Question

The three parallel planes of charge shown in FIGURE $\mathrm{P}24.45$have surface charge densities $-12$,$h$,$h$andlocalid="1649410735638" $-12$,$h$- . Find the electric fields localid="1649410752965" $Eu1$to localid="1649410757308" $Eu4$in regions localid="1649410763257" $1$to localid="1649410765846" $4$.

Short answer

The areas in Zones $1$to localid="1649410567945" $4$are irrigated.

${\overrightarrow{E}}_1=\left(-E_{\mathrm{A}}+E_{\mathrm{B}}-E_{\mathrm{C}}\right)\widehat{\mathbf{j}}=\overrightarrow{0}$

${\overrightarrow{E}}_2=\left(E_{\mathrm{A}}+E_{\mathrm{B}}-E_{\mathrm{C}}\right)\widehat{\mathbf{j}}=\frac{\eta }{2{\epsilon }_0}\widehat{\mathbf{j}}$

${\overrightarrow{E}}_3=\left(E_{\mathrm{A}}-E_{\mathrm{B}}-E_{\mathrm{C}}\right)\widehat{\mathbf{j}}=-\frac{\eta }{2{\epsilon }_0}\widehat{\mathbf{j}}$

${\overrightarrow{E}}_4=\left(E_{\mathrm{A}}-E_{\mathrm{B}}+E_{\mathrm{C}}\right)\widehat{\mathbf{j}}=\overrightarrow{0}$

Step-by-step solution

Step :1 Introduction 

To compute the sector, we will utilize the collocation method, which stipulates that the net horizontal component $\overrightarrow{E}$in a point equals the magnitude of the vector of the piezoelectric effect emitted by all diverse perspectives.

Distinct electrified orthogonal planes serve as the generators in this example. Each of these provides a horizontal to the flat usually focusing whose size is the zeta potential densities divided by $2{\epsilon }_0$. The field points towards the plane if the plane is neutralized, and away from it if the plane is polarised.

Step :2 Magnitude of the field 

The amplitudes of the energies emitted by planes $A$,$B$, andlocalid="1649410440703" $C$are then calculated.

localid="1649410460704" $E_{\mathrm{A}}=\frac{\eta }{4{\epsilon }_0}$

role="math" $E_B=\frac{\eta }{2{\epsilon }_0}$

$E_{\mathrm{C}}=\frac{\eta }{4{\epsilon }_0}$

Step :3 Filed of planes 

The fields due to these planes in regions$1$,$2$,$3$and $4$are

${\overrightarrow{E}}_1=\left(-E_{\mathrm{A}}+E_{\mathrm{B}}-E_{\mathrm{C}}\right)\widehat{\mathbf{j}}=\overrightarrow{0}$

${\overrightarrow{E}}_2=\left(E_{\mathrm{A}}+E_{\mathrm{B}}-E_{\mathrm{C}}\right)\widehat{\mathbf{j}}=\frac{\eta }{2{\epsilon }_0}\widehat{\mathbf{j}}$

${\overrightarrow{E}}_3=\left(E_{\mathrm{A}}-E_{\mathrm{B}}-E_{\mathrm{C}}\right)\widehat{\mathbf{j}}=-\frac{\eta }{2{\epsilon }_0}\widehat{\mathbf{j}}$

${\overrightarrow{E}}_4=\left(E_{\mathrm{A}}-E_{\mathrm{B}}+E_{\mathrm{C}}\right)\widehat{\mathbf{j}}=\overrightarrow{0}$

${\overrightarrow{E}}_3=\left(E_{\mathrm{A}}-E_{\mathrm{B}}-E_{\mathrm{C}}\right)\widehat{\mathbf{j}}=-\frac{\eta }{2{\epsilon }_0}\widehat{\mathbf{j}}$