Question

Two large thin plane sheets, each having surface charge density \(\sigma\), are held close and parallel to each other in air. What is the net electric field at a point (i) inside and (ii) outside the sheets ?

Hint:

For two large parallel sheets carrying charges \(+\sigma\) and \(-\sigma\): \[ \boxed{ E_{\text{inside}} = \frac{\sigma}{\varepsilon_0} } \] and \[ \boxed{ E_{\text{outside}}=0 } \] This arrangement forms a parallel-plate capacitor, where the electric field is uniform between the plates and zero outside.

Answer 1

Concept:
The field of one large uniformly charged plane sheet of surface charge density \(\sigma\) is \(E=\frac{\sigma}{2\varepsilon_0}\), independent of distance, directed away from the sheet if \(\sigma>0\) and towards it if \(\sigma<0\). The net field of two sheets is the vector sum (superposition).

Step 1: Data as printed:
Each of the two parallel sheets has the same surface charge density \(\sigma\) (the same sign). Take \(\sigma>0\).

Step 2: Field at a point between (inside) the sheets:
The field of the first sheet points away from it, i.e. towards the second sheet; the field of the second sheet points away from it, i.e. towards the first sheet. The two fields are equal in magnitude and opposite in direction:
\[ E_{\text{inside}}=\frac{\sigma}{2\varepsilon_0}-\frac{\sigma}{2\varepsilon_0}=0. \]

Step 3: Field at a point outside the sheets:
On either outer side both fields point in the same direction (away from the sheets), so they add:
\[ E_{\text{outside}}=\frac{\sigma}{2\varepsilon_0}+\frac{\sigma}{2\varepsilon_0}=\frac{\sigma}{\varepsilon_0}. \]
The field is directed away from the sheets (towards the sheets if \(\sigma\) is negative).

Remark:
If the sheets carried equal and opposite densities \(+\sigma\) and \(-\sigma\), the two results would be interchanged (inside \(\frac{\sigma}{\varepsilon_0}\), outside zero). The printed question gives the same \(\sigma\) on both sheets.

Final Answer:
\(E_{\text{inside}}=0\) and \(E_{\text{outside}}=\frac{\sigma}{\varepsilon_0}\) (directed away from the sheets for \(\sigma>0\)).