How Is the Electrical Force Calculated on a Charge at the Center of a Hexagon?

[sandplasma]

I'm having a little trouble with this one..

Three pointlike charges Q are located on three successive vertices of a reguar hexagon with sides "l". Find the electrical force on another charge q located at the center of the hexagon. Assume all the charges are like charges. ( all positive )

I know I'm supposed to use Coulombs law as well as the principle of superposition but I'm having trouble with the direction.

Thanks in advance for the help!

 

[Gza]

The first step of any physics problem is defining a coordinate system. A "regular" coordinate system (no rotations or anything) with its origin at the center of the hexagon makes the most sense for this problem since it utilizes a lot of simplifying symmetry. Next find the contribution each individual charge on the vertices of the hexagon makes to the net force on the charge at the origin. This of course is done with as you mentioned, Coulomb's law. Sticking with the vectors throughout your solution makes it much easier to determine directions for forces.

Coulombs Law

$$\vec{F_{q2q1}} = \frac{k q_1q_2}{r^2} \hat{r_{q1q2}}$$

where $$\hat{r_{q1q2}}$$ represents a relative unit vector, or going by the subscripts, the position of charge 1 relative to charge 2. This vector simply "points" from charge 2 to charge 1. Charge 2 will represent the three charges on the vertices of the hexagon since we want to find their force which points at charge 1 at the origin; in other words the force of q2 on q1, as stated on the left side of coulombs law.

Have a look at the picture to get a feel for the setup. By the geometry of a hexagon, the distance from the vertex to the center is simply "l". (verify this) So the only important task left is finding the relative unit vectors for each charge on the hexagon in $$\hat{i},\hat{j}$$ components, sticking them into Coulombs law and adding the whole thing up, by principal of superposition.

The easiest technique I've found for determining relative vectors is through the use of something my mechanics teacher taught me a few years back, which he simply called the neumonic (i think that's how you spell it.) It looks like this:

$$\vec{r_{12}} = \vec{r_{1O}} + \vec{r_{O2}}$$
where O represents the origin of your coord sys.

This is easy for me to remember because the origin shows up on the "inside" of the subscripts of the two added vectors.


or flipping around the second vector being added(since it makes more sense in problem solving to find the position of an object relative to the origin, not vice versa)

$$\vec{r_{12}} = \vec{r_{1O}} - \vec{r_{2O}}$$

of course by dividing through by the magnitude (which is "l") you obtain the unit vector, which is what you need. I hope this helped you get started!

 

[M98Ranger]

Coulomb's Law states that the force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. In this case, we have three point charges, Q, located on three vertices of a regular hexagon with sides "l" and another charge, q, located at the center of the hexagon.

To find the electrical force on q, we need to use the principle of superposition, which states that the total force on a charge is the vector sum of the individual forces exerted by each charge. This means that we will need to calculate the force between q and each of the charges Q separately, and then add them together.

To do this, we can use Coulomb's Law, which is given by F = k * (q1 * q2)/r^2, where k is the Coulomb's constant, q1 and q2 are the two charges, and r is the distance between them. In this case, since all the charges are like charges, we can use the formula F = k * (q^2)/r^2, where q is the magnitude of the charge and r is the distance between the two charges.

Since all the charges Q are located at the vertices of the hexagon, we can calculate the distance between q and each of the charges Q by using the Pythagorean theorem. The distance between q and each of the charges Q will be l/2, since the hexagon is regular and the distance between any two vertices is equal to the length of the side divided by 2.

Now, we can plug in the values into the formula F = k * (q^2)/r^2 and calculate the force between q and each of the charges Q. Remember to include the direction of the force, which will be along the line connecting q and each of the charges Q.

Once you have calculated the force between q and each of the charges Q, you can add them together to find the total force on q. Remember to use vector addition, which means adding the forces as vectors, taking into account their magnitudes and directions.

I hope this helps to clarify the steps needed to solve this problem. Remember to always pay attention to the direction of the forces, and to use the principle of superposition when dealing with multiple charges. Good luck!