Question #281696

The lines of electric force of two opposite charges of the same strength at (-1,0) and (1,0) are the circles through (-1,0) and (1,0). Show that these circles are given by x^2 + (y - c)^2 = 1 + c^2. Show that the equipotential lines (which are orthogonal trajectories of those circles) are the circles given by (x + c*)^2 + y^2 = c*^2 - 1.


1
Expert's answer
2021-12-21T18:42:09-0500

Solution:

Setting y=0 gives from x2+(yc)2=1+c2x^{2}+(y-c)^{2}=1+c^{2} the equation x2+c2=1+c2x^{2}+c^{2}=1+c^{2} ; hence x=-1 and x=1, which verifies that those circles all pass through -1 and 1 , each of them simultaneously through both points. Subtracting c2c^{2} on both sides of the given equation, we obtain

x2+y22cy=1,x2+y21=2cy,x21y+y=2c.x^{2}+y^{2}-2 c y=1, \quad x^{2}+y^{2}-1=2 c y, \quad \frac{x^{2}-1}{y}+y=2 c .

Emphasize to your class that the ODE for the given curves must always be free of c. Having accomplished this, we can now differentiate. This gives

2xy(x21y21)y=0\frac{2 x}{y}-\left(\frac{x^{2}-1}{y^{2}}-1\right) y^{\prime}=0

This is the ODE of the given curves. Replacing y with1/y~y^{\prime}\ with -1 / \widetilde{y}^{\prime} and y with y~\tilde{y} , we obtain the ODE of the trajectories:

2xy~(x21y~21)/(y~)=0\frac{2 x}{\widetilde{y}}-\left(\frac{x^{2}-1}{\tilde{y}^{2}}-1\right) /\left(-\widetilde{y}^{\prime}\right)=0

Multiplying this by yy^{\prime} , we get

2xy~y~+x21y~21=0.\frac{2 x \tilde{y}^{\prime}}{\tilde{y}}+\frac{x^{2}-1}{\tilde{y}^{2}}-1=0 .

Multiplying this by y~2/x2\widetilde{y}^{2} / x^{2} , we obtain

2y~y~x+11x2y~2x2=ddx(y~2x)+11x2=0.\frac{2 \widetilde{y} \tilde{y}^{\prime}}{x}+1-\frac{1}{x^{2}}-\frac{\tilde{y}^{2}}{x^{2}}=\frac{d}{d x}\left(\frac{\tilde{y}^{2}}{x}\right)+1-\frac{1}{x^{2}}=0 .

By integration,

y~2x+x+1x=2c. Thus, y~2+x2+1=2cx\frac{\tilde{y}^{2}}{x}+x+\frac{1}{x}=2 c^{*} . \quad \text { Thus, } \quad \tilde{y}^{2}+x^{2}+1=2 c^{*} x .

We see that these are the circles

y~2+(xc)2=c1\tilde{y}^{2}+\left(x-c^{*}\right)^{2}=c^{*}-1


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