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Introduction to Electrodynamics

David J. Griffiths

Physics

4th edition Edition 路 613 pages

ISBN: 9780321856562

Chapter 1: Q15P (page 18)

Calculate the divergence of the following vector functions:
$(a) v_{a} = x^{2} \hat{x} + 3 x z^{2} \hat{y} - 2 x z \hat{z} (b) v_{b} = x y \hat{x} + 2 y z \hat{y} + 3 z x \hat{z} (c) v_{c} = y^{2} \hat{x} + (2 x y + z^{2}) + 2 y z \hat{z}$

Short Answer

Expert verified

(a) The divergence is 0.

(b) The divergence is $^{3 x + y + 2 z}$ .

(c) The divergence is $^{2 (x + y)}$ .

Step by step solution

01

Write the expression for the divergence of the function.

Determine the divergence of the function.

$鈭� . v = (\hat{x} \frac{鈭�}{鈭� x} + \hat{y} \frac{鈭�}{鈭� y} + \hat{z} \frac{鈭�}{鈭� z}) (v_{x} \hat{x} + v_{y} \hat{y} + v_{z} \hat{z}) = \frac{鈭� v_{x}}{鈭� x} + \hat{y} \frac{鈭� v_{y}}{鈭� y} + \hat{z} \frac{鈭� v_{z}}{鈭� z}$

02

Solve for the divergence of part (a).

Consider the given vector is,

$v_{a} = x^{2} \hat{x} + 3 x z^{2} y - 2 x z \hat{z}$

Solve for the divergence.

$鈭� . v_{a} = \frac{鈭�}{鈭� x} x^{2} + \frac{鈭�}{鈭� y} 3 x z^{2} - \frac{鈭�}{鈭� z} 2 x z = 2 x + 0 - 2 x = 0$

Therefore, the divergence is 0.

03

Solve for the divergence of part (b).

Consider the given vector is,

$v_{b} = x y \hat{x} + 2 y z^{2} y + 3 z x \hat{z}$

Solve for the divergence.

$鈭� . v_{a} = \frac{鈭�}{鈭� x} x y + \frac{鈭�}{鈭� y} 2 y z - \frac{鈭�}{鈭� z} x z = 3 x + y + 2 z$

Therefore, the divergence is $^{3 x + y + 2 z}$

04

Solve for the divergence of part (c).

Consider the given vector is,

$v_{a} = y^{2} \hat{x} + (2 x y + z^{2}) + 2 y z \hat{z}$

Solve for the divergence.

$鈭� . v_{a} = \frac{鈭�}{鈭� x} y^{2} + \frac{鈭�}{鈭� y} (2 x y + z^{2}) + \frac{鈭�}{鈭� z} 2 y z = 0 + (2 x + 0) + 2 y = 2 (x + y)$

Therefore, the divergence is $^{2 (x + y)}$ .

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Most popular questions from this chapter

(a) If A and B are two vector functions, what does the expression $(A 路鈭�) B$ mean?(That is, what are its x, y, and z components, in terms of the Cartesian componentsof A, B, and V?)

(b) Compute $(\hat{r} 路鈭�) \hat{r}$ , where r is the unit vector defined in Eq. 1.21.

(c) For the functions in Prob. 1.15, evaluate $(v_{a} 路鈭�) v_{b}$ .

A uniform current density $J = J_{0} \hat{z}$ fills a slab straddling the $yz$ plane, from $x = - a to x = + a$ to . A magnetic dipole $m = m_{0} x$ is situated at the origin.

(a) Find the force on the dipole, using Eq. 6.3.

(b) Do the same for a dipole pointing in the direction: $m = m_{0} y$ .

(c) In the electrostatic case, the expressions $F = 鈭� (p . E)$ and $F = (p . 鈭�) E$ are equivalent (prove it), but this is not the case for the magnetic analogs (explain why). As an example, calculate $(m . 鈭�)$ for the configurations in (a) and (b).

(a) If A and B are two vector functions, what does the expression $(A 路鈭�) B$ mean?(That is, what are its x, y, and z components, in terms of the Cartesian componentsof A, B, and V?)

(b) Compute $(\hat{r} 路鈭�) \hat{r}$ , where $\hat{r}$ is the unit vector defined in Eq. 1.21.

(c) For the functions in Prob. 1.15, evaluate $(v_{a} 路鈭�) v_{b}$ .

Prove that. $[A 脳 (B 脳 C)] + [B 脳 (C 脳 A)] + [C 脳 (A 脳 B)] = 0$ Under what conditions does $A 脳 (B 脳 C) = (A 脳 B) 脳 C$ ?

Calculate the Laplacian of the following functions:

$(a) T_{a} = x^{2} + 2 xy + 3 z + 4 (b) T_{b} = sinx siny sinz (c) T_{c} = e^{- 5} \sin 4 y \cos 3 z . (d) v = x^{2} \hat{x} + 3 {xz}^{2^{}} \hat{y} + - 2 xz \hat{z}$

See all solutions

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