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電荷の種別ごとの電界、電界ベクトル算出式

| 電荷種別 | 電界(スカラ) | 電界(ベクトル) |
| ------------ | :--------------------------------------------------------------------------------------------------------------------------------------------------: | :----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------: |
| 点電荷 | $ E(r) = \frac{Q}{4 \pi \varepsilon_0 r^{2}} $ | $ \bm{E}(\bm{r}) = \frac{Q}{4 \pi \varepsilon_0 \lvert\bm{r}\rvert^{3}} \bm{r} $ |
| 無限長線電荷 | $ E(r) = \frac{\lambda}{2 \pi \varepsilon_0 r} $ | $ \bm{E}(\bm{r}) = \frac{\lambda}{2 \pi \varepsilon_0\lvert\bm{r}\rvert^{2}} \bm{r} $ |
| 無限面電荷 | $ E(r) = \frac{\sigma}{2 \varepsilon_0} $ | $ \bm{E}(\bm{r}) = \frac{\sigma}{2 \varepsilon_0 \lvert\bm{r}\rvert} \bm{r}$ |
| 球表面電荷 | $ \begin{cases} E(r) = \frac{\sigma a^{2}}{\varepsilon_0 r^{2}} & (a \leq r) \\ E(r) = 0 & (0 \leq r < a) \end{cases} $ | $ \begin{cases} \bm{E}(\bm{r}) = \frac{\sigma a^{2}}{\varepsilon_0 \lvert\bm{r}\rvert^{3}} \bm{r} & (a \leq \lvert\bm{r}\rvert) \\ \bm{E}(\bm{r}) = \bm{0} & (0 \leq \lvert\bm{r}\rvert < a) \end{cases} $ |
| 球体積電荷 | $ \begin{cases} E(r) = \frac{\rho a^{3}}{3 \varepsilon_0 r^{2}} & (a \leq r) \\ E(r) = \frac{\rho r}{3 \varepsilon_0} & (0 \leq r < a) \end{cases} $ | $ \begin{cases} \bm{E}(\bm{r}) = \frac{\rho a^{3}}{3 \varepsilon_0 \lvert\bm{r}\rvert^{3}} \bm{r} & (a \leq \lvert\bm{r}\rvert) \\ \bm{E}(\bm{r}) = \frac{\rho}{3 \varepsilon_0} \bm{r} & (0 \leq \lvert\bm{r}\rvert < a) \end{cases} $ 　 |
| 円筒表面電荷 | $ \begin{cases} E(r) = \frac{\sigma a}{\varepsilon_0 r} & (a \leq r) \\ E(r) = 0 & (0 \leq r < a) \end{cases} $ | $ \begin{cases} \bm{E}(\bm{r}) = \frac{ \sigma a}{\varepsilon_0 \lvert\bm{r}\rvert^{2}} \bm{r} & (a \leq \lvert\bm{r}\rvert) \\ \bm{E}(\bm{r}) = \bm{0} & (0 \leq \lvert\bm{r}\rvert < a) \end{cases} $ |
| 円筒体積電荷 | $ \begin{cases} E(r) = \frac{\rho a^{2}}{2 \varepsilon_0 r} & (a \leq r) \\ E(r) = \frac{\rho r}{2 \varepsilon_0} & (0 \leq r < a) \end{cases} $ | $ \begin{cases} \bm{E}(\bm{r}) = \frac{\rho a^{2}}{2 \varepsilon_0 \lvert\bm{r}\rvert^{2}} \bm{r} & (a \leq \lvert\bm{r}\rvert) \\ \bm{E}(\bm{r}) = \frac{\rho}{2 \varepsilon_0} \bm{r} & (0 \leq \lvert\bm{r}\rvert < a) \end{cases} $ |

| 値 | 意味 | 単位 |
| :---------------: | :------------------: | :----------------: |
| $ \varepsilon_0 $ | 真空中の誘電率 | $ \mathrm{F/m} $ |
| $ q $ | 電荷量 | $ \mathrm{C} $ |
| $ \lambda $ | 線電荷密度 | $ \mathrm{C/m} $ |
| $ \sigma $ | 面電荷密度 | $ \mathrm{C/m^2} $ |
| $ \rho $ | 体積電荷密度 | $ \mathrm{C/m^3} $ |
| $ r $ | 電荷との距離 | $ \mathrm{m} $ |
| $ \bm{r} $ | 電荷との距離ベクトル | $ (\mathrm{m}) $ |
| $ a $ | 半径 | $ \mathrm{m} $ |
| 電荷種別 | 電界(スカラ) | 電界(ベクトル) |
| ------------ | :------------------------------------------------------------------------------------------------------------------------------------------------: | :-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------: |
| 点電荷 | $E(r) = \frac{Q}{4 \pi \varepsilon_0 r^{2}}$ | $\bm{E}(\bm{r}) = \frac{Q}{4 \pi \varepsilon_0 \lvert\bm{r}\rvert^{3}} \bm{r}$ |
| 無限長線電荷 | $E(r) = \frac{\lambda}{2 \pi \varepsilon_0 r}$ | $\bm{E}(\bm{r}) = \frac{\lambda}{2 \pi \varepsilon_0\lvert\bm{r}\rvert^{2}} \bm{r}$ |
| 無限面電荷 | $E(r) = \frac{\sigma}{2 \varepsilon_0}$ | $\bm{E}(\bm{r}) = \frac{\sigma}{2 \varepsilon_0 \lvert\bm{r}\rvert} \bm{r}$ |
| 球表面電荷 | $\begin{cases} E(r) = \frac{\sigma a^{2}}{\varepsilon_0 r^{2}} & (a \leq r) \\ E(r) = 0 & (0 \leq r < a) \end{cases}$ | $\begin{cases} \bm{E}(\bm{r}) = \frac{\sigma a^{2}}{\varepsilon_0 \lvert\bm{r}\rvert^{3}} \bm{r} & (a \leq \lvert\bm{r}\rvert) \\ \bm{E}(\bm{r}) = \bm{0} & (0 \leq \lvert\bm{r}\rvert < a) \end{cases}$ |
| 球体積電荷 | $\begin{cases} E(r) = \frac{\rho a^{3}}{3 \varepsilon_0 r^{2}} & (a \leq r) \\ E(r) = \frac{\rho r}{3 \varepsilon_0} & (0 \leq r < a) \end{cases}$ | $\begin{cases} \bm{E}(\bm{r}) = \frac{\rho a^{3}}{3 \varepsilon_0 \lvert\bm{r}\rvert^{3}} \bm{r} & (a \leq \lvert\bm{r}\rvert) \\ \bm{E}(\bm{r}) = \frac{\rho}{3 \varepsilon_0} \bm{r} & (0 \leq \lvert\bm{r}\rvert < a) \end{cases}$　 |
| 円筒表面電荷 | $\begin{cases} E(r) = \frac{\sigma a}{\varepsilon_0 r} & (a \leq r) \\ E(r) = 0 & (0 \leq r < a) \end{cases}$ | $\begin{cases} \bm{E}(\bm{r}) = \frac{ \sigma a}{\varepsilon_0 \lvert\bm{r}\rvert^{2}} \bm{r} & (a \leq \lvert\bm{r}\rvert) \\ \bm{E}(\bm{r}) = \bm{0} & (0 \leq \lvert\bm{r}\rvert < a) \end{cases}$ |
| 円筒体積電荷 | $\begin{cases} E(r) = \frac{\rho a^{2}}{2 \varepsilon_0 r} & (a \leq r) \\ E(r) = \frac{\rho r}{2 \varepsilon_0} & (0 \leq r < a) \end{cases}$ | $\begin{cases} \bm{E}(\bm{r}) = \frac{\rho a^{2}}{2 \varepsilon_0 \lvert\bm{r}\rvert^{2}} \bm{r} & (a \leq \lvert\bm{r}\rvert) \\ \bm{E}(\bm{r}) = \frac{\rho}{2 \varepsilon_0} \bm{r} & (0 \leq \lvert\bm{r}\rvert < a) \end{cases}$ |

| 値 | 意味 | 単位 |
| :-------------: | :------------------: | :--------------: |
| $\varepsilon_0$ | 真空中の誘電率 | $\mathrm{F/m}$ |
| $q$ | 電荷量 | $\mathrm{C}$ |
| $\lambda$ | 線電荷密度 | $\mathrm{C/m}$ |
| $\sigma$ | 面電荷密度 | $\mathrm{C/m^2}$ |
| $\rho$ | 体積電荷密度 | $\mathrm{C/m^3}$ |
| $r$ | 電荷との距離 | $\mathrm{m}$ |
| $\bm{r}$ | 電荷との距離ベクトル | $(\mathrm{m})$ |
| $a$ | 半径 | $\mathrm{m}$ |

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