A semicircular loop of radius a carries positive charge Q....

[robren]

Homework Statement

A semicircular loop of radius a carries positive charge Q distributed uniformly...

Find the electric field at the loop's center (point P in the figure). Hint: Divide the loop into charge elements dq as shown in the figure, and write dq in terms of the angle dθ, then integrate over θ.Express your answer in terms of i^, j^, k, Q, a.

Homework Equations

I know dq = Q/pi * dtheta

i know dE = (k*dq)/a^2 => [(k*dq)/(pi*a^2) ] * dtheta

The Attempt at a Solution

I do everything up until the integral dE... then solving for E
solving the integral of E = integral from 0 to pi | dE * dtheta

I've looked up how to do that and I see how it is done, (probably not understanding it correctly though) because the integral of dE * dtheta apparently gets you 2[(k*q)/(pi*a^2)] and I have no idea how the 2 got there...

Also, the answer I'm looking for is in i^, j^ forms apparently so the way I'm doing it I think is a little different from how my instructor want's me to do it, or maybe just the conversion through trig in the end could give me x/y components but I simply don't know how since a ring charge doesn't give any y-component field... (maybe I am wrong?)

So please explain how the 2 was gotten in the integral, or how it was done (really I'm not looking for the answer I already have it if i need it so please don't do hints or anything as the i^, j^ was pretty much a hint...

Also I wan't to know exactly what the i^ and j^ constitute as a charge for a ring charge... (i can probably assume those from understanding how the integral could be done but also just have no idea how to get i and j from this other than doing trig but idk how to apply that with the integral equations. JUST SUPER CONFUSED BUT NO REALLY...

 

[Orodruin]

integral from 0 to pi | dE * dtheta

There are two problems here that you are not considering. First of all, you are adding the contributions to the electric field as if they were scalar quantities. The electric field is a vector field and takes vector values. Therefore you must add the contributions as vectors, including their magnitude as well as direction. Second, you have two different differentials in a one dimensional integral, which does not make sense.

 

[robren]

that's why I'm confused... I don't think I explained it properly but...

I know the vector adding is what I'm looking for.. I just don't know how to get there from the integral equation and...

I saw the two differentials dE and dtheta in my integral, that's why I'm confused because I understand how E can = dE*dtheta but am I then doing u sub or some form of sub to get the integral? I just really would like to have some help of setting up and why... I'm trying to work through this as well and telling me two things I did wrong is simply making me more confused because I KNOW IM WRONG and I can't get anywhere by looking at an improperly set up process..

 

[robren]

There are two problems here that you are not considering. First of all, you are adding the contributions to the electric field as if they were scalar quantities. The electric field is a vector field and takes vector values. Therefore you must add the contributions as vectors, including their magnitude as well as direction. Second, you have two different differentials in a one dimensional integral, which does not make sense.

if that makes sense

 

[Orodruin]

I saw the two differentials dE and dtheta in my integral, that's why I'm confused because I understand how E can = dE*dtheta

Then this is the first piece of understanding you need to work on because it is wrong.

What is the vectorial contribution to the E field from a small element dθ located at angle θ?