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Question Number 33912 by ajfour last updated on 27/Apr/18

Commented by ajfour last updated on 27/Apr/18

$$Findtheelectricfieldatthecentreof$$$$apartofthesurfaceofaspherical$$$$shellhavingchargedensity+\sigma .$$

Commented by ajfour last updated on 27/Apr/18

Commented by ajfour last updated on 27/Apr/18

$$dE=\frac{\sigma \left(R\sin \varphi d\varphi \right)\left(Rd\theta \right)}{4\pi {ϵ}_0{R}^2}=\frac{\sigma \sin \varphi d\theta d\varphi }{4\pi {ϵ}_0}$$$$E_{centre}=E_c=\int \int dE\sin \varphi \cos \theta$$$$E_c={\int }_{-\theta /2}^{\theta /2}{\int }_0^{\pi }\frac{\sigma d\theta d\varphi }{4\pi {ϵ}_0}\cos \theta \sin {}^2\varphi$$$$=\left[\frac{\mathit{\sigma }}{4\mathit{\pi }{\mathit{ϵ}}_0}\left(2\sin \frac{\theta }{2}\right)\right]\left(\frac{(\varphi -\frac{1}{2}\cos 2\varphi }{2}\right){\mid }_0^{\pi }$$$$E_c=\frac{\sigma \sin \frac{\theta }{2}}{4{ϵ}_0}.$$

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