   Chapter 16.9, Problem 23E

Chapter
Section
Textbook Problem

Verify that div E = 0 for the electric field E ( x )   =   ε Q | x | 3 x .

To determine

To verify: The divergence of electric field is zero if the electric field E(x)=εQ|x|3x .

Explanation

Given Data:

Write the given electric field as follows.

E(x)=εQ|x|3x (1)

Formula used:

Write the expression to find divergence of vector field F(x,y,z)=Pi+Qj+Rk .

divF=xP+yQ+zR (2)

Consider the vector x as follows.

x=xi+yj+zk

Substitute (xi+yj+zk) for x in equation (1) and rewrite the electric field as follows.

E=εQ|(xi+yj+zk)|3(xi+yj+zk)=εQ(x2+y2+z2)3(xi+yj+zk)=εQ(x2+y2+z2)32(xi+yj+zk)=εQx(x2+y2+z2)32i+εQy(x2+y2+z2)32j+εQz(x2+y2+z2)32k

Calculation of divE :

Substitute εQx(x2+y2+z2)32 for P , εQy(x2+y2+z2)32 for Q , and εQz(x2+y2+z2)32 for R in equation (3),

divE=[x(εQx(x2+y2+z2)32)+y(εQy(x2+y2+z2)32)+z(εQz(x2+y2+z2)32)]=εQ[x(x(x2+y2+z2)32)+y(y(x2+y2+z2)32)+z(z(x2+y2+z2)32)]

Simplify the expression as follows

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