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

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



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*}


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

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.


Potencial Elétrico – Parte 1

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