Path Independence of Conservative Forces — Question 7

Question 7

Cross-coupled components and a rectangular loop

A field is $\vec F=(ay\,\hat i+bx\,\hat j)$, where a and b have units N/m. Follow the rectangle (0,0), (L,0), (L,H), (0,H), (0,0), with L,H positive.

1. Find the work around the rectangle.

2. What relation between a and b is necessary for every such loop to give zero?

3. When that relation holds, construct a potential to show path independence.

Question 7 - Solution

Cross-coupled components and a rectangular loop

(a) Evaluate the four directed segments.

Evaluate the four oriented sides: $$W_{\mathrm{bottom}}=0,\quad W_{\mathrm{right}}=\int_0^H bL\,dy=bLH,$$ $$W_{\mathrm{top}}=\int_L^0 aH\,dx=-aLH,\quad W_{\mathrm{left}}=0.$$ $$\boxed{W_{\mathrm{loop}}=(b-a)LH}.$$

(b) Require zero for arbitrary positive dimensions.

Since $L,H>0$, $$(b-a)LH=0\Rightarrow\boxed{a=b}.$$ The condition is necessary for zero work on every such nondegenerate rectangle.

(c) Show the condition is sufficient for this field.

When $a=b$, construct $$\boxed{U(x,y)=-axy+C},\qquad-\nabla U=(ay,ax)=\vec F.$$ The potential exists throughout the plane, making this condition sufficient as well as necessary for the stated field.