Electrostatics: Charge Distributions, Electric Fields, and Gauss's Law – Study Notes

Electrostatics: Charge Distributions, Electric Fields, and Gauss's Law

Instructions and Exam Policy

This section outlines the rules for a closed-book, closed-notes physics exam. Students are required to show all work, including diagrams and equations, to receive full credit. Only calculators are allowed as electronic devices, and answers must be clearly circled or boxed.

Charge Distributions and Surface Charge Density

Surface Charge and Total Charge

When a large number of protons are spread evenly over a thin circular sheet, the total charge can be calculated by multiplying the number of protons by the elementary charge:

• Total charge (q): $q = N \times e$

• Example: $q = (2.50 \times 10^{12})(1.6 \times 10^{-19}\ \mathrm{C}) = 4 \times 10^{-7}\ \mathrm{C} = 0.4\ \mu\mathrm{C}$

Surface Charge Density

The surface charge density ($\sigma$) is the charge per unit area on a surface:

• $\sigma = \frac{q}{A}$

• For a circular sheet: $A = \pi r^2$

• Example: $\sigma = \frac{4 \times 10^{-7}\ \mathrm{C}}{78.5\ \mathrm{m}^2} = 5.1 \times 10^{-9}\ \mathrm{C/m}^2$

Electric Field Near a Charged Sheet

The electric field very close to a large charged sheet is given by:

• $E = \frac{\sigma}{2\varepsilon_0}$

• Where $\varepsilon_0$ is the vacuum permittivity ($8.85 \times 10^{-12}\ \mathrm{C}^2/\mathrm{N}\cdot\mathrm{m}^2$)

• Example: $E = \frac{5.1 \times 10^{-9}}{2 \times 8.85 \times 10^{-12}} = 288\ \mathrm{N/C}$

Electric Force and Field from Point Charges

Electric Field from a Point Charge

The electric field ($\vec{E}$) at a distance $r$ from a point charge $q$ is:

• $\vec{E} = \frac{1}{4\pi\varepsilon_0} \frac{q}{r^2} \hat{r}$

Superposition Principle

The net electric field at a point due to multiple charges is the vector sum of the fields from each charge.

• For charges at the corners of a square, use symmetry and vector addition to find the net field at the center.

• Example: For two positive and two negative charges at the corners, the net field at the center can be found by summing the $x$ and $y$ components.

Electric Force on a Charge

The force ($\vec{F}$) on a charge $q$ in an electric field $\vec{E}$ is:

• $\vec{F} = q\vec{E}$

• Direction depends on the sign of $q$.

Electric Field Diagrams and Qualitative Analysis

Interpreting Field Diagrams

• Field lines point away from positive charges and toward negative charges.

• The density of field lines indicates the strength of the field.

• Comparing field strengths at different points can be done by counting the number of lines per area.

Gauss's Law and Symmetry

Gauss's Law

Gauss's Law relates the electric flux through a closed surface to the charge enclosed:

• $\Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{q_{\text{enc}}}{\varepsilon_0}$

Application to Planar Symmetry

• For an infinite sheet: $E = \frac{\sigma}{2\varepsilon_0}$ (on either side of the sheet)

• For a charged cylinder: Use a cylindrical Gaussian surface to find $E$ inside and outside.

Electric Field of a Uniformly Charged Rod

To find the field at a point along the axis of a uniformly charged rod, integrate the contributions from each segment:

• $dE = \frac{1}{4\pi\varepsilon_0} \frac{dq}{(d + x)^2}$

• Integrate over the length of the rod to find the total field.

Electric Flux Through Surfaces

Definition of Electric Flux

Electric flux ($\Phi$) through a surface is:

• $\Phi = \vec{E} \cdot \vec{A}$

• For a flat surface perpendicular to $\vec{E}$: $\Phi = EA$

• Example: For $E = 20\ \mathrm{N/C}$ and $A = 2\ \mathrm{m}^2$, $\Phi = 40\ \mathrm{N}\cdot\mathrm{m}^2/\mathrm{C}$

Potential Energy and Work in Electric Fields

Electric Potential Difference

The potential difference ($\Delta V$) between two points in a uniform field is:

• $\Delta V = -Ed$

• Where $d$ is the distance in the direction of the field.

Work and Kinetic Energy

The work done by the electric field on a charge $q$ moving through a potential difference $\Delta V$ is:

• $\Delta U = q\Delta V$

• Conservation of energy: $\Delta E = \Delta K + \Delta U = 0$ (if no non-conservative forces)

• Minimum speed for an electron to reach a point: $v = \sqrt{\frac{2q\Delta V}{m}}$

Potential Energy of Charge Configurations

Pairwise Potential Energy

The potential energy between two point charges $q_1$ and $q_2$ separated by distance $r$:

• $U = k_e \frac{q_1 q_2}{r}$

• For multiple charges, sum over all unique pairs.

Electric Field of Cylindrical Charge Distributions

Field Inside and Outside a Cylinder

• For $r < R$ (inside): $E = \frac{\rho r}{2\varepsilon_0}$

• For $r > R$ (outside): $E = \frac{\rho R^2}{2\varepsilon_0 r}$

• Where $\rho$ is the volume charge density.

Conducting vs. Insulating Cylinders

• Inside a conductor, the electric field is zero.

• Outside, the field is the same as for an insulator with the same total charge.

Summary Table: Key Electrostatics Formulas

Quantity	Formula	Description
Surface charge density	$\sigma = \frac{q}{A}$	Charge per unit area
Electric field (point charge)	$E = \frac{1}{4\pi\varepsilon_0} \frac{q}{r^2}$	Field at distance $r$ from charge $q$
Electric field (infinite sheet)	$E = \frac{\sigma}{2\varepsilon_0}$	Field near a large charged sheet
Electric flux	$\Phi = \vec{E} \cdot \vec{A}$	Flux through a surface
Potential energy (pair of charges)	$U = k_e \frac{q_1 q_2}{r}$	Energy between two point charges
Gauss's Law	$\oint \vec{E} \cdot d\vec{A} = \frac{q_{\text{enc}}}{\varepsilon_0}$	Relates flux to enclosed charge

Additional info:

• Some context and explanations have been expanded for clarity and completeness.

• Examples and formula derivations are based on standard undergraduate physics curriculum.