Chapter 32: Q52P (page 970)

Measurements in mines and boreholes indicate that Earth’s interior temperature increases with depth at the average rate of $30 ° C / km$ . Assuming a surface temperature of $10 ° C$ , 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 $d = 25 km$ .

Step by step solution

Listing the given quantities

Temperature gradient is $\frac{ΔT}{Δd} = \frac{30^{0} C}{km}$

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

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:

$\frac{ΔT}{Δd} = \frac{30^{0} C}{km}$

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

We use the equation of temperature gradient.

$\frac{ΔT}{Δd} = \frac{30^{0} C}{km}$

We can rearrange it for $Δd$ we get

$Δd = \frac{ΔT}{(\frac{30^{0} C}{km})}$

We integrate it to find d. We get

$\int Δd =_{T_{s}}^{T_{c}} \frac{ΔT}{(\frac{30^{0} C}{km})}$

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

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

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

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Short Answer

Step by step solution

Listing the given quantities

Understanding the concepts of temperature gradient

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

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Measurements in mines and boreholes indicate that Earth’s interior temperature increases with depth at the average rate of 30°C/km . Assuming a surface temperature of10°C , at what depth does iron cease to be ferromagnetic? (The Curie temperature of iron varies very little with pressure.)

Short Answer

Step by step solution

Listing the given quantities

Understanding the concepts of temperature gradient

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Nuclear Physics

Rotational Dynamics

Mechanics and Materials

Scientific Method Physics

Kinematics Physics

Medical Physics

Study anywhere. Anytime. Across all devices.