Question

Two infinite, uniformly charged, flat surfaces are mutually perpendicular. One of the sheets has a charge density of + 20.0 pC/m2, and the other carries a charge density of -50.0 pC/m2. What is the magnitude of the electric field at any point not on either surface?

Answer 1

Two infinite, uniformly charged, flat surfaces are mutually perpendicular. One of the sheets has a charge density of $+20.0 \, \text{pC/m}^2$ , and the other carries a charge density of $-50.0 \, \text{pC/m}^2$ . What is the magnitude of the electric field at any point not on either surface?

Solution

We have the configuration of problem's system shows in picture, where $\sigma_{1} = +20.0\mathrm{pC / m2}$ , $\sigma_{2} = -50.0\mathrm{pC / m2}$ .

At point not on either surface with coordinates $(x,y)$ the surfaces with a charge density $\sigma_{1}$ create the electric field $E_{x} = \frac{\sigma_{1}}{x}$ , the surfaces with a charge density $\sigma_{2}$ create the electric field $E_{y} = \frac{\sigma_{1}}{y}$ .

The electric field at point $(x,y)$ is

$$\vec {E} = \vec {E} _ {y} + \vec {E} _ {x} = \frac {\sigma_ {1}}{y} \vec {e} _ {y} + \frac {\sigma_ {2}}{x} \vec {e} _ {x}$$

Where $\vec{e}_y, \vec{e}_x$ are basis vectors of coordinate system.

From hence

$$E = \sqrt {E ^ {2} y + E ^ {2} x} = \sqrt {\left(\frac {\sigma_ {1}}{y}\right) ^ {2} + \left(\frac {\sigma_ {2 ^ {2}}}{x}\right) ^ {2}}$$

Answer

$$E = \sqrt {E ^ {2} y + E ^ {2} x} = \sqrt {\left(\frac {\sigma_ {1}}{y}\right) ^ {2} + \left(\frac {\sigma_ {2 ^ {2}}}{x}\right) ^ {2}}$$ $$\vec {\mathrm {E}} = \vec {\mathrm {E}} _ {y} + \vec {\mathrm {E}} _ {x} = \frac {\sigma_ {1}}{y} \vec {\mathrm {e}} _ {y} + \frac {\sigma_ {2}}{x} \vec {\mathrm {e}} _ {x},$$

Where $\sigma_{1} = +20.0\mathrm{pC / m2}$, $\sigma_{2} = -50.0\mathrm{pC / m2}$.