Consider triangle ABC,the continues charges distribution lies on the side of BC with the linear charge density such as m,prove that the direction of electric force in the point A lies on the bisector of the viewing angle BAC
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You can prove that almost without knowing anything about electric field! We only need to know that field is determined by electric charge and that laws of physics are the same in all non-accelerated coordinate systems.
We will use this simple theory:
If we rotate the triangle (around any axis) by an angle alfa then the field will also rotate by the same angle (around the same axis). This can be proved by solving this problem in a rotated coordinate system (by alfa) where the rotated triangle seems the same as original triangle in original system (so the solution is the same). Then we transform the field vector back into original sistem and we find out it has rotated by alfa.
Proof by contadiction:
Let's suppose that the field does not lie on the bisector. If we rotate the triangle around bisector by alfa=180 degrees, then the field will also rotate, resulting in changed direction of the field. However this rotation transforms triangle back into itself! We got a different electric field from the same charge distribution! This is imposible, so the assumption that field does not lie on bisector is wrong.
Sorry, I missread the question. Here is proof for general triangle:
Use polar coordinates, with the bisector for fi=0 axes. It is enough to show, that
sections of charged line on intervals (fi,fi+dfi) and (-fi,-fi-dfi) exactly cancel each other
out (as far as perpendicular component of E is concerned). Since sin(-fi)=sin(fi), it is enough to show that the magnitudes are the same.
Contributions of these sections are:
r^-2*dl (times a constant)
We can prove that both magnitudes of dE are the same by proving
r^-2*dl/dfi=const (independent of fi) (1)
This is easy: if delta is angle between r and BC, then
where d is the shortest distance between (infinitely extended) BC line and point A.
If you put dl=dfi*d/cos(delta)^2 and r=d/cos(delta) into equation (1),
you find out the expression is realy independent of fi.
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