Electric Field Due to a Point Charge

[tatiana_eggs]

Homework Statement

In below figure, the three particles are fixed in place and have charges q1 = q2 = +e and
q3 = +2e. Distance a = 6.00 μm. What are the magnitude of the
net electric field at point P due to the particles?
http://www.geocities.com/imfrrreal/22-36.jpg

Homework Equations

|E| = k * |q| / r²

The Attempt at a Solution

Well, I've got as far as understanding how to set up my vectors, (blue, above) realizing that E₁ and E₂ cancel. So now I set up my E_net = E₃ = k * 2e / r².

What is r in this case?! I think my real problem here could be geometry, and not physics, but who knows I could be missing something vital from the beginning of the E-fields chapter. I though r would be sqrt 2 * a = 8μm, making r = 4μm. But when I evaluate E_net = k * 2e / (4x10^-6)² I do not get the right answer.

In the solutions manual the final eq. they evaluate is: k * 2e / (a/sqrt 2)²

Where do they get r = a / sqrt 2 ?

Is r not the blue line, but twice as long since the third point charge is twice the charge of the others?

If this ends up being a geometry problem a) slap me b) accept my apologizes and move post.

 

[Doc Al]

I though r would be sqrt 2 * a...

Realize that "a" is the hypotenuse of a right triangle, so it must be the largest side of that triangle.

Where do they get r = a / sqrt 2 ?

Use the pythagorean theorem.

(It's a geometry problem.)

 

[nlightner]

I would first like to commend you on your attempt to solve the problem on your own and for recognizing the potential issue with geometry. It is always important to critically think about the problem and understand the underlying concepts before seeking help.

Now, let's break down the problem and see where the confusion lies. The electric field at point P is the vector sum of the electric fields due to all three point charges. As you correctly pointed out, the electric fields due to q1 and q2 will cancel each other out since they are equal in magnitude and opposite in direction. This leaves us with only the electric field due to q3.

To calculate the magnitude of the electric field, we use the equation |E| = k * |q| / r^2, where k is the Coulomb's constant, q is the charge of the particle, and r is the distance between the point charge and the point of interest. In this case, q3 is twice the charge of q1 and q2, so we can rewrite the equation as |E| = 2k * |q1| / r^2.

Now, let's look at the distance r. You are correct in thinking that r is the distance between the point charge and the point of interest. However, since the point charge is not directly above or below point P, we cannot simply use the distance a as r. Instead, we need to use the distance between the point charge and the projection of point P onto the line connecting the point charges. This distance is given by a/√2, as shown in the diagram.

Therefore, the correct equation for the magnitude of the electric field at point P is |E| = 2k * |q1| / (a/√2)^2. Plugging in the given values, we get |E| = 2*(9x10^9)*(1.6x10^-19) / (6x10^-6 / √2)^2 = 1.5x10^5 N/C.

I hope this helps clarify the confusion and demonstrates the importance of understanding the underlying concepts in physics problems. Keep up the good work!