Physics 2321: Electric Fields, Potentials, and Charge Distributions – Guided Study Notes

Q1. Consider a collection of 8 point charges arranged on the corners of a cube of side-length 50 centimeters. If each charge has a magnitude of 100 nC, and the electric potential is defined to be 0 at a point infinitely far from the cube, what is the potential at the center of the cube?

Background

Topic: Electric Potential due to Point Charges

This question tests your understanding of how to calculate the electric potential at a point due to multiple point charges, using the principle of superposition.

Key Terms and Formulas

• Electric potential due to a point charge: $V = \frac{q}{4\pi\varepsilon_0 r}$

• Superposition principle: The total potential at a point is the algebraic sum of the potentials due to each charge.

• $\varepsilon_0$ (epsilon naught): Permittivity of free space, $8.854 \times 10^{-12} \ \mathrm{C}^2/\mathrm{N}\cdot\mathrm{m}^2$

Step-by-Step Guidance

1. Identify the distance from each corner of the cube to the center. For a cube of side $a$, this distance is $r = \frac{a\sqrt{3}}{2}$.

2. Calculate the potential at the center due to one charge using $V_1 = \frac{q}{4\pi\varepsilon_0 r}$.

3. Since all charges are identical and equidistant from the center, multiply the potential from one charge by 8 to get the total potential.

4. Plug in the values: $q = 100 \times 10^{-9} \ \mathrm{C}$, $a = 0.5 \ \mathrm{m}$, $\varepsilon_0 = 8.854 \times 10^{-12} \ \mathrm{C}^2/\mathrm{N}\cdot\mathrm{m}^2$.

Try solving on your own before revealing the answer!

Final Answer: 4.2 × 104 Volts

By plugging in the values and summing the contributions from all 8 charges, you get approximately $4.2 \times 10^4$ V at the center.

Q2. Consider a charged line segment with charge density λ = 34 nano-Coulombs per meter extending along the positive x axis from the origin to the position x = 50 centimeters. At which of the following points does the electric field point parallel to the z axis?

Background

Topic: Electric Field due to a Line of Charge

This question tests your understanding of the symmetry of the electric field produced by a finite line of charge and how to determine the direction of the field at various points in space.

Key Terms and Formulas

• Linear charge density: $\lambda = \frac{dq}{dx}$

• Electric field due to a line of charge (by integration): $\vec{E} = \frac{1}{4\pi\varepsilon_0} \int \frac{\lambda dx}{r^2} \hat{r}$

• Symmetry: The direction of the electric field depends on the geometry and the observation point.

Step-by-Step Guidance

1. Visualize the line segment along the x-axis from $x = 0$ to $x = 0.5$ m.

2. Consider the points given: (A) $x = 1$, $y = 0$, $z = 0$; (B) $x = 0$, $y = 25$ cm, $z = 0$; (C) $x = 0$, $y = 0$, $z = 25$ cm.

3. Recall that the electric field points away from the line (for positive charge) and is perpendicular to the line at points along the z-axis.

4. Determine at which point the field has only a z-component (i.e., is parallel to the z-axis).

Try solving on your own before revealing the answer!

Final Answer: C. $x = 0$, $y = 0$, $z = 25$ cm

At this point, by symmetry, the electric field points directly along the z-axis.

Q3. Consider four point charges, each with charge equal to 7.0 nC, arranged in a square with side length 20 centimeters. What is the electric potential at the center of one of the sides of the square?

Background

Topic: Electric Potential from Multiple Point Charges

This question tests your ability to use the superposition principle to find the electric potential at a specific point due to several charges arranged in a square.

Key Terms and Formulas

• Electric potential due to a point charge: $V = \frac{q}{4\pi\varepsilon_0 r}$

• Superposition: Add the potentials from each charge at the point of interest.

Step-by-Step Guidance

1. Draw the square and label the positions of the charges and the point of interest (center of one side).

2. Calculate the distance from each charge to the center of the side.

3. Sum the potentials from all four charges at this point, using the formula for each distance.

4. Plug in the values: $q = 7.0 \times 10^{-9}$ C, $a = 0.2$ m, $\varepsilon_0 = 8.854 \times 10^{-12}$ C$^2$/N·m$^2$.

Try solving on your own before revealing the answer!

Final Answer: 216 Volts

By calculating the contributions from each charge and summing, you find the total potential at the specified point.