Problem 63 - University Physics with Modern Physics

Question

Two infinite sheets of charge are separated by \(10.0 \mathrm{~cm}\) as shown in the figure. Sheet 1 has a surface charge distribution of \(\sigma_{1}=3.00 \mu \mathrm{C} / \mathrm{m}^{2}\) and sheet 2 has a surface charge distribution of \(\sigma_{2}=-5.00 \mu \mathrm{C} / \mathrm{m}^{2}\). Find the total electric field (magnitude and direction) at each of the following locations: a) at point \(P, 6.00 \mathrm{~cm}\) to the left of sheet 1 b) at point \(P^{\prime} 6.00 \mathrm{~cm}\) to the right of sheet 1

Step-by-Step Solution

Answer

Answer: The total electric field at point P is 113.00 N/C to the left, and the total electric field at point P' is 451.98 N/C to the right.

Step 1: Write down the electric field formula for an infinite sheet

For an infinite sheet of charge with surface charge density \(\sigma\), the electric field is given by the following formula: $$E = \frac{\sigma}{2 \epsilon_0}$$ Where \(E\) is the electric field in the perpendicular direction of the sheet and \(\epsilon_0\) is the vacuum permittivity constant, with a value of \(8.85 × 10^{-12} \mathrm{C}^2/\mathrm{N} \cdot \mathrm{m}^2\).

Step 2: Calculate the electric fields produced by each sheet at point \(P\)

At point \(P\), which is \(6.00 \mathrm{~cm}\) to the left of sheet 1, both sheets produce electric fields in the same direction (leftwards), as they both have charges with opposite signs. Calculate the electric fields produced by each sheet individually and then combine them by adding the magnitudes of their electric fields at point \(P\). For sheet 1: $$E_1 = \frac{3.00 × 10^{-6} \mathrm{C} / \mathrm{m}^{2}}{2 × 8.85 × 10^{-12} \mathrm{C}^2/\mathrm{N} \cdot \mathrm{m}^2} = 169.49 \mathrm{N/C}$$ For sheet 2: $$E_2 = \frac{-5.00 × 10^{-6} \mathrm{C} / \mathrm{m}^{2}}{2 × 8.85 × 10^{-12} \mathrm{C}^2/\mathrm{N} \cdot \mathrm{m}^2} = -282.49 \mathrm{N/C}$$ As they are both acting in the same direction (leftwards), the total electric field at point \(P\) will be the sum of their magnitudes: $$E_P = E_1 + E_2 = 169.49 \mathrm{N/C} - 282.49 \mathrm{N/C} = -113.00 \mathrm{N/C}$$ The total electric field at point \(P\) is \(113.00 \mathrm{N/C}\) to the left.

Step 3: Calculate the electric fields produced by each sheet at point \(P'\)

At point \(P'\), which is \(6.00 \mathrm{~cm}\) to the right of sheet 1, both sheets produce electric fields in opposite directions (sheet 1 produces an electric field to the right, and sheet 2 produces an electric field to the left). Calculate the electric fields produced by each sheet individually and then combine them by subtracting the magnitudes of their electric fields at point \(P'\). For sheet 1: $$E_1 = 169.49 \mathrm{N/C}$$ (already calculated in step 2) For sheet 2: $$E_2 = -282.49 \mathrm{N/C}$$ (already calculated in step 2) As they are both acting in opposite directions, the total electric field at point \(P'\) will be the difference between their magnitudes: $$E_{P'} = E_1 - E_2 = 169.49 \mathrm{N/C} - (-282.49 \mathrm{N/C}) = 451.98 \mathrm{N/C}$$ The total electric field at point \(P'\) is \(451.98 \mathrm{N/C}\) to the right.

Key Concepts

Infinite Sheets of Charge

An infinite sheet of charge is a theoretical concept used in physics to simplify electromagnetic calculations. This assumes the sheet extends infinitely in two dimensions, and that it has a uniformly distributed charge across its surface. Due to its infinite nature, the edge effects can be ignored, which simplifies calculations considerably. The electric field generated by such a sheet is constant in magnitude at any point and perpendicular to the plane of the sheet.

In reality, no sheet can truly be infinite, but this concept is useful in approximations where the physical size of the sheet is much larger than the distance from which observations are made. This approximation is commonly used in various fields of electromagnetic studies and helps in understanding the behavior of charge distributions on flat surfaces.

Surface Charge Density

Surface charge density, represented by the symbol \( \sigma \), describes how much charge is distributed over a unit area on the surface of a conductor or dielectric. It is measured in units of coulombs per square meter \( \text{C/m}^{2} \).

For example, in the provided exercise, the surface charge densities of the two sheets are given as \( \sigma_{1}=3.00 \, \mu \text{C/m}^{2} \) for the first sheet and \( \sigma_{2}=-5.00 \, \mu \text{C/m}^{2} \) for the second sheet. Note that one is positively charged, while the other is negatively charged. This is crucial as it determines the direction and interaction of the electric fields. A positive surface charge density represents an outward electric field, while a negative one signifies an inward electric field relative to the sheet's surface.

Electric Field Calculation

To calculate the electric field produced by an infinite sheet of charge, we use a straightforward formula:

\[ E = \frac{\sigma}{2 \epsilon_0} \]

Here, \( E \) is the electric field , \( \sigma \) is the surface charge density, and \( \epsilon_0 = 8.85 \times 10^{-12} \text{C}^2/\text{N} \cdot \text{m}^2 \) is the vacuum permittivity constant. This formula gives a constant electric field that is directed perpendicular to the surface of the sheet, irrespective of the distance from the sheet.

In addressing the original problem statement:

• For point \( P \), both sheets' fields add up because the charges are opposite, amplifying their collective effect.
• For point \( P' \), the fields have to be subtracted because the directions are opposite.

This showcases how understanding the direction of the electric fields can predict the resulting field strength at any point in space affected by these sheets.

Electromagnetism

Electromagnetism is a fundamental branch of physics that deals with the interactions between electric charges and magnetic fields. It encompasses concepts such as electric forces, electric fields (as we see in this exercise), magnetic fields, and ultimately, the unified nature of electromagnetism as described by Maxwell's equations.

Electromagnetic theories are essential for explaining how electric charges in motion create magnetic fields and how these fields can exert forces on other charges. This principle is the basis for technologies such as electric motors, generators, and transformers.

Understanding the behavior of infinite sheets of charge and their fields is just one piece of the broader puzzle harnessed by electromagnetic theory, showing its application ranging from simple circuit predictions to complex phenomena in various modern electronic devices.