Electric field
Continuous charge distributions

The superposition principle allows for the calculation of the electric field due to a distribution of charge density ρ (r) {\displaystyle \rho (\mathbf {r})}. By considering the charge ρ (r ′) d v {\displaystyle \rho (\mathbf {r} ')dv} in each small volume of space d v {\displaystyle dv} at point r ′ {\displaystyle \mathbf {r} '} as a point charge, the resulting electric field, d E (r) {\displaystyle d\mathbf {E} (\mathbf {r})}, at point r {\displaystyle \mathbf {r} } can be calculated as d E (r) = ρ (r ′) 4 π ε 0 r ^ ′ | r ′ | 2 d v = ρ (r ′) 4 π ε 0 r ′ | r ′ | 3 d v {\displaystyle d\mathbf {E} (\mathbf {r})={\frac {\rho (\mathbf {r} ')}{4\pi \varepsilon _{0}}}{{\hat {\mathbf {r} }}' \over {|\mathbf {r} '|}^{2}}dv={\frac {\rho (\mathbf {r} ')}{4\pi \varepsilon _{0}}}{\mathbf {r} ' \over {|\mathbf {r} '|}^{3}}dv} where

The total field is found by summing the contributions from all the increments of volume by integrating the charge density over the volume V {\displaystyle V}: E (r) = 1 4 π ε 0 ∭ V ρ (r ′) r ′ | r ′ | 3 d v {\displaystyle \mathbf {E} (\mathbf {r})={\frac {1}{4\pi \varepsilon _{0}}}\iiint _{V}\,\rho (\mathbf {r} '){\mathbf {r} ' \over {|\mathbf {r} '|}^{3}}dv}

Similar equations follow for a surface charge with surface charge density σ (r ′) {\displaystyle \sigma (\mathbf {r} ')} on surface S {\displaystyle S} E (r) = 1 4 π ε 0 ∬ S σ (r ′) r ′ | r ′ | 3 d a, {\displaystyle \mathbf {E} (\mathbf {r})={\frac {1}{4\pi \varepsilon _{0}}}\iint _{S}\,\sigma (\mathbf {r} '){\mathbf {r} ' \over {|\mathbf {r} '|}^{3}}da,} and for line charges with linear charge density λ (r ′) {\displaystyle \lambda (\mathbf {r} ')} on line L {\displaystyle L} E (r) = 1 4 π ε 0 ∫ L λ (r ′) r ′ | r ′ | 3 d ℓ. {\displaystyle \mathbf {E} (\mathbf {r})={\frac {1}{4\pi \varepsilon _{0}}}\int _{L}\,\lambda (\mathbf {r} '){\mathbf {r} ' \over {|\mathbf {r} '|}^{3}}d\ell.}
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