Chapter 8: Q24P (page 461)

a) Show that $\nabla \cdot (e_{r} / r^{2}) = 0$ for $r > 0$ .
(b) Show that $\nabla (1 / r) = - e_{r} / r^{2}$ .

Short Answer

Expert verified

(a) The solution is $\frac{2 F (r)}{r} + r F^{''} (r)$

(b) The solution is $- e_{r} / r^{2}$

Step by step solution

Given information

The given expressions is

(a)

The vectors $e_{r} / r^{2}$ and $e_{r} F (r)$ .

(b)

The function $f (r) = 1 / r$ .

Formula of Spherical Coordinates

The divergence of a vector in spherical coordinates is given as

$\nabla \cdot V = \frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} V_{r}) + \frac{1}{r s i n θ} \frac{\partial}{\partial θ} (V_{θ} s i n θ) + \frac{1}{r s i n θ} \frac{\partial}{\partial θ} (V_{ϕ}) .$

The gradient of a function in spherical coordinates is given as

$\nabla f = e_{r} \frac{\partial f}{\partial r} + e_{o} \frac{1}{r} \frac{\partial f}{\partial θ} + e_{ϕ} \frac{1}{r s i n ϕ} \frac{\partial f}{\partial θ}$

Show the given function ∇⋅(er/r2)=0

Solve the equation

\nabla . (e_{r} / r^{2}) = \nabla . (1 / r^{2} . e_{r} + 0 . e_{0} + 0 . e_{ϕ}) = \frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} . 1 / r^{2}) + \frac{1}{r \sin θ} \frac{\partial}{\partial θ} (0 . \sin θ) + \frac{1}{r \sin θ} \frac{\partial}{\partial θ} (0) = \frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} . 1 / r^{2}) + 0 + 0 = \frac{1}{r^{2}} \frac{\partial}{\partial r} (1)

On further calculating,

$\nabla . (e_{r} / r^{2}) = \frac{1}{r^{2}} . 0 = 0$

Thus, $\nabla \cdot (e_{r} / r^{2}) = 0$ for r>0.

And,

$\nabla . [e_{r} F (r)] = \nabla . (F (r) e_{r} + 0 . e_{θ} + 0 . e_{ϕ}) = \frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} . F (r)) + \frac{1}{r s in θ} \frac{\partial}{\partial θ} (0 . \sin θ) + \frac{1}{r \sin θ} \frac{\partial}{\partial θ} (0) = \frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} . F (r)) + 0 + 0 = \frac{1}{r^{2}} [2 r F (r) + r^{2} F^{''} (r)] = \frac{2 F (r)}{r} + r F^{''} (r)$

Which is a function ofr only.

Show the given function

Solve the equation

$\nabla (1 / r) = e_{r} \frac{\partial}{\partial r} (1 / r) + e_{0} \frac{1}{r} \frac{\partial}{\partial θ} (1 / r) + e_{ϕ} \frac{1}{r \sin ϕ} \frac{\partial}{\partial ϕ} (1 / r) = e_{r} (- \frac{1}{r^{2}}) + e_{0} \frac{1}{r} . 0 + e_{ϕ} \frac{1}{r \sin ϕ} . 0 = e_{r} (- \frac{1}{r^{2}}) + 0 + 0 = - e_{r} / r^{2}$

Thus, the solution is $- e_{r} / r^{2}$ .

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a) Show that $\nabla \cdot (e_{r} / r^{2}) = 0$ for $r > 0$ .
(b) Show that $\nabla (1 / r) = - e_{r} / r^{2}$ .

Short Answer

Step by step solution

Given information

Formula of Spherical Coordinates

Show the given function ∇⋅(er/r2)=0

Show the given function

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a) Show that∇⋅(er/r2)=0forr>0.(b) Show that∇(1/r)=-er/r2.

Short Answer

Step by step solution

Given information

Formula of Spherical Coordinates

Show the given function ∇⋅(er/r2)=0

Show the given function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Famous Physicists

Wave Optics

Medical Physics

Atoms and Radioactivity

Electric Charge Field and Potential

Kinematics Physics

Study anywhere. Anytime. Across all devices.

a) Show that $\nabla \cdot (e_{r} / r^{2}) = 0$ for $r > 0$ .
(b) Show that $\nabla (1 / r) = - e_{r} / r^{2}$ .