Solving Electric Field: From (3) to (4)

[jumi]

So there's this situation going on:
http://imageshack.us/a/img826/7398/physicsforums.png

Going from the definition of an electric field:
(1) $\vec{E} ( \vec{x} ) = \frac{1}{4\pi\epsilon_{0}} ∫ \frac{\vec{x} - \vec{x'}}{| \vec{x} - \vec{x'} | ^3} ρ( \vec{x'}) d^3x'$


(2) The $ρ(\vec{x'})d^3x'$ reduces to $λdz'$. And $\vec{x} - \vec{x'} = x \hat{i} - z' \hat{k} = \sqrt{x^2 + z'^{2}}$.


(3) Now, plugging this information into the electric field equation yields:
$\vec{E} ( \vec{x} ) = \frac{1}{4\pi\epsilon_{0}} \int^{l}_{-l} \frac{\sqrt{x^2 + z'^{2}}}{(\sqrt{x^2 + z'^{2}})^3} λdz'$


(4) However, the book (Electromagnetism by Pollack and Stump) shows:
$E_{x}(x,0,0) = \frac{1}{4\pi\epsilon_{0}} \int^{l}_{-l} \frac{x}{(\sqrt{x^2 + z'^{2}})^3} λdz'$

How do we get from (3) to (4)? Why is z' only removed from the numerator?

Thanks in advance.

 

[vanhees71]

Your formula (3) is obviously wrong, because on the left-hand side is a vector and on the right-hand side a scalar quantity.

From (1) and (2) you immediately write down (4). So your book is correct. There is no z' in the numerator of the x component!

 

[andrien]

so the charged wire is along z axis so by symmetry only x component will survive so
z will be absent in numerator and beware it is a vector.

 

[jumi]

Your formula (3) is obviously wrong, because on the left-hand side is a vector and on the right-hand side a scalar quantity.

From (1) and (2) you immediately write down (4). So your book is correct. There is no z' in the numerator of the x component!

I don't understand how I could just go from (1) and (2) to (4)...

so the charged wire is along z axis so by symmetry only x component will survive so
z will be absent in numerator and beware it is a vector.

Or why the numerator doesn't have a vector magnitude, whereas the demoninator does...


I was, however, able to get the correct answer, just not using the full notation from equation (1).

So I start with: $\vec{E} ( \vec{x} ) = \frac{1}{4\pi\epsilon_{0}} ∫ \frac{\vec{x} - \vec{x'}}{| \vec{x} - \vec{x'} | ^3} ρ( \vec{x'}) d^3x'$

And since $d\vec{E}$ generated by $dq$ on the line will be symmetric (i.e. only the x-component will survive), we can say $dE_{x} = d\vec{E}cos(\theta)$.

Therefore: $dE = \frac{1}{4\pi\epsilon_{0}} \frac{1}{r^2} dq$, where $r = | \vec{x} - \vec{x'}| = \sqrt{x^2 + z'^2}$

Therefore: $dE_{x} = \frac{1}{4\pi\epsilon_{0}} \frac{1}{x^2 + z'^2} \frac{x}{\sqrt{x^2 + z'^2}} λdz'$ from $cos(\theta) = \frac{x}{\sqrt{x^2 + z'^2}}$ and $dq = λdz'$

Which gives the result in the book: $E_{x}(x,0,0) = \frac{1}{4\pi\epsilon_{0}} \int^{l}_{-l} \frac{x}{(\sqrt{x^2 + z'^{2}})^3} λdz'$.

So was I supposed to recognize that I needed to use the cos(θ) relationship from the beginning? I'm also still confused on how the vector difference only acts on the denominator, if I were to just use equation (1).

Thank in advance.

 

[sardar]

Based on the information provided, it appears that the situation being discussed is related to calculating the electric field at a point due to a line of charge. The initial equation (1) is the general definition of the electric field, which takes into account the distance between the point of interest and the source of charge, as well as the distribution of charge.

In (2), the distribution of charge is simplified to a linear charge density (λ) multiplied by a differential length (dz'). This is likely done for convenience in the calculation, as it reduces the integral to a one-dimensional problem.

In (3), the equation is further simplified by substituting the expression for the distance between the point of interest and the source (x and z') into the denominator of the equation. This results in a simpler integral to solve for the electric field at a given point.

In (4), the electric field is being calculated at a specific point along the x-axis (x,0,0). This means that z' = 0, and thus the term for z' is removed from the equation. This is because the point of interest is on the same axis as the line of charge, so the distance between the two is simply the value of x.

In summary, the simplifications made in (2) and (3) are done to make the integral easier to solve, and the difference between (3) and (4) is due to the specific point at which the electric field is being calculated. I hope this helps clarify the situation.