Prove Electric Force on Triangle ABC Lies on Bisector

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In summary, the conversation discusses how to prove that the direction of electric force at point A in triangle ABC lies on the bisector of angle BAC. The theory of electric fields and rotation is used to show that this is true for any triangle, not just when AB=AC. The proof is done using polar coordinates and the contribution of charged sections on intervals around the bisector. The conversation also mentions the rules of the forum, reminding users to give their own attempts at solutions and not ask for full solutions.
  • #1
sma
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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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  • #2
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.
 
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  • #3
I think your assumpion is not right.why If we rotate the triangle around bisector by alfa=180 degrees, then rotation transforms riangle back into itself! It is true only for AB=AC!
 
  • #4
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

dl=r*dfi/cos(delta), r=d/cos(delta)

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 really independent of fi.
 
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  • #5
excuse me,I don't underestand your solution
 
  • #6
Sma: You have posted this question TWICE in a very short period of time

and don't ask for solutions, and also you MUST give attempt to soultion.

Lojzek: never give full solutions

Please read and follow the rules of this forum.
 

Related to Prove Electric Force on Triangle ABC Lies on Bisector

1. How is the electric force on triangle ABC determined?

The electric force on triangle ABC can be determined by using Coulomb's Law, which states that the force between two point charges is equal to the product of the charges divided by the square of the distance between them.

2. What is the significance of the electric force lying on the bisector of triangle ABC?

The bisector is a line that divides an angle into two equal parts. When the electric force on triangle ABC lies on the bisector, it tells us that the force is acting equally on both sides of the angle, creating a balanced and stable system.

3. How do you prove that the electric force on triangle ABC lies on the bisector?

To prove that the electric force on triangle ABC lies on the bisector, we can use the concept of vector addition. By breaking down the force into its components and using trigonometric functions, we can show that the force is acting equally on both sides of the bisector.

4. Can the electric force on triangle ABC ever not lie on the bisector?

Yes, there are situations where the electric force may not lie on the bisector. This can happen if there are other external forces acting on the system, or if the charges are not evenly distributed on the triangle.

5. What are the real-life applications of understanding the electric force on triangle ABC lying on the bisector?

Understanding the electric force on triangle ABC can be applied in various fields such as engineering, physics, and technology. It can help in designing stable structures, analyzing the behavior of electric fields, and creating efficient electrical systems.

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