \frac{1}{2}b\theta^2$$

where $a$ and $b$ are thermoelectric constants of the pair ($b$ is negative). The $E$–$\theta$ graph is a parabola.

   E
   ^
   |              .  -  -  .
   |          .'      :      '.
   |        /         :         \
   |      /           :           \
   |    /             :             \
   |  /               :               \
   +------------------+-----------------+-----> θ
  θc                  θn                θi
                   (neutral)       (inversion)

1. Increase: At first $E$ increases with $\theta$, but its rate of increase gets smaller.
2. Neutral temperature ($\theta_n$): This is the hot-junction temperature at which the thermo-emf is maximum. Setting $\dfrac{dE}{d\theta} = a + b\theta = 0$ gives $\theta_n = -\dfrac{a}{b}$. It is a constant for a given pair of metals and does not depend on $\theta_c$.
3. Decrease and inversion temperature ($\theta_i$): Above $\theta_n$ the emf decreases and becomes zero at the inversion temperature $\theta_i$. Beyond $\theta_i$ the emf changes sign (the current reverses).
4. The curve is symmetric about $\theta_n$:

$$\theta_n = \frac{\theta_c + \theta_i}{2}$$

For Cu–Fe with $\theta_c = 0^\circ$C: $\theta_n \approx 270^\circ$C and $\theta_i \approx 540^\circ$C.

(i) Thermoelectric effect

When the two junctions of a closed circuit of two different metals are kept at different temperatures, an emf (thermo-emf) is produced and a current flows in the circuit. This is the Seebeck effect.

(ii) $\theta_n$ and $\theta_i$

• $\theta_n$ is the neutral temperature: the temperature of the hot junction at which the thermo-emf is maximum.
• $\theta_i$ is the temperature of inversion: the temperature of the hot junction at which the thermo-emf becomes zero and then reverses its direction.

(iii) Effect of shifting the cold junction to $-10^\circ$C

The neutral temperature stays the same, and $\theta_i = 2\theta_n - \theta_c$, so $\theta_i$ increases by $10^\circ$C.

(iv) Application

It can be used as a thermocouple thermometer to measure temperatures (for example of furnaces), and a combination of thermocouples forms a thermopile to detect radiant heat.

(A) 2%

Using $\theta_i = 2\theta_n - \theta_c$, and noting that $\theta_n = 500^\circ$C does not change with $\theta_c$:

• with $\theta_c = 0^\circ$C: $\theta_i = 1000^\circ$C
• with $\theta_c = 20^\circ$C: $\theta_i = 980^\circ$C

$$\text{Percentage change} = \frac{1000 - 980}{1000} \times 100\% = 2\%$$

(c) remains the same

The neutral temperature $\theta_n = -\dfrac{a}{b}$ depends only on the two metals of the thermocouple, not on the cold-junction temperature. Changing the cold junction changes only the inversion temperature, since $\theta_i = 2\theta_n - \theta_c$.