Q61P The magnetic field of Earth can ... [FREE SOLUTION]

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Chapter 32: Q61P (page 971)

The magnetic field of Earth can be approximated as the magnetic field of a dipole. The horizontal and vertical components of this field at any distance r from Earth鈥檚 center are given by $B_{H} = \frac{渭_{0} 渭}{4 {蟺谤}^{3}} 脳 {肠辞蝉位}_{m}$ , $B_{v} = \frac{渭_{0} 渭}{2 {蟺谤}^{3}} 脳 {蝉颈苍位}_{m}$ where l_m is the magnetic latitude (this type of latitude is measured from the geomagnetic equator toward the north or south geomagnetic pole). Assume that Earth鈥檚 magnetic dipole moment has magnitude $渭 = 8.00 脳 1022 A m^{2}$ . (a) Show that the magnitude of Earth鈥檚 field at latitude lm is given by $B = \frac{渭_{0} 渭}{4 {蟺谤}^{3}} 脳 \sqrt{1 + 3 \sin^{2} 位_{m}}$
(b) Show that the inclination $蠒_{i}$ of the magnetic field is related to the magnetic latitude $位_{m}$ by tan $蠒_{i} = 2 {迟补苍位}_{m}$ .

Short Answer

Expert verified

a. $B = \frac{渭_{0} 渭}{4 {蟺谤}^{3}} 脳 \sqrt{1 + 3 \sin^{2} 位_{m}}$

b. ${迟补苍蠒}_{i} = 2 \tan 位_{m}$

Step by step solution

Listing the given quantities

$B_{H} = \frac{渭_{0} 渭}{4 {蟺谤}^{3}} 脳 {肠辞蝉位}_{m}$

$B_{v} = \frac{渭_{0} 渭}{2 {蟺谤}^{3}} 脳 {蝉颈苍位}_{m}$

Understanding the concepts of magnetic field

Here, we have to use Pythagoras theorem to find the magnitude of the earth鈥檚 magnetic field. The inclination of the magnetic field is found using the equation of the tangent ratio and the vertical and the horizontal component of the magnetic field.

Formula:

$B = \sqrt{B_{h}^{2} + B_{v}^{2}}$

$迟补苍蠒 = \frac{B_{v}}{B_{H}}$

(a) Calculations of the B

$\begin{matrix} B = \sqrt{{(\frac{渭_{0} 渭}{4 {蟺谤}^{3}} 脳 {肠辞蝉位}_{m})}^{2} + {(\frac{渭_{0} 渭}{2 {蟺谤}^{3}} 脳 {蝉颈苍位}_{m})}^{2}} \\ = \frac{渭_{0} 渭}{4 {蟺谤}^{3}} 脳 \sqrt{{({肠辞蝉位}_{m})}^{2} + {(2 {蝉颈苍位}_{m})}^{2}} \\ = \frac{渭_{0} 渭}{4 {蟺谤}^{3}} 脳 \sqrt{(1 鈭� \sin^{2} 位_{m}) + 4 \sin^{2} 位_{m}} \\ = \frac{渭_{0} 渭}{4 {蟺谤}^{3}} 脳 \sqrt{1 + 3 \sin^{2} 位_{m}} \end{matrix}$

$B = \frac{渭_{0} 渭}{4 {蟺谤}^{3}} 脳 \sqrt{1 + 3 \sin^{2} 位_{m}}$

(b) Calculations of the inclination ϕi of the magnetic field

$\begin{matrix} {迟补苍蠒}_{i} = \frac{B_{v}}{B_{h}} \\ = \frac{\frac{渭_{0} 渭}{2 {蟺谤}^{3}} 脳 {蝉颈苍位}_{m}}{\frac{渭_{0} 渭}{4 {蟺谤}^{3}} 脳 {肠辞蝉位}_{m}} \\ = 2 {迟补苍位}_{m} \end{matrix}$