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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 21.8 Practice - Charge Distributions

Subsection Explanation Practice

Explanation 21.8.1. Three Wires.

Three straight wires (with one end at \(x = –L\) and the other end at \(x = +L\)) each have the same total charge \(+Q\) distributed differently. Wire A has charge density \(\lambda_A(x) = +\alpha x^2\text{.}\) Wire B has charge density \(\lambda_B(x) = +\beta \cos\left(\frac{\pi x}{2L}\right)\text{.}\) Wire C has uniform charge density \(+\lambda_C\text{.}\)
You want to predict how the electric fields of these three wires will compare a distance \(L\) above the center of each rod (along the \(y\)-axis). Rank the three wires by the magnitude of the electric field produced at this location from smallest to largest.
Tip.
You will need to make a qualitative argument; do not solve for the magnitude of each electric field.

Explanation 21.8.2. Trio of Charges.

You have two point charges at different locations along the \(x\)-axis. Each charge can have either \(+q_o\) or \(-q_o\text{.}\) Sketch a charge diagram showing the configuration and sign of the charges if the electric field at the origin is zero.

Explanation 21.8.3. Charged Wires.

A charged wire has one end at \(x = -L\) and the other end at \(x = L\text{.}\) The wire has a known, non-uniform charge density \(\lambda(x) = \frac{q_o}{L^2} x\text{.}\) You have an electric field generator that can make different electric fields, but when you set it to create a uniform electric field you find that the net force on the wire is 0! Which of the following alternate settings for your electric field generator would result in a nonzero net force on the wire?
\begin{equation*} \vec{E}_1 = - E_o\frac{x}{L}\hat{y} \end{equation*}
\begin{equation*} \vec{E}_2 = - E_o\left(\frac{x}{L}\right)^2 \hat{y} \end{equation*}

Explanation 21.8.4. Three Electric Fields.

A charged wire has one end at \(x = –L\) and the other end at \(x = L\text{.}\) The wire has a known, non-uniform charge density \(\lambda(x) = \frac{q_o}{L} \cos\left(\frac{\pi x}{2L}\right)\text{.}\) You have an electric field generator that can make the following three electric fields:
\begin{equation*} \vec{E}_A = E_o\hat{y} \end{equation*}
\begin{equation*} \vec{E}_B = E_o\frac{x}{L} \hat{y} \end{equation*}
\begin{equation*} \vec{E}_C = E_o\left(\frac{x}{L}\right)^2 \hat{y} \end{equation*}
Which electric field will give you the smallest net force on the wire? Which will give you the largest net force on the wire? (You may calculate the net forces to check your answer, but your explanation should be qualitative.)
As part of your explanation, you should include and use the following:
  • a charge diagram of the wire showing the charge density, using \(+\) and \(-\) symbols to represent positive and negative charge
  • an electric field vector map for each electric field

Explanation 21.8.5. Distributed Charges.

Figure 21.8.1. A distribution of point charges.
In the figure above, the charge on the left is \(+q_L\) and the total charge on the right is \(+q_R\text{.}\) The charges on the right are all identical and are spread out uniformly along an arc of radius \(R\text{.}\) Is the magnitude of the net electric force on \(+q_L\) greater than, less than, or equal to \(k\frac{q_L q_R}{R^2}\text{?}\)

Subsection A*R*C*S Practice

A*R*C*S 21.8.6. Three Charges on an Arc.

In the figure below, the top charge has has a charge \(+q\text{,}\) and the two bottom charges have identical charge \(+Q\text{.}\) Determine the net electric force on the upper charge.
Figure 21.8.2. Three point charges.
Tip.
Evaluate your answer in one or two special cases.

A*R*C*S 21.8.7. Square of Charges.

Find the electric field at the center of a square with charges of \(+q\text{,}\) \(+2q\text{,}\) \(+3q\text{,}\) and \(+4q\) on the corners (going clockwise starting from the upper right corner).

A*R*C*S 21.8.8. Torque on an Electric Dipole.

An electric dipole consists of two opposite charges \(\pm q\) separated by a small distance \(d\text{.}\) The product \(p=qd\) is called the dipole moment. The figure below shows an electric dipole perpendicular to an electric field \(\vec{E}\text{.}\) Determine an expression in terms of p and E for the magnitude of the torque that the electric field exerts on the dipole.
Figure 21.8.3. A dipole in an electric field.

A*R*C*S 21.8.9. Launching an Electron.

An electron is launched at a \(45^{\circ} \) angle and a speed of \(5.0×10^6\) \(m/s\) from the positive plate of the parallel-plate capacitor shown below. The electron lands \(4.0 \) \(cm\) away.
  1. Determine the electric field inside the parallel place capacitor. Hint: are there approximations that are useful here?
  2. What is the minimum spacing \(d \) between the plates for the electron to land \(4.0 \) \(cm\) away?
Figure 21.8.4. An electron Launched in a Capacitor.

A*R*C*S 21.8.10. Lines of Charges.

A positively charged wire with uniform charge density \(+\lambda\) lies along the \(x\)-axis and a negatively charged wire with uniform charge density \(-\lambda\) lies along the \(y\)-axis (both lines are infinitely long). Find the electric field at the point \((x, y)\text{.}\) You may look up the electric field due to an infinitely long wire with uniform charge density.
Tip 1.
Sketch an electric field vector map for the xy-plane.
Tip 2.
Use your sketch of the electric field vector map to help interpret your answer.

A*R*C*S 21.8.11. Split Wire.

A straight wire with one end at \(x = –L\) and the other end at \(x = +L\) has a total charge \(+Q_o\) distributed uniformly along the right side and total charge \(-Q_o\) distributed uniformly along the left side. Find the electric field at a point on the \(y\)-axis a distance \(a\) from the center of the wire.

Subsection Numerical Practice

Calculation 21.8.12. Charges on a Circle.

For each of the questions in the next activity, consider two charges placed at the locations shown in the image below (the circle is drawn for clarity -- it is imaginary). Let \(R = 5.0\) cm, \(q_1 = 3.0 \ \mu\)C, and \(q_2 = 1.0 \ \mu\)C. Now suppose that a proton is placed at the center of the circle.
Figure 21.8.5. Two point charges on a circle of radius \(R\text{.}\)
  1. What is the magnitude of the electric force by \(q_1\) on the proton?
  2. What is the magnitude of the electric force by \(q_2\) on the proton?
  3. What is the magnitude of the net electric force on the proton?
  4. What angle does the net electric force on the proton make with respect to the positive x-axis (using a standard coordinate system)?
Answer 1.
\(1.73 \times 10^{-12}\) N
Answer 2.
\(5.76 \times 10^{-13}\) N
Answer 3.
\(1.82 \times 10^{-12}\) N
Answer 4.
\(108^\circ\)

Calculation 21.8.13. Electric Field of Many Charges.

Consider a negative charge \(q_1\) = -9.00 nC that exists in a space far away from other charges. Let the charge \(q_1\) lie at the origin of a standard coordinate system. What is the electric field at a point P in space that has coordinates (2, -3, 6) m?
Answer.
\(\vec{E} = (-0.472 \ \mathrm{N}/\mathrm{C})\hat{x} + (0.708 \ \mathrm{N}/\mathrm{C})\hat{y} + (-1.42 \ \mathrm{N}/\mathrm{C})\hat{z}\)

Calculation 21.8.14. Charges in a Square.

Figure 21.8.6. Multiple point charges within a square.
Consider the charge configuration in the diagram above.
  1. Determine the direction of the electric field at the center of the square in the figure, given that \(q_a\) = \(q_b\) = -1.00 \(\mu\)C and \(q_c\) = \(q_d\) = +1.00 \(\mu\)C.
  2. Calculate the magnitude of the electric field at the location of \(q\text{,}\) given that the square is 5.00 cm on a side.
Answer 1.
The electric field at the center of the square will be straight up.
Answer 2.
\(|\vec{E}| = 2.04 \times 10^7 \mathrm{N}/\mathrm{C}\)

Calculation 21.8.15. Protons Around a Circle.

Consider 16 protons that are placed evenly along the circumference of a circle.
  1. What is the direction of the electric field directly at the center of the circle?
  2. Now suppose that the proton at the top of the circle is removed. Now what is the direction of the electric field directly at the center of the circle?
Answer 1.
No direction; the electric field is zero.
Answer 2.
Up

References References

[1]
Practice activities provided by BoxSand: https://boxsand.physics.oregonstate.edu/welcome.
[2]
Practice activities provided by Ling, S. J., Moebs, W., & Sanny, J. (2016). Electric Charges and Fields. In University Physics Volume 2. OpenStax.
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