Φ=E→⋅A→=EAcosθ=∫E→⋅dA→=kqr2⋅4πr2=q4πr24πϵr2=qϵ{\displaystyle \Phi ={\vec {E}}\cdot {\vec {A}}=EA\cos \theta =\int {\vec {E}}\cdot d{\vec {A}}={\frac {kq}{r^{2}}}\cdot 4\pi r^{2}={\frac {\quad q\;\;4\pi r^{2}}{4\pi \epsilon \,r^{2}\quad \;\;}}={\frac {q}{\epsilon }}}
圖23.2
E=∫dE=k∫dqr2=kλ∫al+adxx2=kλ[−1x]al+adq=λdx=kQl(−1l+a+1a)=kQa(l+a){\displaystyle {\begin{aligned}E&=\int dE=k\int {\frac {dq}{r^{2}}}=k\lambda \int _{a}^{l+a}{\frac {dx}{x^{2}}}=k\lambda \left[-{\frac {1}{x}}\right]_{a}^{l+a}\\&\qquad \qquad \qquad dq=\lambda dx\\&=k{\frac {Q}{l}}\left(-{\frac {1}{l+a}}+{\frac {1}{a}}\right)={\frac {kQ}{a\left(l+a\right)}}\end{aligned}}}
圖23.3
Ex=∫dEcosθ=∫kdqr2xr=∫kxdqr3=kxQx2+a23{\displaystyle E_{x}=\int dE\,\cos \theta =\int {\frac {kdq}{r^{2}}}{\frac {x}{r}}=\int {\frac {kxdq}{r^{3}}}={\frac {kxQ}{{\sqrt {x^{2}+a^{2}}}^{3}}}}
Ex=∫kxσrdrdθx2+r23=kx2πσ∫0Rrdrx2+r23=kxπσ∫0Rd(x2+r2)x2+r23dq=σdA=σrdrdθ=2πσrdr=kxπσ∫0R(x2+r2)−32d(x2+r2)=−2kxπσ[1x2+r2]0R=−2kxπσ(1x2+R2−1x2)=2kxπσ(1x−1x2+R2)=2kπσ(1−xx2+R2){\displaystyle {\begin{aligned}E_{x}=\int {\frac {kx\sigma rdrd\theta }{{\sqrt {x^{2}+r^{2}}}^{3}}}=kx2\pi \sigma \int _{0}^{R}{\frac {rdr}{{\sqrt {x^{2}+r^{2}}}^{3}}}=kx\pi \sigma \int _{0}^{R}{\frac {d\left(x^{2}+r^{2}\right)}{{\sqrt {x^{2}+r^{2}}}^{3}}}\\dq=\sigma dA=\sigma rdrd\theta =2\pi \sigma rdr\qquad \qquad \qquad \qquad \qquad \qquad \quad \\=kx\pi \sigma \int _{0}^{R}\left(x^{2}+r^{2}\right)^{-{\frac {3}{2}}}d\left(x^{2}+r^{2}\right)=-2kx\pi \sigma \left[{\frac {1}{\sqrt {x^{2}+r^{2}}}}\right]_{0}^{R}\\=-2kx\pi \sigma \left({\frac {1}{\sqrt {x^{2}+R^{2}}}}-{\frac {1}{\sqrt {x^{2}}}}\right)=2kx\pi \sigma \left({\frac {1}{x}}-{\frac {1}{\sqrt {x^{2}+R^{2}}}}\right)\\=2k\pi \sigma \left(1-{\frac {x}{\sqrt {x^{2}+R^{2}}}}\right)\end{aligned}}}
圖23.9
ΦE=0{\displaystyle \Phi _{E}=0}
E(r>a)=kQr2ΦE=E∮dA=Qϵ=E4πr2E(r<a)=kQra3E4πr2=Qinϵ=Qr3a3/ϵ=Qr3ϵa3E=kQinr2Qin=Q43πr343πa3=Qr3a3{\displaystyle {\begin{aligned}E\left(r>a\right)={\frac {kQ}{r^{2}}}\\\Phi _{E}=E\oint dA={\frac {Q}{\epsilon }}=E4\pi r^{2}\\E\left(r<a\right)={\frac {kQr}{a^{3}}}\\E4\pi r^{2}={\frac {Q_{in}}{\epsilon }}=Q{\frac {r^{3}}{a^{3}}}{\big /}\epsilon ={\frac {Qr^{3}}{\epsilon a^{3}}}\\E={\frac {kQ_{in}}{r^{2}}}\qquad Q_{in}=Q{\frac {{\frac {4}{3}}\pi r^{3}}{{\frac {4}{3}}\pi a^{3}}}=Q{\frac {r^{3}}{a^{3}}}\end{aligned}}}
E2πrl=λlϵE=λ2πϵr=2kλr{\displaystyle E2\pi rl={\frac {\lambda l}{\epsilon }}\qquad E={\frac {\lambda }{2\pi \epsilon r}}={\frac {2k\lambda }{r}}}
+++++
2EA=σAϵE=σ2ϵ{\displaystyle 2EA={\frac {\sigma A}{\epsilon }}\qquad E={\frac {\sigma }{2\epsilon }}}