Q3.36P Show that the electric field of ... [FREE SOLUTION]

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Introduction to Electrodynamics

David J. Griffiths

Physics

4th edition Edition 路 613 pages

ISBN: 9780321856562

Chapter 3: Q3.36P (page 160)

Show that the electric field of a (perfect) dipole (Eq. 3.103) can be written in the coordinate-free form
$E_{d i p} (r) = \frac{1}{4 蟺蔚_{0}} \frac{1}{4 蟺蔚_{0}} \frac{1}{r^{3}} [3 (p \hat{路} r) r - p]$

Short Answer

Expert verified

Answer

The given relation is proved.

Step by step solution

Define functions

Write the expression for electric field.

$E_{d i p o l e} (r, 胃) = \frac{1}{4 蟺蔚_{0}} \frac{1}{r^{3}} 蚁 [2 c o s \hat{胃} r + s i n \hat{胃} 胃]$ 鈥︹€� (1)

Here, $蚁$ is the dipole moment, $胃$ is the orientation of dipole electric field and $蔚_{0}$ is the permittivity for the free space.

Determine electric field

Write the expression for the electric field.

$E_{d i p o l e} (r, 胃) = \frac{1}{4 蟺蔚_{0}} \frac{1}{r^{3}} [2 p \cos \hat{胃} r + p \sin \hat{胃} 胃] = \frac{1}{4 蟺蔚_{0}} \frac{1}{r^{3}} [2 p \cos \hat{胃} r - p \cos \hat{胃} r + p \sin \hat{胃} 胃] = \frac{1}{4 蟺蔚_{0}} \frac{1}{r^{3}} [3 p \cos \hat{胃} r - (p \cos \hat{胃} r + p \sin \hat{胃} 胃)]$ 鈥︹€� (2)

Write the dipole moment vector.

$p = p \cos \hat{胃} 胃$ 鈥︹€� (3)

$p \hat{路} r = (p \cos \hat{胃} r - p \sin \hat{胃} 胃) \hat{路} r = p \cos 胃$ 鈥︹€� (4)

Substitute $p \cos \hat{胃} r - p \sin \hat{胃} 胃$ for $蚁$ and $p \cos 胃$ for $p \hat{路} r$ in equation (2).

$E_{d i p o l e} (r, 胃) = \frac{1}{4 蟺蔚_{0}} \frac{1}{r^{3}} [3 p \cos \hat{胃} r - (p \cos \hat{胃} r + p \sin \hat{胃} 胃)] E_{d i p o l e} (r, 胃) = \frac{1}{4 蟺蔚_{0}} \frac{1}{r^{3}} [3 (p \hat{路} r) r - p]$

Thus, the given relation is proved.

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91影视

Show that the electric field of a (perfect) dipole (Eq. 3.103) can be written in the coordinate-free form
$E_{d i p} (r) = \frac{1}{4 蟺蔚_{0}} \frac{1}{4 蟺蔚_{0}} \frac{1}{r^{3}} [3 (p \hat{路} r) r - p]$

Short Answer

Step by step solution

Define functions

Determine electric field

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91影视

Show that the electric field of a (perfect) dipole (Eq. 3.103) can be written in the coordinate-free formEdip(r)=14蟺蔚014蟺蔚01r3[3p路^rr-p]

Short Answer

Step by step solution

Define functions

Determine electric field

One App. One Place for Learning.

Most popular questions from this chapter

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Nuclear Physics

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Show that the electric field of a (perfect) dipole (Eq. 3.103) can be written in the coordinate-free form
$E_{d i p} (r) = \frac{1}{4 蟺蔚_{0}} \frac{1}{4 蟺蔚_{0}} \frac{1}{r^{3}} [3 (p \hat{路} r) r - p]$