(a) In Example 3 of Section 15.1 we showed that(x, y) = c(x2 + y2)1/2is a potential

Chapter 15, Problem 34

(choose chapter or problem)

(a) In Example 3 of Section 15.1 we showed that

\(\phi(x,\ y)=-\frac{c}{\left(x^2+y^2\right)^{1/2}}\)

is a potential function for the two-dimensional inverse-square field  

\(F(x,\ y)=\frac{c}{\left(x^2+y^2\right)^{3/2}}(xi+yj)\)

but we did not explain how the potential function \(\phi(x,\ y)\) was obtained. Use Theorem 15.3.3 to show that the two-dimensional inverse-square field is conservative everywhere except at the origin, and then use the method of Example 4 to derive the formula for \(\phi(x,\ y)\) .

(b) Use an appropriate generalization of the method of Example 4 to derive the potential function  

\(\phi(x,\ y,\ z)=-\frac{c}{\left(x^2+y^2+z^2\right)^{1/2}}\)

for the three-dimensional inverse-square field given by Formula (5) of Section 15.1.

Equation Transcription:

Text Transcription:

phi(x, y)=- frac c / (x^2+y^2)^1/2

F(x, y) = frac c / (x^2+y^2)^3/2 (xi + y j)

phi(x, y)

phi(x, y)

phi(x, y, z)=- frac c / x^2+y^2+z^2)^1/2