A long line carrying a uniform linear charge density$\mathbf{}\mathbf{+}\mathbf{50}\mathbf{.}\mathbf{0}\mathbf{\mu C}\mathbf{/}\mathbf{m}$runs parallel to and 10.0 cmfrom the surface of a large, flat plastic sheet that has a uniform surface charge density of$\mathbf{-}\mathbf{100}\mathbf{\mu C}\mathbf{/}{\mathbf{m}}^{\mathbf{2}}$ on one side. Find the location of all points where an$\mathit{\alpha }$particle would feel no force due to this arrangement of charged objects.

An electric property associated with each point in space when charge is present in any form. The magnitude and direction of the electric field are expressed by the value of E, called electric field strength or electric field intensity or simply the electric field.

Each element applies an electric field on the other as shown in the figure below, where for an infinite sheet of charge the electric field is given by

$E=\frac{\sigma }{2{\displaystyle \stackrel{\text{'}}{o_{°}}}}$

While for an infinite line charge the electric field is given by

$E=\frac{I}{2\mathrm{\pi }{\displaystyle \stackrel{\text{'}}{{\mathrm{o}}_{°}}}}\frac{\lambda }{r}$

Where r is the distance between the line and the sheet, The point which each electric field cancel each other is when the summation of both electric fields is zero at the distance r

$$\begin{gathered} \frac{\sigma }{2{\displaystyle \stackrel{\text{'}}{o_{°}}}}+\frac{1}{2\mathrm{\pi }\stackrel{\text{'}}{{\mathrm{o}}_{°}}}\frac{\lambda }{r}=0 \\ \stackrel{\text{'}}{\frac{\sigma }{2{\displaystyle \stackrel{\text{'}}{o_{°}}}}=-\frac{1}{2\mathrm{\pi }\stackrel{\text{'}}{{\mathrm{o}}_{°}}}}\frac{\lambda }{r} \\ r=-\frac{\lambda }{\mathrm{\pi \sigma }} \end{gathered}$$

(Solve for r )

Now we can plug our values for$\sigma$and$\lambda$into equation (3) to get the distance r

$$\begin{gathered} r=-\frac{\lambda }{\mathrm{\pi \sigma }} \\ =\frac{+50.0/m}{-\mathrm{\pi }100/{\mathrm{m}}^2} \\ =0.160m \\ =16.0cm \end{gathered}$$

At distance r=16.0 cm, the particle would feel no force and this location is outside the distance between the sheet and the line, right to the sheet or left to the line.