# Coulomb's law, balancing a ball with point charges

• theneedtoknow
In summary, the radial force exerted on the bead of charge Qb is dependent on its position on the z-axis.
theneedtoknow
3 equal charges q are at the vertices of an equilateral triangle as shown, with the z-axis running through the midpoint of the triangle (such that the distance from each charge to the midpoint is d)

http://img111.imageshack.us/img111/7159/graphieew7.th.jpg

a bead of charge Qb (of equal sign as the 3 charges q) is supposed to be levitated on the positive z-azis (coming out of the midpoint of the triangle). Derive an expression for the coloumb force exerted on the bead as a function of its position on the positve z-axis

## Homework Equations

F = (K * |q| * |Qb| ) / r^2 is the force in the radial direction (straight line connecting the 2 charges)

## The Attempt at a Solution

OK I'm kind of having trouble here...

The force from each charge would simply be F = (K * |q| * |Qb| ) / (d^2 + z^2)
(by pythagorean theorem, the square of the line connecting each charge to any point on the positive z-azis would be d^2 + z^2)
So far so good
because of the way the charges are positioned positioned , the x and y components of the vectors will cancel each other in between the 3 charges (I think) so we're left toworry only about the z-component of each radial vector...but how do i do calculate for this z-component?

If i break down the radial force into 2 components, with one in the x-y plane and the other in the z-direction, then the angle etween the xy-plane component and the radial component is tan^-1 (z/d)...but now hat i have the hypotenuse ((K * |q| * |Qb| ) / (d^2 + z^2) ) and the angle tan^-1 (z/d), how do i isolate for just the z-component of the radial vector?
z-component of the vector would be the hypotenuse times cos of the angle...but i have the angle expressed as inverse tan of (z/d) so how can i take the cos of something i only have expressed as that? I am totally lost as to how to derive this expression...

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Radial force can be written as F = i*Fx + j*Fy + k*Fz
Direction of the F is given by direction cosines. z component of the F = Fz = F*cos(gamma)= F*z/r. This is due to one charge.

So how does cos(gamma) come out to z/r?

OH never midn i udnerstand where it comes from...If i do it by similar triangles than
Fz / Ftot = z / r
Fz = Ftot * z/r

So i guess Fz = kqQb * z / (Z^2+D^2)^2

thanks for the help :)

## 1. What is Coulomb's law?

Coulomb's law is a fundamental law of electrostatics that describes the force of attraction or repulsion between two electrically charged particles. It states that the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

## 2. How does Coulomb's law apply to balancing a ball with point charges?

In the context of balancing a ball with point charges, Coulomb's law is used to calculate the force of repulsion between the charged ball and the charged point charges. By carefully arranging the charges, the forces can be balanced and the ball can be suspended in mid-air.

## 3. What is the difference between positive and negative charges in Coulomb's law?

In Coulomb's law, positive charges attract negative charges and repel other positive charges. Similarly, negative charges attract positive charges and repel other negative charges. This is due to the principle of like charges repel and opposite charges attract.

## 4. How does the distance between charges affect the force in Coulomb's law?

In Coulomb's law, the force between two charges is inversely proportional to the square of the distance between them. This means that as the distance between the charges increases, the force decreases. Similarly, as the distance decreases, the force increases.

## 5. How is Coulomb's law related to other laws of physics?

Coulomb's law is closely related to other laws of physics, such as Newton's law of gravitation and the inverse-square law. It is also related to the concept of electric potential energy, which is the energy stored in a system of charged particles. Additionally, Coulomb's law is a fundamental building block of electromagnetism and is used in various fields such as engineering and physics.

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