Mathematical Physics Vol 1

Chapter 4. Field theory

170

Exercise 106 A force field is given by

F =( y 2 cos x + z 3 ) i +( 2 y sin x − 4 ) j +( 3 xz 2 + 2 ) k .

a) Prove that F is a conservative force. b) Find the scalar potential φ of the force F . c) Find the work required to move a particle in this field from point A ( 0 , 1 , − 1 ) to point B ( π / 2 , − 1 , 2 ) .

Solution b) φ = y 2 sin x + xz 3 − 4 y + 2 z + const, c)15 + 4 π .

Exercise 107 Prove that F = r 2 r is a conservative force and find the scalar potential.

Solution

r 4 4

φ =

+ const.

Exercise 108

Let E = r r . a) Is there a function φ that satisfies E = − ∇ φ ? If there is, find it. b) Calculate I c E · d r , if c is any unambiguous closed curve.

Solution

r 3 3

a) φ = −

+ const, b) 0.

Exercise 109

Showthat

( 2 x cos y + z sin y ) d x +( xz cos y − x 2 sin y ) d y + x sin y d z ,

is a total differential. From here, solve the differential equation

( 2 x cos y + z sin y ) d x +( xz cos y − x 2 sin y ) d y + x sin y d z = 0 .