Electrostatics: Forces, Fields, and Potentials – Study Notes

Electrostatics: Forces, Fields, and Potentials

Constants and Conversion Factors

In electrostatics problems, several physical constants and conversion factors are frequently used:

• Elementary charge: $e = 1.60 \times 10^{-19}\ \mathrm{C}$

• Electron mass: $m_e = 9.11 \times 10^{-31}\ \mathrm{kg}$

• Proton mass: $m_p = 1.67 \times 10^{-27}\ \mathrm{kg}$

• Coulomb's constant: $k_e = 8.85 \times 10^9\ \mathrm{N\cdot m^2/C^2}$

Electrostatic Forces and Free Body Diagrams

Forces Between Point Charges

Electrostatic forces between point charges are governed by Coulomb's Law. The direction and magnitude of these forces depend on the sign and magnitude of the charges involved.

• Coulomb's Law: The force between two point charges $q_1$ and $q_2$ separated by a distance $r$ is given by:

$F = k_e \frac{|q_1 q_2|}{r^2}$

• Direction: Like charges repel; unlike charges attract.

• Superposition Principle: The net force on a charge is the vector sum of the forces exerted by all other charges.

Example: Three Charges in a Triangle

• Given three charges arranged in a triangle, the net force on one charge can be found by resolving the forces from the other two into components and summing them.

• For the net force to be vertical, the horizontal components from the other two charges must cancel.

• If $q_3$ is negative and the net force is vertical, $q_1$ must be positive and $q_2$ negative (or vice versa, depending on the arrangement).

Sample Calculation

• For charges $q_1$ and $q_2$ equidistant from $q_3$, and net force vertical:

$\sum F = m a = 2k_e \frac{q_1 q_3}{r^2} \cos(\theta)$

$q_1 = q_2 = \frac{ma r^2}{2k_e q_3 \cos(\theta)}$

Continuous Charge Distributions

Linear Charge Density

For a rod of length $a$ and total charge $+Q$, the linear charge density $\lambda$ is:

$\lambda = \frac{Q}{a}$

Electric Field from a Charged Rod

• To find the electric field at a point $P$ due to a rod, divide the rod into infinitesimal segments $dy$.

• The charge on a segment is $dQ = \lambda dy$.

• The distance from $dy$ to $P$ is $r = \sqrt{y^2 + d^2}$.

• The infinitesimal electric field at $P$ is:

$dE = k_e \frac{dQ}{r^2} = k_e \frac{Q}{a} \frac{dy}{y^2 + d^2}$

• To find the $y$-component, use $\sin(\theta) = \frac{y}{\sqrt{y^2 + d^2}}$:

$dE_y = k_e \frac{Q}{a} \frac{y}{(y^2 + d^2)^{3/2}} dy$

$E_y = k_e \frac{Qd}{a} \int_{0}^{a} \frac{dy}{(y^2 + d^2)^{3/2}}$

Gauss's Law and Spherical Charge Distributions

Surface and Volume Charge Densities

• Surface charge density on a shell of radius $a$ with charge $Q$:

$\sigma = \frac{Q}{4\pi a^2}$

• Volume charge density for a shell with inner radius $b$ and outer radius $c$ and total charge $Q$:

$\rho = \frac{Q}{\frac{4}{3}\pi (c^3 - b^3)}$

Electric Field Using Gauss's Law

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

$\oint \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\epsilon_0}$

• For $r < a$ (inside a conducting shell): $E = 0$

• For $a < r < b$ (between shells): $E = \frac{1}{4\pi\epsilon_0} \frac{Q_{inner}}{r^2}$

• For $b < r < c$ (inside the outer shell): $E = \frac{1}{4\pi\epsilon_0} \frac{Q_{inner} + Q_{outer}(r^3 - b^3)/(c^3 - b^3)}{r^2}$ Additional info: The last formula is inferred for a shell with non-uniform charge distribution.

• For $r > c$ (outside both shells): $E = \frac{1}{4\pi\epsilon_0} \frac{Q_{total}}{r^2}$

Table: Electric Field in Spherical Shells

Region	Electric Field $E$
$r < a$	$0$
$a < r < b$	$\frac{1}{4\pi\epsilon_0} \frac{Q_{inner}}{r^2}$
$b < r < c$	$\frac{1}{4\pi\epsilon_0} \frac{Q_{inner} + Q_{outer}(r^3 - b^3)/(c^3 - b^3)}{r^2}$ Additional info: Formula inferred for non-uniform shell.
$r > c$	$\frac{1}{4\pi\epsilon_0} \frac{Q_{total}}{r^2}$

Electric Potential

Potential Due to Point Charges

The electric potential $V$ at a point due to a point charge $q$ at distance $r$ is:

$V = k_e \frac{q}{r}$

• For multiple charges, potentials add algebraically:

$V_{total} = \sum_i k_e \frac{q_i}{r_i}$

Potential Energy and Conservation of Energy

When a charge moves in an electric field, its change in potential energy is related to the change in electric potential:

$\Delta U = q \Delta V$

By conservation of energy:

$\Delta K = -\Delta U$

or

$\frac{1}{2} m v^2 = q \Delta V$

Zero Electric Field vs. Zero Potential

• If the electric field is zero at a point, the potential is constant in that region, but not necessarily zero.

• It is possible for the electric field to be zero at a point where the potential is non-zero, depending on the charge configuration.

Summary Table: Key Electrostatics Formulas

Quantity	Formula
Force between point charges	$F = k_e \frac{|q_1 q_2|}{r^2}$
Electric field (point charge)	$E = k_e \frac{|q|}{r^2}$
Electric potential (point charge)	$V = k_e \frac{q}{r}$
Gauss's Law	$\oint \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\epsilon_0}$
Surface charge density	$\sigma = \frac{Q}{A}$
Volume charge density	$\rho = \frac{Q}{V}$

Applications and Problem-Solving Tips

• Always draw a free body diagram for force problems.

• Use symmetry to simplify calculations for fields and potentials.

• For continuous charge distributions, set up integrals carefully and identify limits.

• Check units and signs for all quantities.