# 电阻

$R\ \stackrel{def}{=}\ \frac{V}{I}$

## 導體與電阻器

### 直流電

$\mathbf{E} = \rho \mathbf{J}$

$V_{gh}\stackrel{def}{=}\ \frac{\mathrm{d}w}{\mathrm{d}q}=\int_g^h {\mathbf{E} \cdot \mathrm{d}\mathbf{l}}=\rho\int_g^h {\mathbf{J} \cdot \mathrm{d}\mathbf{l}}$

$\mathrm{d} \mathbf{l}=\mathrm{d} l\hat{\mathbf{l}}$

$V_{gh}=J\rho l$

$J = I/a$

$V=V_{gh}= I \rho l/a$

$R = \rho l/a$

## 非歐姆元件

$\mathfrak{r}\ \stackrel{def}{=}\ \frac{\mathrm{d}V}{\mathrm{d}I}$

## 溫度對電阻的影響

### 導电体

$R =R_* [\alpha(T - T_*) + 1]$

$\alpha$是電阻變化百分比每單位溫度。每一種物質都有其特定的$\alpha$。實際而言，上述關係式只是近似，真實的物理是非線性的；換句話說，$\alpha$本身會隨著溫度的改變而變化。因此，通常會在$\alpha$字尾添加測量時的溫度。例如，$\alpha_{15}$是在溫度為15 °C時測量的電阻溫度係數；使用$\alpha_{15}$為電阻溫度係數，則參考溫度$T_*$為15 °C，參考電阻為金屬在參考溫度為15 °C時的參考電阻，而且上述關係式只適用於計算溫度在15 °C附近的電阻$R$ [6]

$\alpha=\frac{R - R_*}{R_*(T - T_*)}$

$R - R_*\to 0$的極限，則可得到微分方程式[4]

$\alpha=\frac{1}{R_*}\left(\frac{\mathrm{d}R}{\mathrm{d}T}\right)_*$

$\rho=\rho_d+\rho_p$

$\rho_d$與金屬內部的缺陷密度有關，是電阻率對溫度的曲線外推至0K時的電阻率。因此，$\rho_d$與溫度無關。$\rho_p$等於$\rho - \rho_d$。假若缺陷密度不高，則$\rho_p$通常與缺陷密度無關。$\rho_p$與電子跟聲子的碰撞率有關，而碰撞率與聲子密度成正比。假設溫度高於德拜溫度，則聲子密度與溫度成正比，所以，$\rho_p$與溫度成正比：

$\rho_p= C_h T$
$\rho= \rho_d+C_h T$

$\rho_p= C_l T^5$

### 半導体

$R= R_0 e^{-aT}$

## 參考文獻

1. ^ Alexander, Charles; Sadiku, Matthew, fundamentals of Electric Circuits 3, revised, McGraw-Hill, pp. 9–10, 2006, ISBN 9780073301150
2. ^ Seymour J, Physical Electronics, pp 48–49, Pitman, 1972
3. ^ Horowitz, Paul; Winfield Hill. The Art of Electronics 2nd. Cambridge University Press. 1989: 13. ISBN 0-521-37095-7.
4. ^ 4.0 4.1 Pender, Harold & Del Mar, William (编), Handbook for Electrical Engineers:a reference book for practicing engineers and students 2nd, New York: John Wiley & Sons, Inc., pp. 1350, 2094, 1922
5. ^ Bird, John, Electrical and electronic principles and technology, Newnes, pp. 22–24, 2006, ISBN 9780750685566
6. ^ Ward, MR, Electrical Engineering Science, pp36–40, McGraw-Hill, 1971.
7. ^ Kittel, Charles, Introduction to Solid State Physics 8th, John Wiley & Sons, Inc., 148–152, 2005, ISBN 9780471415268
8. ^ A. Matthiessen, Rep. Brit. Ass. 32, 144 (1862)
9. ^ A. Matthiessen, Progg. Anallen, 122, 47 (1864)
10. ^ Enss, Christian; Hunklinger, Siegfried, Low-temperature physics illustrated, Springer, pp. 216–218, 2005, ISBN 9783540231646
11. ^
12. ^ Seymour J, Physical Electronics, chapter 2, Pitman, 1972

## 外部連結

• 克萊門森大學車輛電子實驗室網頁：電阻計算機（英文）