Calculation 9.23.1. Net Charge I.
In a laboratory, a system is constructed that contains 7 protons and 4 electrons. What is the net charge of this system?
Answer.
\(4.8 \times 10^{-19} \mathrm{~C}\)
Section 9.23 Practice, Study, and Apply - Electricity
Subsection Practice
Calculation 9.23.2. Net Charge II.
A system of protons and electrons has a net charge of \(-1.12 \times 10^{-18} \mathrm{~C}\text{.}\) If the system contains 15 protons, how many electrons are in the system?
Answer.
\(22\) electrons
Calculation 9.23.3. Net Charge III.
An amoeba has \(1 \times 10^{16}\) protons and a net charge of \(0.300 \mathrm{~pC}\text{.}\) How many fewer electrons are there than protons?
Answer.
\(1.88 \times 10^6\) fewer electrons
Calculation 9.23.4. Dryer Charge.
Consider a cotton sock inside a dryer machine whose drum (inner container) is made of steel. After tumbling around in the dryer’s drum for a while, the cotton sock has a net charge on it now. Which of the following charge transfers most likely happened in this sock-drum system?
- The steel drum lost electrons while the cotton sock gained those electrons.
- The steel drum lost protons while the cotton sock gained those protons.
- The cotton sock lost electrons while the steel drum gained those electrons.
- The cotton sock lost protons while the steel drum gained those protons.
- Both the cotton sock and steel drum lost electrons.
- Both the cotton sock and the steel drum gained electrons.
Calculation 9.23.5. Proton and Electron.
An electron and a proton are \(3.00 \mathrm{~\mu m}\) apart from each other. What is the magnitude of the electric force by the proton on the electron?
Answer.
\(2.56 \times 10^{-17} \mathrm{~N}\)
Calculation 9.23.6. Charged Spheres.
Problem Statement: Consider two small, charged spheres, \(q_1\) and \(q_2\text{,}\) a distance \(\Delta x\) apart.
- If \(\Delta x\) increases by a factor of \(16/9\text{,}\) and \(q_1\) and \(q_2\) remain the same, by what factor does the force between the two change?
- If \(\Delta x\) increases by a factor of \(4/\sqrt{2}\text{,}\) \(q_1\) increases by a factor of \(2\text{,}\) and \(q_2\) increases by a factor of \(\sqrt{2}\text{,}\) by what factor does the force between the two change?
Answer.
- \(\displaystyle 81/256\)
- \(\displaystyle \sqrt{2}/4\)
Subsection Study
A*R*C*S 9.23.7. A Piece of Pollen.
A piece of pollen typically has both a small mass (about \(5 \mathrm{~ng}\)) and a small amount of charge (about \(1 \mathrm{~fC}\)). Estimate the electric field that a honeybee needs to create to carry a typical piece of pollen.
Explanation 9.23.8. 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 9.23.9. 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.
A*R*C*S 9.23.10. 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 9.23.11. 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.
A*R*C*S 9.23.12. Charge on a Spring.
An ideal spring has spring constant \(k_s\) (to distinguish it from the electrostatic constant \(k\)) and equilibrium length \(l\text{.}\) Then, you glue two identical negative point charges to the ends of the spring and observe that the equilibrium length doubles. Determine the amount of charge on each end of the spring.
Explanation 9.23.13. Distributed Charges.
In the figure below, 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 \(k\frac{q_L q_R}{R^2}\text{?}\)
A*R*C*S 9.23.14. 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.
Explanation 9.23.15. Flux Practice.
The figure above shows an imaginary square loop centered between two points \(A\) and \(B\text{.}\)
(a)
Choose an area vector for the loop.
(b)
Suppose a point charge \(+Q_o\) is located at point \(A\text{.}\) Is the electric flux through the loop positive, negative, or zero?
(c)
Suppose a point charge \(+Q_o\) is located at point \(A\) and a point charge \(+2Q_o\) is located at point \(B\text{.}\) Is the electric flux through the loop positive, negative, or zero?
(d)
Suppose a point charge \(+Q_o\) is located at point \(A\) and a point charge \(-2Q_o\) is located at point \(B\text{.}\) Is the electric flux through the loop positive, negative, or zero?
Subsection Apply
A*R*C*S 9.23.16. The Gecko.
One reason that a gecko (a small lizard with a mass of around \(75\) grams) can walk across a ceiling is the electric force. If we assume the magnitude of the electric field acting on the gecko is \(10^6 \mathrm{~N/C}\text{,}\) determine the net charge on the gecko’s feet.
Tip.
Numerical sensemaking: explain why, for example, an elephant might have a tough time walking across a ceiling.
Explanation 9.23.17. 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 9.23.18. 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 9.23.19. Three Charges on an Arc.
The three point charges shown below have identical charge \(+Q\text{.}\) Determine the net electric force on the upper charge.
Tip.
Evaluate your answer in one or two special cases.
Explanation 9.23.20. Charges and Arcs.
Shown below are three different situations. In case A, both point charges are negative. In case B, the left point charge is positive and the right point charge is negative. In case C, the negative charge is uniformly distributed across the circular arc. Rank the three situations by the magnitude of the electric field at point P. (Focus on A vs. B and A vs. C: treat B vs. C as a challenge problem only!)
A*R*C*S 9.23.21. The End of the Line.
A straight wire with one end at \(x = -L\) and the other end at \(x = +L\) has a total charge \(+Q_o\) distributed uniformly. Find the electric field at a point on the \(x\)-axis a distance a from the center of the wire (where \(a \gt L\)).
Tip.
Sensemaking suggestion (3c): Evaluate your answer in the special case that \(a \gt\gt L\text{.}\)
Explanation 9.23.22. Charges and Arcs.
Two positive point charges \(+q\) are located a distance \(L\) apart, as shown above. A Gaussian surface (a cube with side length \(L\)) is drawn with its center at the location of the left charge.
(a)
Is the electric flux through the right face of the cube positive, negative, or zero?
(b)
b) Suppose the charge on the right were replaced by a charge of \(-q\text{.}\) Would the net electric flux through the entire cube increase, decrease, or stay the same?
