91影视

The cells carrying charges resemble a parallel plate capacitor.

For the medium air, the expression of electric field is,

$\begin{array}{r}{E}_0=\frac{蟽}{{蔚}_0}\\ =\frac{0.50脳{10}^{鈭�3}\mathrm{C}/{\mathrm{m}}^2}{8.854脳{10}^{鈭�12}{\mathrm{C}}^2/\left(\mathrm{N}鈰�{\mathrm{m}}^2\right)}\\ =5.6脳{10}^7\mathrm{V}/\mathrm{m}\end{array}$

Thus, magnitude of $\stackrel{鈫�}{E}$in the wall between two layers of charge is $5.6x{10}^{7}V/m$.

The potential energy expression is given as,

$\begin{array}{r}{V}_0=E_{0}d\\ =\left(5.6脳{10}^7\mathrm{V}/\mathrm{m}\right)\left(5.0脳{10}^{鈭�9}\mathrm{m}\right)\\ =0.28\mathrm{V}\end{array}$

The energy density expression is,

$u=\frac{1}{2}{蔚}_0{E}_0^2$ 鈥�(颈)

Volume of the cell is equal to volume of a sphere (assumption),

$V_{\text{cell}}=\frac{4}{3}蟺r^3$

Solve for r,

$\begin{array}{r}r={\left(\frac{3V_{\text{cell}}}{4蟺}\right)}^{\frac{1}{3}}\\ ={\left(\frac{3\left({10}^{鈭�10}{m}^3\right)}{4蟺}\right)}^{\frac{1}{3}}\\ =2.9脳{10}^{鈭�6}\mathrm{m}\end{array}$

The volume of the cell wall with thickness t is given as,

$\begin{array}{r}{V}_{\text{waII}}=4蟺R^{2}t\\ =4蟺{\left(2.9脳{10}^{鈭�6}\mathrm{m}\right)}^2\left(5.0脳{10}^{鈭�9}\mathrm{m}\right)\\ =5.3脳{10}^{鈭�19}{\mathrm{m}}^3\end{array}$

Substitute all the values in equation (i),

$\begin{array}{r}u=\frac{1}{2}\left[8.854脳{10}^{鈭�12}{C}^2/\left(N鈰�m^2\right)\right]{\left[5.6脳{10}^7\mathrm{V}/\mathrm{m}\right]}^2\\ =1.39脳{10}^4\mathrm{J}/{\mathrm{m}}^3\end{array}$

Thus, the total field energy on the cell wall is,

$\begin{array}{r}U=u脳V_{\text{wal}}\\ =\left(1.39脳{10}^4\mathrm{J}/{\mathrm{m}}^3\right)\left(5.3脳{10}^{鈭�19}{\mathrm{m}}^3\right)\\ =7脳{10}^{鈭�15}\mathrm{J}\end{array}$

When air is replaced by dielectric the change in electric field is,

$\begin{array}{r}E=\frac{E_0}{K}\\ =\frac{5.6脳{10}^7\mathrm{V}/\mathrm{m}}{5.4}\\ =1.0脳{10}^7\mathrm{V}/\mathrm{m}\end{array}$

Similarly, for potential difference,

$\begin{array}{r}V=\frac{V_0}{K}\\ =\frac{0.28\mathrm{V}}{5.4}\\ =0.052\mathrm{V}\end{array}$

Thus, the changed electric field and potential difference are $1.0x{10}^{7}V/m$and $0.052V$respectively.