All Problems

Constant Electric Field in a Vacuum

Problem 3.45

An electric capacitor consists of thin round parallel plates, each of radius R,R, separated by a distance l(l≪R)l(l \ll R) and uniformly charged with surface densities σ\sigma and −σ.-\sigma . Find the potential of the electric field and the magnitude of its strength vector at the axes of the capacitor as functions of a distance xx from the plates if xx ≫l\gg l Investigate the obtained expressions at x≫Rx \gg R.

Reveal Answer
3.45. φ≈±σl2ε0(1−xx2+R2),E≈σlR22ε0(x2+R2)3/2.\varphi \approx \pm \frac{\sigma l}{2 \varepsilon_{0}}\left(1-\frac{x}{\sqrt{x^{2}+R^{2}}}\right), E \approx \frac{\sigma l R^{2}}{2 \varepsilon_{0}\left(x^{2}+R^{2}\right)^{3 / 2}} . If x≫Rx \gg R then φ≈±p4σε0x2\varphi \approx \pm \frac{p}{4 \sigma \varepsilon_{0} x^{2}} and E≈p2πε0x3,E \approx \frac{p}{2 \pi \varepsilon_{0} x^{3}}, where p=πR2σl.p=\pi R^{2} \sigma l . In the formulas for the potential φ\varphi the plus sign corresponds to the space adjoining the positively charged plate and the minus sign to the space adjoining the negatively charged pläte.
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