Why Does SQRT(3) Factor into the Coulomb Force in an Equilateral Triangle?

[Chronos000]

Homework Statement

3 identically charged charges are placed at corners of an equilateral triangle of sides a. I need to find the force felt by each charge.

To do this I have calculated the force from charges - "2 on 1" and then "3 on 1". The answer i get is (2*q^2)/4*Pi*e*a^2. This is not correct however. A factor of SQRT(3) times the answer i got is correct. I don't see how to get to this

 

[Mr.A.Gibson]

The charges are not pushing in the same direction so they cannot be added together, you'll have to add vectors and do some trig.

If you draw out the diagram then draw the direction of force on each particle, i.e. two forces on each, all forces the same size, this'll help you see they are pushing in different directions and hence cannot be added.

You then draw a vector diagram for the force on one of the particles. Draw the arrow for one force, where that arrow ends start the arrow for the other force. This should form an isosceles triangle with an angel of 120 between the two force vectors.

Since you know the size of one of the force vectors, and hence two sides of the triangle and the angle between them you can now calculate the length of the other side of the triangle. This will be the size of the resultant force on each charge.

Hope this is clear it's much easier to draw it than explain it words.

 

[Chronos000]

Could you please explain how I get an angle of 120. I can only see that it would be 60.

 

[Mr.A.Gibson]

The green line is the resultant force, the angle in black is the 120 degree angle.

 

[rwisz]

The concept of superposition of charges states that the total force experienced by a charge due to multiple other charges can be found by adding up the individual forces from each charge. In this case, we have three identically charged charges placed at the corners of an equilateral triangle.

To find the force experienced by each charge, we can use the Coulomb's Law equation, which states that the force between two charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

In the first calculation, "2 on 1," you correctly used the equation to find the force between two charges, but this only accounts for the force between two of the three charges. To find the total force experienced by each charge, we need to consider the forces from all three charges.

In the second calculation, "3 on 1," you are finding the total force experienced by the charge at the center of the triangle due to the other two charges. However, this does not account for the forces between the two charges at the corners.

To include the forces between all three charges, we can use vector addition. This means we need to consider the direction and magnitude of each force. Since the triangle is equilateral, the forces between the two charges at the corners and the charge at the center will be equal in magnitude, but opposite in direction.

Therefore, to find the total force experienced by each charge, we can use the equation F = (2*q^2)/4*Pi*e*a^2 * SQRT(3), where the factor of SQRT(3) accounts for the vector addition of the forces.

I hope this explanation helps in understanding how to arrive at the correct answer. Remember to always consider the direction and magnitude of forces when dealing with multiple charges.