Chapter 5.5 - Calculating Electric Fields of Charge Distributions

Line Charge

• $ L $: length of the line charge
• $ \lambda $: linear charge density ($ \rm C/m $)
• $ z $: distance from the line charge
• $ dl $: differential length

Electric field for a line charge:

$$ \vec{E}(P) = \frac{1}{4 \pi \varepsilon_0} \int_{\text{line}} \frac{\lambda , dl}{r^2} \hat{r} $$

For a finite line charge (charged wire), where $z \gg L$, this simplifies to:

$$ \vec{E} \approx \frac{1}{4 \pi \varepsilon_0} \frac{\lambda L}{z^2} \hat{k}. $$

For an infinite line charge, the electric field is:

$$ \vec{E}(z) = \frac{1}{4 \pi \varepsilon_0} \frac{2 \lambda}{z} \hat{k} $$

Surface Charge

Electric field due to an infinite plane of charge:

$$ \vec{E} = \frac{\sigma}{2 \varepsilon_0} \hat{k} $$

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