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.
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Expert's answer
2021-12-21T18:42:09-0500
Solution:
Setting y=0 gives from x2+(y−c)2=1+c2 the equation x2+c2=1+c2 ; 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 c2 on both sides of the given equation, we obtain
x2+y2−2cy=1,x2+y2−1=2cy,yx2−1+y=2c.
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
y2x−(y2x2−1−1)y′=0
This is the ODE of the given curves. Replacing y′with−1/y′ and y with y~ , we obtain the ODE of the trajectories:
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