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Chapter 32: Q52P (page 970)

Measurements in mines and boreholes indicate that Earth鈥檚 interior temperature increases with depth at the average rate of $30 掳 C / km$ . Assuming a surface temperature of $10 掳 C$ , at what depth does iron cease to be ferromagnetic? (The Curie temperature of iron varies very little with pressure.)

Short Answer

Expert verified

The depth at which iron does cease to be ferromagnetic is $d = 25 鈥� km$ .

Step by step solution

01

Listing the given quantities

Temperature gradient is $\frac{螖罢}{螖诲} = \frac{30^{0} C}{km}$

Surface temperature is $T_{s} = 10^{0} C$

02

Understanding the concepts of temperature gradient

We use the concept of temperature gradient. Using the equation of temperature gradient and integrating it over the limit fromthesurface temperature to Curie temperature, we can find the depth at which iron ceases to be ferromagnetic.

Formulae:

$\frac{螖罢}{螖诲} = \frac{30^{0} C}{km}$

03

Calculations of the depth at which iron does cease to be ferromagnetic

We use the equation of temperature gradient.

$\frac{螖罢}{螖诲} = \frac{30^{0} C}{km}$

We can rearrange it for $螖诲$ we get

$螖诲 = \frac{螖罢}{(\frac{30^{0} C}{km})}$

We integrate it to find d. We get

$鈭� 螖诲 =_{T_{s}}^{T_{c}} \frac{螖罢}{(\frac{30^{0} C}{km})}$

Where, $T_{c}$ - is Curie temperature.

$\begin{matrix} d = \frac{1}{(\frac{30^{0} C}{km})}_{T_{s}}^{T_{c}} 螖罢 \\ = \frac{1}{(\frac{30^{0} C}{km})} {| T |}_{T_{s}}^{T_{c}} \\ = \frac{1}{(\frac{30^{0} C}{km})} (T_{c} 鈭� T_{s}) \\ = \frac{1}{(\frac{30^{0} C}{km})} (770^{0} C - 10^{0} C) \\ = \frac{760^{0} C}{(\frac{30^{0} C}{km})} \\ = 25.3 \\ = 25 km \end{matrix}$

The depth at which iron does cease to be ferromagnetic is $d = 25 鈥� km$ .

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Most popular questions from this chapter

In Fig. 32-36, a capacitor with circular plates of radius $R = 18.0 cm$ is connected to a source of emf $尉 = 尉_{m} s i n 蝇 t$ , where $尉_{m} = 220 V$ and $蝇 = 130 rad / s$ . The maximum value of the displacement current is $i_{d} = 7.60 渭 A$ . Neglect fringing of the electric field at the edges of the plates. (a) What is the maximum value of the currenti in the circuit? (b) What is the maximum value of $d 蠒_{E} / dt$ , where $蠒_{E}$ is the electric flux through the region between the plates? (c) What is the separation dbetween the plates? (d) Find the maximum value of the magnitude of $\overset{鈬赌}{B}$ between the plates at a distance $r = 11.0 cm$ from the center.

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