Problemas Resolvidos: Fundamentos do Eletromagnetismo

Esta é uma secção que pode ser útil para os estudantes Física, Matemática, Engenharia que estão cursando a disciplina de FUNDAMENTOS DE ELETROMAGNETISMO.

Nesta disciplina, tenho utilizado como bibliografia básica  o livro volume 3 e 4 dos autores Halliday, Hesnick e Walker da 10ª edição.  

Ementa da disciplina

  • Cargas Elétricas;
  • Campos Elétricos;
  • Lei de Gauss;
  • Potencial Elétrico;
  • Capacitância;
  • Corrente e Resistência;
  • Circuitos;
  • Campos Magnéticos;
  • Campos Magnéticos produzidos por correntes;
  • Indução e Indutância;
  • Oscilações Eletromagnéticas e corrente alternada;
  • Equações de Maxwell; Magnetismo da Matéria;
  • Ondas Eletromagnéticas;
  • Óptica Geométrica

Provas aplicadas em semestre anteriores:

1ª prova , 2ª prova ,   3ª prova ,   4ª prova

Notas de aulas

Capítulo 22: Campo Elétrico Gerado por uma Linha de Carga de Comprimento L

Problemas resolvidos

Capítulo 21: 21,7 21.10 , 21.13 , 21.42

Capítulo 22: 22.19 , 22.21 , 22.24 , 22.25 , 22.31 ,

Capítulo 23: 22.53, 23.24, 23.29

Capítulo 24: 24.11, 24.31, 24.47, 24.66

Capítulo 25: 25.48, 25.49

Capítulo 26: 26.34, 26.35, 26.2,

Capítulo 27: 27.10, 27.18, 27.29, 27.30, 27.33, 27.57, 27.97,

Potencial Elétrico

Potencial Elétrico Gerado por uma Linha de Carga

Consideremos na Figura abaixo uma barra fina, isolante e de comprimento L, uniformemente carregada eletricamente.
Potencial elétrico gerado por uma linha de carga.
Um elemento do potencial elétrico dV gerado no ponto P é dado por

(1)   \begin{equation*}      dV=\frac{1}{4 \pi \varepsilon _0}\frac{dQ}{r}=\frac{1}{4 \pi \varepsilon_0}\frac{dQ}{\sqrt{x^2+z^2}}    \end{equation*}

Considerando que a carga esteja uniformemente distribuída ao longo da barra, a densidade linear de carga será constante e dada por:

(2)   \begin{equation*}      \lambda=\frac{dQ}{dx}.    \end{equation*}

Substituindo a Equação 8 na Equação 1, teremos

(3)   \begin{equation*}   dV=\frac{1}{4 \pi \varepsilon_0}\frac{\lambda dx}{\sqrt{x^2+z^2}}. \end{equation*}

Observando a Figura podemos escrever

    \[x=z\,tg\theta\,\,\,\ \Rightarrow \,\,\, dx=z\,sec^2 \theta d \theta\]

    \[cos\theta=\frac{z}{\sqrt{x^2+z^2}}\]

    \[\sqrt{x^2+z^2}=z\,sec\theta\]

Sendo assim, da Equação 8 podemos obter

    \[dV=\frac{\lambda}{4\pi \varepsilon _0} \frac{z\,sec^2 \theta \,d\theta}{z\,sec\theta}\]

(4)   \begin{equation*}   V=\frac{\lambda}{4\pi \varepsilon _0}\int_{\theta_1}^{\theta_2}sec\theta \,d\theta \end{equation*}

Uma das técnicas para solucionar a Equação 4 consiste em multiplicar e dividir o integrando por tg\theta+sec\theta. Assim 4 assumirá a forma

(5)   \begin{equation*}   V=\frac{\lambda}{4\pi \varepsilon _0}\int_{\theta_1}^{\theta_2} \frac{(tg\theta +sec\theta)sec\theta}{tg\theta+sec\theta}   \,d\theta \end{equation*}

Fazendo

    \[tg\theta+sec\theta=u \,\,\,\Rightarrow \,\,\, du=sec\theta(tg\theta+sec\theta)d\theta\]

e substituindo em 5 teremos

(6)   \begin{equation*}   V=\frac{\lambda}{4\pi \varepsilon _0}\int_{\theta_1}^{\theta_2} \frac{(tg\theta +sec\theta)sec\theta}{tg\theta+sec\theta}   \,d\theta= \frac{\lambda}{4\pi \varepsilon _0}\int_{u_1}^{u_2}\frac{du}{u} =  \frac{\lambda}{4\pi \varepsilon _0}\,ln\,u \Big|_{u_1}^{u_2} . \end{equation*}

Mas u=tg\theta+sec\theta, então

    \[V=\frac{\lambda}{4\pi \varepsilon _0} \int_{\theta_1}^{\theta_2}=\frac{\lambda}{4\pi \varepsilon _0}\,ln\,(tg\theta+sec\theta) \Big|_{\theta_1}^{\theta_2},\]

ou ainda

    \[V=\frac{\lambda}{4\pi \varepsilon _0}\,ln\,\left(\frac{x}{z}+ \frac{\sqrt{x^2+z^2}}{z} \right) \Big|_{x=0}^{x=L}\]

E finalmente, teremos a Equação

(7)   \begin{equation*}   V=\frac{\lambda}{4\pi \varepsilon _0}\,ln\,\left(\frac{L+\sqrt{L^2+z^2}}{z} \right) \end{equation*}

que determina o potencial elétrico gerado pela distribuição linear de carga no ponto P conforme ilustrado pela Figura.

Vídeos





Potencial Elétrico – Parte 1

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  55. Cottage out enabled was entered greatly prevent message. No procured unlocked an likewise. Dear but what she been over gay felt body. Six principles advantages and use entreaties decisively. Eat met has dwelling unpacked see whatever followed. Court in of leave again as am. Greater sixteen to forming colonel no on be. So an advice hardly barton. He be turned sudden engage manner spirit.

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