Coulomb's Law and Integration

[nealh149]

Homework Statement

A rod is bent into a semi circle of 180 degrees. The bottom half has a charge of -q and the top half a charge of q. Find the electric field at the center of the circle, point P.

Homework Equations

E = kq/r^2

The Attempt at a Solution

I tried to set up an integral for this. I know that the horizontal components will cancel out, and the vertical will be additive, pointing in the negative y direction.

S = integral

0 - 180 S((2kq/r^2)sin(theta))d(theta), which is (2kq/r^2)(-2)

4kq/r^2

which seems wrong.

Any help?

 

[anathema]

Hello! Thank you for posting your question. I am a scientist and I would be happy to help you find the electric field at point P.

First, let's start by visualizing the situation. We have a semi-circle with a charge of -q on the bottom half and a charge of q on the top half. The center of the circle, point P, is located at the midpoint of the semi-circle.

To find the electric field at point P, we can use the equation E = kq/r^2, where k is the Coulomb's constant, q is the charge, and r is the distance between the charge and the point where we want to find the electric field.

In this case, we have two charges, -q and q, located at a distance of r/2 from point P. We can calculate the electric field due to each charge separately and then add them together to find the total electric field at point P.

For the charge -q on the bottom half, the electric field at point P will be directed towards the charge and can be calculated as:

E1 = k(-q)/(r/2)^2 = -4kq/r^2

For the charge q on the top half, the electric field at point P will be directed away from the charge and can be calculated as:

E2 = kq/(r/2)^2 = 4kq/r^2

Since the electric fields due to the two charges are in opposite directions, we need to subtract them to find the total electric field at point P:

E = E1 + E2 = (-4kq/r^2) + (4kq/r^2) = 0

Therefore, the electric field at point P is equal to 0.

I hope this helps! Let me know if you have any further questions.

 

[m2003]

I would first commend the effort in trying to use integration to solve this problem. However, there are a few things to consider in your approach.

First, Coulomb's Law states that the electric field is directly proportional to the charge and inversely proportional to the square of the distance. In this case, the distance from the center of the circle to any point on the rod will vary, so you will need to use a variable (such as r) in your integral.

Second, when using integration to find the electric field, it is important to take into account the direction of the field at each point. In this case, you correctly noted that the horizontal components will cancel out, but the vertical components will not simply be additive. Instead, they will have different directions at different points on the rod, and this needs to be accounted for in your integral.

Lastly, your approach to using integration may not be the most efficient method for solving this problem. Instead, you could consider using the superposition principle, which states that the total electric field at a point is the vector sum of the individual electric fields from each charge. In this case, you could find the electric field from the top half of the rod and the bottom half separately, and then add them together to find the total electric field at point P.

In summary, while your attempt at using integration to solve this problem shows a good understanding of the concepts, there are some additional factors to consider in order to arrive at the correct solution.