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Chapter 21: Problem 45

Three parallel sheets of charge, large enough to be treated as infinite sheets, are perpendicular to the \(x\) -axis. Sheet \(A\) has surface charge density \(\sigma_{A}=+8.00 \mathrm{nC} / \mathrm{m}^{2}\). Sheet \(B\) is \(4.00 \mathrm{~cm}\) to the right of sheet \(A\) and has surface charge density \(\sigma_{B}=-4.00 \mathrm{nC} / \mathrm{m}^{2} .\) Sheet \(C\) is \(4.00 \mathrm{~cm}\) to the right of sheet \(B,\) so is \(8.00 \mathrm{~cm}\) to the right of sheet \(A,\) and has surface charge density \(\sigma_{C}=+6.00 \mathrm{nC} / \mathrm{m}^{2}\). What are the magnitude and direction of the resultant electric field at a point that is midway between sheets \(B\) and \(C,\) or \(2.00 \mathrm{~cm}\) from each of these two sheets?

Short Answer

Expert verified
The magnitude of the resultant electric field at the point midway between the sheets B and C can be found by calculating the individual electric fields due to each sheet and then adding them as vectors. The exact value will depend on the calculated values of \(E_A, E_B, and E_C\). The direction of the resultant field will be towards the left or right, depending on which part of the sum dominates.

Step by step solution

01

Determine the Direction of the Electric Fields

The electric field created by a sheet of charge always points away from the sheet if the charge is positive, and towards the sheet if the charge is negative. So the electric field \(E_A\) created by sheet \(A\) points to the right (since it's positively charged), while fields \(E_B\) and \(E_C\) from sheets \(B\) and \(C\) point to the left (B is negatively charged and C being on the right side exerts its positive charge effect to the left).
02

Calculate the Magnitude of the Electric Fields

The magnitude of the electric field created by a sheet of charge is given by \(E = \frac{蟽}{2蔚}\), where \(蟽\) is the surface charge density and \(蔚\) is the permittivity of free space (approximately \(8.85 x 10^{-12} C^2/Nm^2\)). Therefore, \(E_A = \frac{蟽_A}{2蔚}\), \(E_B = \frac{蟽_B}{2蔚}\), and \(E_C = \frac{蟽_C}{2蔚}\). Plug in the given values to find \(E_A\), \(E_B\) and \(E_C\).
03

Combine the Electric Fields at the Point of Interest

The electric field at a point due to multiple charges is the vector sum of the electric fields due to the individual charges. Therefore, the electric field midway between sheets B and C is the resultant of \(E_A + E_B + E_C\). Since everything is along the x-axis, the magnitudes can be added and subtracted as per their direction. \(E_B\) and \(E_C\) are in the same direction (to the left) so they can be added together. Then \(E_A\) (to the right) can be subtracted from their sum to find the resultant field.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Surface Charge Density
Surface charge density, denoted by the Greek letter \(\sigma\), is a measure of how much electric charge is accumulated per unit area on a surface. Imagine a large, flat sheet with electric charges spread evenly across its surface. The surface charge density tells us how much charge sits on each square meter of that sheet.

Understanding surface charge density is crucial because it directly affects the electric field that the sheet creates in the surrounding space. In our exercise, each sheet has a different surface charge density:
  • Sheet A: \(\sigma_A = +8.00 \text{ nC/m}^2\)
  • Sheet B: \(\sigma_B = -4.00 \text{ nC/m}^2\)
  • Sheet C: \(\sigma_C = +6.00 \text{ nC/m}^2\)
These values indicate the strength and the direction of the electric field produced by each sheet. A positive value of \(\sigma\) means the electric field points away from the sheet, while a negative value implies that the field points toward the sheet. This fundamental concept is a cornerstone in calculating the fields and their resultant effects.
Permittivity of Free Space
The permittivity of free space, often denoted as \(\varepsilon_0\), plays a pivotal role in calculating electric fields. It is a constant value that reflects how much electric field can "pass through" a vacuum, and its approximate value is \(8.85 \times 10^{-12} \text{ C}^2/\text{Nm}^2\).

This constant is used in the equation to determine the magnitude of the electric field \(E\) produced by a charged sheet: \[ E = \frac{\sigma}{2\varepsilon_0} \]The equation highlights that the electric field's intensity increases with the surface charge density and decreases with the permittivity of free space. This is integral to our problem because it helps us compute the electric fields generated by the sheets A, B, and C. Remember, this relationship holds true only for infinite sheets, which we assume here to simplify the mathematics and understand the core physics.
Vector Sum of Electric Fields
Electric fields follow the principle of superposition, meaning that when multiple fields exist at a point, we calculate the resultant field by vector addition of the individual fields. This process is called the vector sum of electric fields.

In this exercise, each sheet creates its own electric field at a given point. To find the total field at the point midway between sheets B and C, we combine these fields taking their directions into account. For sheet A, the field \(E_A\) points to the right, while \(E_B\) and \(E_C\) from sheets B and C respectively, point to the left. When performing the vector sum:
  • Add the fields from sheets B and C, since they point in the same direction.
  • Subtract the field from sheet A from the combined field of B and C, as it points in opposition.
By understanding and applying these vector additions correctly, we find the net electric field at our point of interest. This provides insight into how electric fields interact and affect their surroundings.

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Most popular questions from this chapter

A proton is traveling horizontally to the right at \(4.50 \times 10^{6} \mathrm{~m} / \mathrm{s}\). (a) Find the magnitude and direction of the weakest electric field that can bring the proton uniformly to rest over a distance of \(3.20 \mathrm{~cm}\). (b) How much time does it take the proton to stop after entering the field? (c) What minimum field (magnitude and direction) would be needed to stop an electron under the conditions of part (a)?

Positive charge \(Q\) is distributed uniformly along the \(x\) axis from \(x=0\) to \(x=a\). A positive point charge \(q\) is located on the positive \(x\) -axis at \(x=a+r,\) a distance \(r\) to the right of the end of \(Q\) (Fig. P21.79). (a) Calculate the \(x\) - and \(y\) -components of the electric field produced by the charge distribution \(Q\) at points on the positive \(x\) -axis where \(x>a\). (b) Calculate the force (magnitude and direction) that the charge distribution \(Q\) exerts on \(q\). (c) Show that if \(r \gg a\), the magnitude of the force in part (b) is approximately \(Q q / 4 \pi \epsilon_{0} r^{2}\). Explain why this result is obtained.

What is the best explanation for the observation that the electric charge on the stem became positive as the charged bee approached (before it landed)? (a) Because air is a good conductor, the positive charge on the bee's surface flowed through the air from bee to plant. (b) Because the earth is a reservoir of large amounts of charge, positive ions were drawn up the stem from the ground toward the charged bee. (c) The plant became electrically polarized as the charged bee approached. (d) Bees that had visited the plant earlier deposited a positive charge on the stem.

In an experiment in space, one proton is held fixed and another proton is released from rest a distance of \(2.50 \mathrm{~mm}\) away. (a) What is the initial acceleration of the proton after it is released? (b) Sketch qualitative (no numbers!) acceleration-time and velocity-time graphs of the released proton's motion.

A uniform electric field exists in the region between two oppositely charged plane parallel plates. A proton is released from rest at the surface of the positively charged plate and strikes the surface of the opposite plate, \(1.60 \mathrm{~cm}\) distant from the first, in a time interval of \(3.20 \times 10^{-6} \mathrm{~s} .\) (a) Find the magnitude of the electric field. (b) Find the speed of the proton when it strikes the negatively charged plate.
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