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
$E_{d i p} (r) = \frac{1}{4 π ε_{0}} \frac{1}{4 π ε_{0}} \frac{1}{r^{3}} [3 (p \hat{\cdot} r) r - p]$

Short Answer

Expert verified

Answer

The given relation is proved.

Step by step solution

Define functions

Write the expression for electric field.

$E_{d i p o l e} (r, θ) = \frac{1}{4 π ε_{0}} \frac{1}{r^{3}} ρ [2 c o s \hat{θ} r + s i n \hat{θ} θ]$ …… (1)

Here, $ρ$ is the dipole moment, $θ$ is the orientation of dipole electric field and $ε_{0}$ is the permittivity for the free space.

Determine electric field

Write the expression for the electric field.

$E_{d i p o l e} (r, θ) = \frac{1}{4 π ε_{0}} \frac{1}{r^{3}} [2 p \cos \hat{θ} r + p \sin \hat{θ} θ] = \frac{1}{4 π ε_{0}} \frac{1}{r^{3}} [2 p \cos \hat{θ} r - p \cos \hat{θ} r + p \sin \hat{θ} θ] = \frac{1}{4 π ε_{0}} \frac{1}{r^{3}} [3 p \cos \hat{θ} r - (p \cos \hat{θ} r + p \sin \hat{θ} θ)]$ …… (2)

Write the dipole moment vector.

$p = p \cos \hat{θ} θ$ …… (3)

$p \hat{\cdot} r = (p \cos \hat{θ} r - p \sin \hat{θ} θ) \hat{\cdot} r = p \cos θ$ …… (4)

Substitute $p \cos \hat{θ} r - p \sin \hat{θ} θ$ for $ρ$ and $p \cos θ$ for $p \hat{\cdot} r$ in equation (2).

$E_{d i p o l e} (r, θ) = \frac{1}{4 π ε_{0}} \frac{1}{r^{3}} [3 p \cos \hat{θ} r - (p \cos \hat{θ} r + p \sin \hat{θ} θ)] E_{d i p o l e} (r, θ) = \frac{1}{4 π ε_{0}} \frac{1}{r^{3}} [3 (p \hat{\cdot} r) r - p]$

Thus, the given relation is proved.

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

Short Answer

Step by step solution

Define functions

Determine electric field

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Step by step solution

Define functions

Determine electric field

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