- #1

jumi

- 28

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So there's this situation going on:

http://imageshack.us/a/img826/7398/physicsforums.png [Broken]

Going from the definition of an electric field:

(1) [itex]\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'[/itex]

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

(3) Now, plugging this information into the electric field equation yields:

[itex]\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'[/itex]

(4) However, the book (

[itex]E_{x}(x,0,0) = \frac{1}{4\pi\epsilon_{0}} \int^{l}_{-l} \frac{x}{(\sqrt{x^2 + z'^{2}})^3} λdz' [/itex]

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

Thanks in advance.

http://imageshack.us/a/img826/7398/physicsforums.png [Broken]

Going from the definition of an electric field:

(1) [itex]\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'[/itex]

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

(3) Now, plugging this information into the electric field equation yields:

[itex]\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'[/itex]

(4) However, the book (

*Electromagnetism*by Pollack and Stump) shows:[itex]E_{x}(x,0,0) = \frac{1}{4\pi\epsilon_{0}} \int^{l}_{-l} \frac{x}{(\sqrt{x^2 + z'^{2}})^3} λdz' [/itex]

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

Thanks in advance.

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