Question

Three infinite non-conducting sheets, with uniform positive surface charge densities$\mathbf{\sigma }\mathbf{,}\mathbf{2}\mathbf{\sigma }\mathbf{,}$and $\mathbf{3}\mathbf{\sigma }\mathbf{,}$are arranged to be parallel like the two sheets in Fig. 23-19a. What is their order, from left to right, if the electric field produced by the arrangement has magnitude$\mathit{E}\mathbf{=}\mathbf{0}$in one region and$\mathbf{E}\mathbf{=}\mathbf{2}\mathbf{\sigma }\mathbf{/}{\mathbf{\epsilon }}_{\mathbf{0}}$in another region?

Short answer

The order of the sheets from left to right for the given electric fields is$\text{Sheet3,sheet1,}$ and $\text{Sheet2}$.

Step-by-step solution

The given data:

Three infinite non-conducting sheets with uniform positive charge densities$\mathrm{\sigma },2\mathrm{\sigma }$ and $3\sigma$are arranged parallel.

Understanding the concept of electric field: 

Here, e can arbitrarily arrange the sheets such that we can get the given values of the electric fields between the regions of the non-conducting sheets.

Formula:

The electric field for an infinite non-conducting sheet,

$\mathrm{E}=\frac{\mathrm{\sigma }}{2{\mathrm{\epsilon }}_0}$ ….. (i)

Calculation of the order of the three sheets:

As the charge densities are positive for all the sheets, thus, the any point at the extreme ends of the arrangements will add up due to repulsion and thus, there will be no minimum value of zero electric field $\left(\mathrm{E}=0\right)$ or $\mathrm{E}=\frac{2\mathrm{\sigma }}{{\mathrm{\epsilon }}_0}$ at either end.

At region I, the net electric field at any point in this region can be given using equation (i) as follows:

(For condition of repulsion of similar charges field is towards left for charge densities $+\sigma ,+2\sigma$ and the field is towards right for $+3\mathrm{\sigma }$).

$\begin{array}{c}{\mathrm{E}}_{\mathrm{I}}=\frac{+3\mathrm{\sigma }}{{\mathrm{\epsilon }}_0}-\frac{+\mathrm{\sigma }}{{\mathrm{\epsilon }}_0}-\frac{+2\mathrm{\sigma }}{{\mathrm{\epsilon }}_0}\\ =0\end{array}$

At region II, the net electric field at any point in this region can be given using equation (i) as follows:

(For condition of repulsion of similar charges field is towards left for charge densities$+2\sigma$ and the field is towards right for role="math" localid="1661872288885" $+\sigma ,+3\sigma$).

$\begin{array}{c}{\mathrm{E}}_{\mathrm{II}}=\frac{+3\mathrm{\sigma }}{{\mathrm{\epsilon }}_0}+\frac{+\mathrm{\sigma }}{{\mathrm{\epsilon }}_0}-\frac{+2\mathrm{\sigma }}{{\mathrm{\epsilon }}_0}\\ =\frac{+2\mathrm{\sigma }}{{\mathrm{\epsilon }}_0}\end{array}$

Hence, the order of the sheets is $\text{Sheet3,sheet1,sheet2}$.