El campo Eléctrico es un campo vectorial en el cual en un punto P donde se encuentra una carga electrica sufre la interacción de una fuerza electrica. Esta interaccion es describe mediante lineas de campo. Las lineas de campo salen de las cargas electricas positivas, mientras que en las cargas electricas negativas entran las lineas de campo.
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| Lineas del campo eléctrico de dos cargas electricas. |
Filamento Infinito
Por Campo Eléctrico
$${\large \begin{align*} \vec{E} &= K\int \frac{dq}{{r'}^{2}} \hat{r'} \\ \vec{E} &= K\lambda \left[ \int_{-\infty}^{\infty} \left(\frac{ -x}{(h^2 + x^2)^{3/2}}\right)dx, \int_{-\infty}^{\infty} \left(\frac{h }{(h^2 + x^2)^{3/2}}\right) dx \right] \end{align*} }$$
Por ser una integral impropia entonces
$${\large \begin{align*} \vec{E} &= K\lambda \lim_{b \to \infty} \left[ \int_{-b}^{b} \left(\frac{ -x}{(h^2 + x^2)^{3/2}}\right)dx, \int_{-b}^{b} \left(\frac{h }{(h^2 + x^2)^{3/2}}\right) dx \right] \\ \vec{E} &= K\lambda \lim_{b \to \infty} \left[ \left. \frac{1}{\sqrt{h^2 + x^2}} \right|_{-b}^{b}, \left. \frac{x}{h\sqrt{h^2 + x^2}} \right|_{-b}^{b} \right] \\ \vec{E} &= K\lambda \lim_{b \to \infty} \left[ 0, \frac{2b}{h\sqrt{h^2 + b^2}} \right] \\ \vec{E} &= K\lambda \left( 0, \lim_{b \to \infty} \left[ \frac{2}{h\sqrt{1 + (h/b)^2}} \right] \right) \\ \vec{E} &= K\lambda \left( 0, \frac{2}{h} \right) \\ \vec{E} &= \frac{\lambda}{2\pi \epsilon_{0}} \left( 0, \frac{1}{h} \right) \\ ||\vec{E}|| &= \frac{\lambda}{2\pi \epsilon_{0} h} \end{align*} }$$
Filamento Finito
Por Campo Eléctrico
$${\large \begin{align*} \vec{E} &= K\int \frac{dq}{{r'}^{2}} \hat{r'} \\ \vec{E} &= K\lambda \left[ \int_{-L/2}^{L/2} \left(\frac{ -x}{(h^2 + x^2)^{3/2}}\right)dx, \int_{-L/2}^{L/2} \left(\frac{h }{(h^2 + x^2)^{3/2}}\right) dx \right] \\ \vec{E} &= K\lambda \left[ \left. \frac{1}{\sqrt{h^2 + x^2}} \right|_{-L/2}^{L/2}, \left. \frac{x}{h\sqrt{h^2 + x^2}} \right|_{-L/2}^{L/2} \right] \\ \vec{E} &= K\lambda \left[ 0 , \frac{L}{h\sqrt{h^2 + (L/2)^2}} \right] \\ \vec{E} &= K\lambda \left[ 0 , \frac{L}{h\sqrt{h^2 + (L/2)^2}} \right] \\ \vec{E} &= K\lambda \left[ 0 , \frac{2\sin{\theta}}{h} \right] \\ \vec{E} &= \frac{\lambda}{2\pi \epsilon_{0}} \left[ 0 , \frac{\sin{\theta}}{h} \right] \\ ||\vec{E}|| &= \frac{\lambda \sin{\theta}}{2\pi \epsilon_{0} h} \end{align*} }$$
Espira Cuadrada
Tramo I
Por Campo Eléctrico
$${\large \begin{align*} \vec{E_{I}} &= K\int \frac{dq}{{r'}^{2}} \hat{r'} \\ \vec{E_{I}} &= K\lambda \left[\int_{-a}^{a}\left( \frac{-a}{(a^2 + h^2 + y^2)^{3/2}} \right)dy, \int_{-a}^{a}\left( \frac{-y}{(a^2 + h^2 + y^2)^{3/2}} \right)dy, \int_{-a}^{a}\left( \frac{h}{(a^2 + h^2 + y^2)^{3/2}} \right)dy \right] \\ \vec{E_{I}} &= K\lambda \left[\left. \frac{-ay}{(a^2 + h^2)\sqrt{a^2 + h^2 + y^2}} \right|_{-a}^{a}, \left. \frac{1}{\sqrt{a^2 + h^2 + y^2}} \right|_{-a}^{a}, \left. \frac{hy}{(a^2 + h^2)\sqrt{a^2 + h^2 + y^2}} \right|_{-a}^{a} \right] \\ \vec{E_{I}} &= K\lambda \left[ \frac{-2a^2}{(a^2 + h^2)\sqrt{2a^2 + h^2}}, 0, \frac{2ah}{(a^2 + h^2)\sqrt{2a^2 + h^2}} \right] \\ \vec{E_{I}} &= \frac{2aK\lambda}{(a^2 + h^2)\sqrt{2a^2 + h^2}} \left[-a, 0, h \right] \end{align*} }$$
Tramo II
Por Campo Eléctrico
$${\large \begin{align*} \vec{E_{II}} &= K\int \frac{dq}{{r'}^{2}} \hat{r'} \\ \vec{E_{II}} &= K\lambda \left[\int_{-a}^{a}\left( \frac{-x}{(a^2 + h^2 + x^2)^{3/2}} \right)dx, \int_{-a}^{a}\left( \frac{-a}{(a^2 + h^2 + x^2)^{3/2}} \right)dx, \int_{-a}^{a}\left( \frac{h}{(a^2 + h^2 + x^2)^{3/2}} \right)dx \right] \\ \vec{E_{II}} &= K\lambda \left[\left. \frac{1}{\sqrt{a^2 + h^2 + x^2}} \right|_{-a}^{a}, \left. \frac{-ax}{(a^2 + h^2)\sqrt{a^2 + h^2 + x^2}} \right|_{-a}^{a}, \left. \frac{hx}{(a^2 + h^2)\sqrt{a^2 + h^2 + x^2}} \right|_{-a}^{a} \right] \\ \vec{E_{II}} &= K\lambda \left[ 0, \frac{-2a^2}{(a^2 + h^2)\sqrt{2a^2 + h^2}}, \frac{2ah}{(a^2 + h^2)\sqrt{2a^2 + h^2}} \right] \\ \vec{E_{II}} &= \frac{2aK\lambda}{(a^2 + h^2)\sqrt{2a^2 + h^2}} \left[0, -a, h \right] \end{align*} }$$
Tramo III
Por Campo Eléctrico
$${\large \begin{align*} \vec{E_{III}} &= K\int \frac{dq}{{r'}^{2}} \hat{r'} \\ \vec{E_{III}} &= K\lambda \left[\int_{-a}^{a}\left( \frac{a}{(a^2 + h^2 + y^2)^{3/2}} \right)dy, \int_{-a}^{a}\left( \frac{-y}{(a^2 + h^2 + y^2)^{3/2}} \right)dy, \int_{-a}^{a}\left( \frac{h}{(a^2 + h^2 + y^2)^{3/2}} \right)dy \right] \\ \vec{E_{III}} &= K\lambda \left[\left. \frac{ay}{(a^2 + h^2)\sqrt{a^2 + h^2 + y^2}} \right|_{-a}^{a}, \left. \frac{1}{\sqrt{a^2 + h^2 + y^2}} \right|_{-a}^{a}, \left. \frac{hy}{(a^2 + h^2)\sqrt{a^2 + h^2 + y^2}} \right|_{-a}^{a} \right] \\ \vec{E_{III}} &= K\lambda \left[ \frac{2a^2}{(a^2 + h^2)\sqrt{2a^2 + h^2}}, 0, \frac{2ah}{(a^2 + h^2)\sqrt{2a^2 + h^2}} \right] \\ \vec{E_{III}} &= \frac{2aK\lambda}{(a^2 + h^2)\sqrt{2a^2 + h^2}} \left[a, 0, h \right] \end{align*} }$$
Tramo IV
Por Campo Eléctrico
$${\large \begin{align*} \vec{E_{IV}} &= K\int \frac{dq}{{r'}^{2}} \hat{r'} \\ \vec{E_{IV}} &= K\lambda \left[\int_{-a}^{a}\left( \frac{-x}{(a^2 + h^2 + x^2)^{3/2}} \right)dx, \int_{-a}^{a}\left( \frac{a}{(a^2 + h^2 + x^2)^{3/2}} \right)dx, \int_{-a}^{a}\left( \frac{h}{(a^2 + h^2 + x^2)^{3/2}} \right)dx \right] \\ \vec{E_{IV}} &= K\lambda \left[\left. \frac{1}{\sqrt{a^2 + h^2 + x^2}} \right|_{-a}^{a}, \left. \frac{ax}{(a^2 + h^2)\sqrt{a^2 + h^2 + x^2}} \right|_{-a}^{a}, \left. \frac{hx}{(a^2 + h^2)\sqrt{a^2 + h^2 + x^2}} \right|_{-a}^{a} \right] \\ \vec{E_{IV}} &= K\lambda \left[ 0, \frac{2a^2}{(a^2 + h^2)\sqrt{2a^2 + h^2}}, \frac{2ah}{(a^2 + h^2)\sqrt{2a^2 + h^2}} \right] \\ \vec{E_{IV}} &= \frac{2aK\lambda}{(a^2 + h^2)\sqrt{2a^2 + h^2}} \left[0, a, h \right] \end{align*} }$$
Tramo Total
El campo eléctrico total es la suma de los campos eléctricos de los distintos tramos.
$${\large \begin{align*} \vec{E} &= \vec{E_{I}} + \vec{E_{II}} + \vec{E_{III}} + \vec{E_{IV}} \\ \vec{E} &= \frac{8aK\lambda}{(a^2 + h^2)\sqrt{2a^2 + h^2}} \left[0, 0, h \right] \\ \vec{E} &= \frac{2aK\lambda}{\pi \epsilon_{0} (a^2 + h^2)\sqrt{2a^2 + h^2}} \left[0, 0, h \right] \\ ||\vec{E}|| &= \frac{2ah\lambda}{\pi \epsilon_{0} (a^2 + h^2)\sqrt{2a^2 + h^2}} \end{align*} }$$
Plano Infinito
Por Campo Eléctrico
$${\large \begin{align*} \vec{E} &= K\int \frac{dq}{{r'}^{2}} \hat{r'} \\ \vec{E} &= K\lambda \left[ \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \left(\frac{ -x}{(h^2 + x^2 + y^2)^{3/2}}\right)dydx, \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \left(\frac{-y}{(h^2 + x^2 + y^2)^{3/2}}\right) dydx, \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \left(\frac{h}{(h^2 + x^2 + y^2)^{3/2}}\right) dydx \right] \end{align*}}$$
Por ser una integral impropia entonces
$$ {\large \begin{align*} \vec{E} &= K\sigma \lim_{b \to \infty}\left[ \int_{-b}^{b} \int_{-b}^{b} \left(\frac{ -x}{(h^2 + x^2 + y^2)^{3/2}}\right)dydx, \int_{-b}^{b} \int_{-b}^{b} \left(\frac{-y}{(h^2 + x^2 + y^2)^{3/2}}\right) dydx, \int_{-b}^{b} \int_{-b}^{b} \left(\frac{h}{(h^2 + x^2 + y^2)^{3/2}}\right) dydx \right] \\ \vec{E} &= K\sigma \lim_{b \to \infty}\left[ \int_{-b}^{b} \left(\lim_{b \to \infty} \left[ \frac{-2xb}{(h^2 + x^2)\sqrt{h^2 + x^2 + b^2}} \right] \right)dx, 0, \int_{-b}^{b} \left( \lim_{b \to \infty} \left[\frac{2hb}{(h^2 + x^2)\sqrt{h^2 + x^2 + b^2}}\right] \right)dx \right] \\ \vec{E} &= K\sigma \lim_{b \to \infty}\left[ \int_{-b}^{b} \left(\lim_{b \to \infty} \left[ \frac{-2x}{(h^2 + x^2)\sqrt{(h/b)^2 + (x/b)^2 + 1}} \right] \right)dx, 0, \int_{-b}^{b} \left( \lim_{b \to \infty} \left[\frac{2h}{(h^2 + x^2)\sqrt{(h/b)^2 + (x/b)^2 + 1}}\right] \right)dx \right] \\ \vec{E} &= K\sigma \lim_{b \to \infty}\left[ \int_{-b}^{b} \left( \frac{-2x}{h^2 + x^2} \right)dx, 0, \int_{-b}^{b} \left(\frac{2h}{h^2 + x^2} \right)dx \right] \\ \vec{E} &= K\sigma \left( 0, 0, 4\lim_{b \to \infty} \left[ \arctan{\left(\frac{b}{h} \right)} \right] \right) \\ \vec{E} &= K\sigma \left( 0, 0, 2\pi \right) \\ \vec{E} &= \frac{\sigma}{2\epsilon_{0}} \left( 0, 0, 1 \right) \\ ||\vec{E}|| &= \frac{\sigma}{2\epsilon_{0}} \end{align*} }$$
Espira Circular de radio R
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Por Campo Eléctrico
$${\large \begin{align*} \vec{E} &= K\int \frac{dq}{{r'}^{2}} \hat{r'} \end{align*}}$$
Pasando a Coordenadas Polares
$${\large \begin{align*} \vec{E} &= K\lambda R \left[ \int_{0}^{2\pi} \left( \frac{-R\cos{\varphi}}{(R^2 + h^2)^{3/2}} \right)d\varphi, \int_{0}^{2\pi} \left(\frac{-R\sin{\varphi}}{(R^2 + h^2)^{3/2}} \right)d\varphi, \int_{0}^{2\pi} \left(\frac{h}{(R^2 + h^2)^{3/2}}\right) d\varphi \right] \\ \vec{E} &= K\lambda R \left[ \left. \frac{-R\sin{\varphi}}{(R^2 + h^2)^{3/2}} \right|_{0}^{2\pi}, \left. \frac{R\cos{\varphi}}{(R^2 + h^2)^{3/2}} \right|_{0}^{2\pi}, \left. \frac{h\varphi }{(R^2 + h^2)^{3/2}}\right|_{0}^{2\pi} \right] \\ \vec{E} &= K\lambda R \left[ 0, 0, \frac{2\pi h }{(R^2 + h^2)^{3/2}} \right] \\ \vec{E} &= \frac{2\pi K\lambda Rh}{(R^2 + h^2)^{3/2}} \left( 0, 0, 1 \right) \\ \vec{E} &= \frac{\lambda Rh}{2\epsilon_{0}(R^2 + h^2)^{3/2}} \left( 0, 0, 1 \right) \\ ||\vec{E}|| &= \frac{\lambda Rh}{2\epsilon_{0}(R^2 + h^2)^{3/2}} \end{align*} }$$
Disco Circular de radio R
Por Campo Eléctrico
$${\large \begin{align*} \vec{E} &= K\int \frac{dq}{{r'}^{2}} \hat{r'} \end{align*}}$$
Pasando a coordenadas polares
$${\large \begin{align*} \vec{E} &= K\sigma \left[ \int_{0}^{2\pi} \int_{0}^{R} \left( \frac{-r^2\cos{\varphi}}{(r^2 + h^2)^{3/2}} \right)dr d\varphi, \int_{0}^{2\pi} \int_{0}^{R} \left(\frac{-r^2\sin{\varphi}}{(r^2 + h^2)^{3/2}} \right)dr d\varphi, \int_{0}^{2\pi} \int_{0}^{R} \left(\frac{hr}{(r^2 + h^2)^{3/2}}\right) dr d\varphi \right] \end{align*}}$$
Aplicando el teorema de Fubini
$${\large \begin{align*} \vec{E} &= K\sigma \left[ \int_{0}^{R} \left( \frac{-r^2}{(r^2 + h^2)^{3/2}} \right)dr \int_{0}^{2\pi} \left( \cos{\varphi} \right)d\varphi, \int_{0}^{R} \left( \frac{-r^2}{(r^2 + h^2)^{3/2}} \right)dr \int_{0}^{2\pi} \left( \sin{\varphi} \right)d\varphi, \int_{0}^{R} \left(\frac{hr}{(r^2 + h^2)^{3/2}}\right) dr \int_{0}^{2\pi} d\varphi \right] \\ \vec{E} &= K\sigma \left[ \int_{0}^{R} \left( \frac{-r^2}{(r^2 + h^2)^{3/2}} \right)dr \left( \left. \sin{\varphi} \right|_{0}^{2\pi} \right), \int_{0}^{R} \left( \frac{-r^2}{(r^2 + h^2)^{3/2}} \right)dr \left( \left. -\cos{\varphi} \right|_{0}^{2\pi} \right), \int_{0}^{R} \left(\frac{hr}{(r^2 + h^2)^{3/2}}\right) dr \left( \left. \varphi \right|_{0}^{2\pi} \right) \right] \\ \vec{E} &= K\sigma \left[ 0, 0, \int_{0}^{R} \left(\frac{2\pi hr}{(r^2 + h^2)^{3/2}}\right) dr \right] \\ \vec{E} &= K\sigma \left[ 0, 0, \left( \left. -\frac{2\pi h}{\sqrt{r^2 + h^2}} \right|_{0}^{R} \right) \right] \\ \vec{E} &= K\sigma \left[ 0, 0, 2\pi\left( 1 - \frac{h}{\sqrt{R^2 + h^2}} \right) \right] \\ \vec{E} &= 2\pi K \sigma \left( 1 - \frac{h}{\sqrt{R^2 + h^2}} \right) \left( 0, 0, 1 \right) \\ ||\vec{E}|| &= \frac{\sigma}{2 \epsilon_{0}} \left( 1 - \frac{h}{\sqrt{R^2 + h^2}} \right) \end{align*}}$$
Planos Paralelos Delgados Infinitos de opuestas Densidad Superficial
Puntos Interiores
Por Campo Eléctrico
$${\large \begin{align*} \vec{E} &= \vec{E_{+\sigma}} + \vec{E_{-\sigma}} \\ \vec{E} &= \frac{\sigma}{2\epsilon_{0}}(0,0,-1) + \frac{\sigma}{2\epsilon_{0}}(0,0,-1) \\ \vec{E} &= -\frac{\sigma}{\epsilon_{0}}(0,0,1)\\ ||\vec{E}|| &= \frac{\sigma}{\epsilon_{0}} \end{align*} }$$
Puntos Exteriores
Por Campo Eléctrico
$${\large \begin{align*} \vec{E} &= \vec{E_{+\sigma}} + \vec{E_{-\sigma}} \\ \vec{E} &= \frac{\sigma}{2\epsilon_{0}}(0,0,\pm 1) + \frac{\sigma}{2\epsilon_{0}}(0,0,\mp1) \\ \vec{E} &= (0,0,0) \\ ||\vec{E}|| &= 0 \end{align*}}$$
Planos Paralelos Delgados Infinitos de igual Densidad Superficial
Puntos Interiores
Por Campo Eléctrico
$${\large \begin{align*} \vec{E} &= \vec{E_{\uparrow \sigma}} + \vec{E_{\downarrow \sigma}} \\ \vec{E} &= \frac{\sigma}{2\epsilon_{0}}(0,0,-1) + \frac{\sigma}{2\epsilon_{0}}(0,0,1) \\ \vec{E} &= (0,0,0) \\ ||\vec{E}|| &= 0 \end{align*}}$$
Puntos Exteriores
Por Campo Eléctrico
$${\large \begin{align*} \vec{E} &= \vec{E_{\uparrow \sigma}} + \vec{E_{\downarrow \sigma}} \\ \vec{E} &= \frac{\sigma}{2\epsilon_{0}}(0,0,\pm 1) + \frac{\sigma}{2\epsilon_{0}}(0,0,\pm 1) \\ \vec{E} &= \frac{\sigma}{\epsilon_{0}}(0,0,\pm 1) \\ ||\vec{E}|| &= \frac{\sigma}{\epsilon_{0}} \end{align*}}$$
Casquete Esférico
Por Campo Eléctrico
$${\large \begin{align*} \vec{E} &= K\int \frac{dq}{{r'}^{2}} \hat{r'} \end{align*}}$$
Pasando a Coordenadas Esféricas
$${\large \begin{align*} \vec{E} &= K\sigma \int_{0}^{2\pi} \int_{0}^{\alpha} \left( \frac{ R^2 \sin{\theta} (-\cos{\varphi} \sin{\theta}, -\cos{\theta}, -\sin{\varphi} \sin{\theta})}{R^2} \right) d\theta d\varphi \\ \vec{E} &= - K\sigma \left[ \int_{0}^{2\pi} \int_{0}^{\alpha} \left( \cos{\varphi} \sin^2{\theta} \right) d\theta d\varphi, \int_{0}^{2\pi} \int_{0}^{\alpha} \left( \sin{\theta} \cos{\theta} \right) d\theta d\varphi, \int_{0}^{2\pi} \int_{0}^{\alpha} \left( \sin{\varphi} \sin^2{\theta} \right) d\theta d\varphi \right] \\ \vec{E} &= - K\sigma \left[ \int_{0}^{2\pi} \left( \cos{\varphi} \left( \left. \frac{\theta - \cos{\theta}\sin{\theta}}{2} \right|_{0}^{\alpha} \right) \right) d\varphi, \int_{0}^{2\pi} \left( \left( \left. \frac{\sin^2{\theta}}{2} \right|_{0}^{\alpha} \right) \right) d\varphi, \int_{0}^{2\pi} \left( \sin{\varphi} \left( \left. \frac{\theta - \cos{\theta}\sin{\theta}}{2} \right|_{0}^{\alpha} \right) \right) d\varphi \right] \\ \vec{E} &= - K\sigma \left[ \int_{0}^{2\pi} \left( \cos{\varphi} \left( \frac{\alpha - \cos{\alpha}\sin{\alpha}}{2}\right) \right) d\varphi, \int_{0}^{2\pi} \left( \frac{\sin^2{\alpha}}{2} \right) d\varphi, \int_{0}^{2\pi} \left( \sin{\varphi} \left( \frac{\alpha - \cos{\alpha}\sin{\alpha}}{2} \right) \right) d\varphi \right] \\ \vec{E} &= - K\sigma \left[ \left( \left. \sin{\varphi} \left( \frac{\alpha - \cos{\alpha}\sin{\alpha}}{2}\right) \right|_{0}^{2\pi} \right) , \left( \left. \varphi \frac{\sin^2{\alpha}}{2} \right|_{0}^{2\pi} \right) , \left( \left. - \cos{\varphi} \left( \frac{\alpha - \cos{\alpha}\sin{\alpha}}{2}\right) \right|_{0}^{2\pi} \right) \right] \\ \vec{E} &= - K\sigma \left( 0, \pi \sin^2{\alpha}, 0 \right) \\ \vec{E} &= - \pi K\sigma \sin^2{\alpha} \left( 0, 1, 0 \right) \\ \vec{E} &= - \frac{\sigma \sin^2{\alpha}}{4 \epsilon_{0}} \left( 0, 1, 0 \right) \\ \vec{E} &= - \frac{\sigma r^2}{4 \epsilon_{0} R^2} \left( 0, 1, 0 \right) \\ || \vec{E} ||&= \frac{\sigma r^2}{4 \epsilon_{0} R^2} \end{align*}}$$
Esfera Hueca de Radio R
Puntos Interiores, $r<R$
Como la carga se encuentra en la superficie esférica, es decir, $r<R$. Entonces el campo eléctrico es igual a cero.
$${\large \begin{align*} \vec{E} &= K\int \frac{dq}{{r'}^{2}} \hat{r'} \\ \vec{E} &= K\int \frac{(0)}{{r'}^{2}} \hat{r'} \\ \vec{E} &= (0,0,0) \end{align*}}$$
Puntos Exteriores, $r>R$
Por Campo Eléctrico
$${\large \begin{align*} \vec{E} &= K\int \frac{dq}{{r'}^{2}} \hat{r'} \\ \vec{E} &= K\sigma \int \frac{dA}{{r'}^{2}} \hat{r'} \end{align*} }$$
Pasando a coordenadas esféricas
$${\large \begin{align*} \vec{E} &= K\sigma \int_{0}^{2\pi} \int_{0}^{\pi} \left(\frac{R^2 \sin{\theta} (-R\cos{\varphi}\sin{\theta}, -R\sin{\varphi}\sin{\theta}, r - R\cos{\theta}) }{(r^2 + R^2 - 2rR\cos{\theta})^{3/2}} \right) d\theta d\varphi \end{align*} }$$
Eje X
$${\large \begin{align*} E_{x} &= -K\sigma \int_{0}^{2\pi} \int_{0}^{\pi} \left(\frac{R^3 \sin^2{\theta}\cos{\varphi}}{(r^2 + R^2 - 2rR\cos{\theta})^{3/2}} \right) d\theta d\varphi \\ E_{x} &= -K\sigma \int_{0}^{\pi} \left(\frac{R^3 \sin^2{\theta}}{(r^2 + R^2 - 2rR\cos{\theta})^{3/2}} \right) d\theta \int_{0}^{2\pi} \left( \cos{\varphi} \right) d\varphi \\ E_{x} &= -K\sigma \int_{0}^{\pi} \left(\frac{R^3 \sin^2{\theta}}{(r^2 + R^2 - 2rR\cos{\theta})^{3/2}} \right) d\theta \left(\left. \sin{\varphi} \right)|_{0}^{2\pi} \right) \\ E_{x} &= -K\sigma \int_{0}^{\pi} \left(\frac{R^3 \sin^2{\theta}}{(r^2 + R^2 - 2rR\cos{\theta})^{3/2}} \right) d\theta \left( 0 \right) \\ E_{x} &= 0 \end{align*}}$$
Eje Y
$${\large \begin{align*} E_{y} &= -K\sigma \int_{0}^{2\pi} \int_{0}^{\pi} \left(\frac{R^3 \sin^2{\theta}\sin{\varphi}}{(r^2 + R^2 - 2rR\cos{\theta})^{3/2}} \right) d\theta d\varphi \\ E_{y} &= -K\sigma \int_{0}^{\pi} \left(\frac{R^3 \sin^2{\theta}}{(r^2 + R^2 - 2rR\cos{\theta})^{3/2}} \right) d\theta \int_{0}^{2\pi} \left( \sin{\varphi} \right) d\varphi \\ E_{y} &= -K\sigma \int_{0}^{\pi} \left(\frac{R^3 \sin^2{\theta}}{(r^2 + R^2 - 2rR\cos{\theta})^{3/2}} \right) d\theta \left(\left. -\cos{\varphi} \right)|_{0}^{2\pi} \right) \\ E_{y} &= -K\sigma \int_{0}^{\pi} \left(\frac{R^3 \sin^2{\theta}}{(r^2 + R^2 - 2rR\cos{\theta})^{3/2}} \right) d\theta \left( 0 \right) \\ E_{y} &= 0 \end{align*} }$$
Eje Z
$${\large \begin{align*} E_{z} &= K\sigma \int_{0}^{2\pi} \int_{0}^{\pi} \left(\frac{R^2 \sin{\theta}(r - R\cos{\theta})}{(r^2 + R^2 - 2rR\cos{\theta})^{3/2}} \right) d\theta d\varphi \\ E_{z} &= K\sigma R^2 \int_{0}^{2\pi} \int_{0}^{\pi} \left(\frac{\sin{\theta}(r - R\cos{\theta})}{(r^2 + R^2 - 2rR\cos{\theta})^{3/2}} \right) d\theta d\varphi \\ E_{z} &= K\sigma R^2 \int_{0}^{2\pi} \left( \left. \frac{R-r\cos{\theta}}{r^2 \sqrt{r^2 + R^2 - 2rR\cos{\theta}}} \right|_{0}^{\pi} \right) d\varphi \\ E_{z} &= K\sigma R^2 \int_{0}^{2\pi} \left( \frac{2}{r^2} \right) d\varphi \\ E_{z} &= \frac{4\pi K \sigma R^2}{r^2} \\ E_{z} &= \frac{\sigma R^2}{\epsilon_{0} r^2} \end{align*} }$$
El Vector Campo Eléctrico es
$${\large \begin{align*} \vec{E} &= \frac{\sigma R^2}{\epsilon_{0} r^2} \left(0, 0, 1 \right) \\ ||\vec{E}|| &= \frac{\sigma R^2}{\epsilon_{0} r^2} \end{align*}} $$
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