Question

Two infinite nonconducting plates are parallel to each other, with a distance \(d=10.0 \mathrm{~cm}\) between them, as shown in the figure. Each plate carries a uniform charge distribution of \(\sigma=4.5 \mu \mathrm{C} / \mathrm{m}^{2}\). What is the electric field, \(\vec{E},\) at point \(P\left(\right.\) with \(\left.x_{p}=20.0 \mathrm{~cm}\right) ?\)

Short answer

Answer: The net electric field at point P due to the two infinite charged plates is \(0 \, \text{V/m}\).

Step-by-step solution

Identify key variables

In this problem, we have the following key variables: - Distance between the plates, \(d = 10.0\,\text{cm}\) - Uniform charge distribution, \(\sigma = 4.5 \, \mu \mathrm{C} / \mathrm{m}^{2}\) - Position of point P, \(x_p = 20.0\,\text{cm}\)

Find electric field due to one plate

As the plates are infinite and nonconducting, we can use the formula for electric field due to an infinite sheet of charge: \(E = \dfrac{\sigma}{2 \epsilon_0}\) where \(\epsilon_0 = 8.85 \times 10^{-12} \,\text{F/m}\) is the vacuum permittivity. Calculate the electric field due to one plate: \(E = \dfrac{4.5 \times 10^{-6} \, \text{C/m}^2}{2(8.85 \times 10^{-12} \,\text{F/m})} = 253.16\,\text{V/m}\)

Determine electric field components

Since point P is located to the right of both plates, we'll have electric field components due to both plates acting at point P. Both electric fields will have the same magnitude but opposite directions. The electric field due to plate 1 (left plate) will be directed to the right (positive x-direction), while the electric field due to plate 2 (right plate) will be directed to the left (negative x-direction).

Apply superposition principle

Applying the superposition principle, the net electric field at point P is the sum of the electric field components due to the two plates: \(\vec{E}_{\text{net}} = \vec{E}_{\text{plate 1}} + \vec{E}_{\text{plate 2}}\) Since the magnitudes of the electric fields are the same, their x-components are equal and opposite. Thus, the x-components will cancel each other out, resulting in a net electric field of zero at point P: \(\vec{E}_{\text{net}} = 0 \,\text{V/m}\)

State the result

The net electric field at point P due to the two infinite charged plates is \(\vec{E}_{\text{net}} = 0 \, \text{V/m}\).