Halliday Physics Chapter 23 · 평생 소원이 누룽지

1. Electric Flux

Electric Flux($\Phi_{\text{E}}$): 전기장선 총 개수에 비례
면적 벡터
- 크기 $A$
- 방향: 면에 수직($\hat{n}$)
- 폐곡면에서 나가는 방향이 $(+)$
Uniform Electric Field & Flat Surface $\Phi_{\text{E}} = EA \cos \theta = \mathbf{E} \cdot \mathbf{A}$
Non-Uniform Electric Field & General Surface $\Phi_{\text{E}} = \iint E \cos \theta\ \mathrm dA = \iint E_{\perp} \cdot \mathrm dA$
Flux Through a Closed Surface $\Phi_{\text{E}} = \oiint E_{\perp} \cdot \mathrm dA$

\Phi_{\text{E}} = \oiint E_{\perp}\ \mathrm dA = \dfrac{q_{\text{enc}}}{\varepsilon_0} \quad \text{(in closed Gaussian surface)}

증명

Gaussian surface를 반지름 $r$인 구면으로 잡으면

\displaystyle \oiint E_{\perp}\ \mathrm dA = \oiint E\ \mathrm dA = E \oiint \mathrm dA = E \cdot 4 \pi r^2 = \dfrac{q_{\text{enc}}}{\varepsilon_0}

따라서 $E = \dfrac{1}{4 \pi \varepsilon_0} \dfrac{q_{\text{enc}}}{r^2}$

\Phi_{\text{E}} = \oiint E \cdot \mathrm dA = E \oiint \mathrm dA = EA = \dfrac{q_{\text{enc}}}{\varepsilon_0} = \dfrac{\sigma A}{\varepsilon_0} \Rightarrow E = \dfrac{\sigma}{\varepsilon_0}

\oiint E \cdot \mathrm dA = E \cdot (2 \pi rL) = \dfrac{q_{\text{enc}}}{\varepsilon_0} = \dfrac{\lambda L}{\varepsilon_0} \Rightarrow E = \dfrac{\lambda}{2 \pi r \varepsilon_0}

\oiint E \cdot \mathrm dA = E \cdot 2A = \dfrac{q_{\text{enc}}}{\varepsilon_0} = \dfrac{\sigma A}{\varepsilon_0} \Rightarrow E = \dfrac{\sigma}{2 \varepsilon_0}

안쪽 껍질의 반지름이 $r_1$, 바깥쪽 껍질의 반지름이 $r_2$일 때

\[E = 0\]

\begin{aligned} \oiint E \cdot \mathrm dA = E \cdot 4 \pi r^2 = \dfrac{q_{\text{enc}}}{\varepsilon_0} &= \dfrac{1}{\varepsilon_0} \left( \int_{r_1}^{r} \rho \cdot 4 \pi r^2\ \mathrm dr \right) \\\\ &= \dfrac{4 \pi \rho}{3 \varepsilon_0} \bigg[ r^3 \bigg]_{r_1}^{r} = \dfrac{4 \pi \rho}{3 \varepsilon_0} (r^3 - {r_1}^3) \\\\ E\ &= \dfrac{\rho}{3 \varepsilon_0 r^2} (r^3 - {r_1}^3) \end{aligned}

\begin{aligned} \oiint E \cdot \mathrm dA = E \cdot 4 \pi r^2 = \dfrac{q_{\text{enc}}}{\varepsilon_0} &= \dfrac{1}{\varepsilon_0} \cdot \rho \left( \dfrac{4}{3} \pi {r_2}^3 - \dfrac{4}{3} \pi {r_1}^3 \right) \\ &= \dfrac{4 \pi \rho}{3 \varepsilon_0} ({r_2}^3 - {r_1}^3) \\\\ E\ &= \dfrac{\rho}{3 \varepsilon_0 r^2} ({r_2}^3 - {r_1}^3) \end{aligned}