Electric Charge, Electric Field, and Gauss's Law: Exam 1 Review Notes

Electric Charge and Electric Force

Fundamental Properties of Electric Charge

Electric charge is a fundamental property of matter that gives rise to electric forces and fields. There are two types of charge: positive and negative. The elementary charge is denoted by e and has a value of $e = 1.602 \times 10^{-19}$ C.

• Conservation of Charge: The total electric charge in an isolated system remains constant.

• Quantization of Charge: Charge exists in discrete packets, multiples of the elementary charge.

• Conductors vs. Insulators: Conductors allow free movement of charge; insulators do not.

Example: Electrons carry negative charge, protons carry positive charge.

Electric Force Between Point Charges

The force between two point charges is described by Coulomb's Law:

• Coulomb's Constant: $k = \frac{1}{4\pi\epsilon_0} = 8.988 \times 10^9$ N m2 C-2

• Force Equation:

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

• If charges have the same sign, the force is repulsive.

• If charges have opposite signs, the force is attractive.

Example: Two electrons separated by 1 nm repel each other with a force calculated using the above formula.

Electric Field

Definition and Properties

The electric field at a point in space is defined as the force per unit charge experienced by a small positive test charge placed at that point:

$\vec{E} = \frac{\vec{F}}{q}$

• Direction: The direction of the electric field is the direction of the force on a positive test charge.

• Units: Newtons per Coulomb (N/C).

• Electric Field Lines: Visual representations showing the direction and strength of the field; lines begin on positive charges and end on negative charges.

Electric Field Due to Point Charges

The magnitude of the electric field produced by a point charge at a distance r is:

$E = \frac{1}{4\pi\epsilon_0} \frac{|q|}{r^2}$

• For multiple point charges, the net electric field is the vector sum:

$\vec{E}_{net} = \sum_i \vec{E}_i$

Electric Field Due to Continuous Charge Distributions

For a continuous distribution, the net electric field is found by integrating:

$\vec{E}_{net} = \frac{1}{4\pi\epsilon_0} \int \frac{dq}{r^2} \hat{r}$

• Line charge: $dq = \lambda d\ell$

• Surface charge: $dq = \sigma dA$

• Volume charge: $dq = \rho dV$

Special Charge Distributions

• Uniformly Charged Sphere (outside): $E = \frac{1}{4\pi\epsilon_0} \frac{|Q|}{r^2}$ for $r \geq R$

• Uniformly Charged Insulating Sphere (inside): $E = \frac{1}{4\pi\epsilon_0} \frac{|Q|}{R^3} r$ for $r \leq R$

• Conducting Sphere (inside): $E = 0$

• Infinite Line of Charge: $E = \frac{1}{2\pi\epsilon_0} \frac{|\lambda|}{r}$

• Infinite Sheet of Charge: $E = \frac{|\sigma|}{2\epsilon_0}$

• Parallel Plates: $E = \frac{|\sigma|}{\epsilon_0}$

Gauss's Law

Electric Flux

Electric flux through a surface quantifies the number of electric field lines passing through that surface:

$\Phi_E = \int \vec{E} \cdot d\vec{A}$

• Area Vector: Perpendicular to the surface; for closed surfaces, points outward.

Gauss's Law Statement

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

$\Phi_{net} = \oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$

• Useful for calculating electric fields of symmetric charge distributions.

Conductors and Insulators in Electric Fields

Properties of Conductors

• Interior Field: The electric field inside an isolated conductor is always zero.

• Surface Charge: Any net charge resides on the outer surface.

• Surface Field: The electric field just outside the surface is perpendicular and has magnitude $E = \frac{|\sigma|}{\epsilon_0}$, where $\sigma = \frac{q}{A}$.

Uniform Electric Fields

Charged Particle in Uniform Field

A charged particle in a uniform electric field experiences a constant acceleration:

$\vec{a} = \frac{q}{m} \vec{E}$

• Direction depends on the sign of the charge.

Electric Dipole in Uniform Field

• Net Force: $F = 0$ (no net force on the dipole as a whole)

• Net Torque: $\vec{\tau} = \vec{p} \times \vec{E}$, where $\vec{p}$ is the dipole moment ($p = qd$)

• Potential Energy: $U = -\vec{p} \cdot \vec{E}$

Example: A water molecule (electric dipole) aligns with an external electric field due to torque.

Summary Table: Electric Field Formulas for Common Charge Distributions

Charge Distribution	Electric Field Magnitude	Direction
Point Charge	$E = \frac{1}{4\pi\epsilon_0} \frac{|q|}{r^2}$	Radially outward (positive), inward (negative)
Uniformly Charged Sphere (outside)	$E = \frac{1}{4\pi\epsilon_0} \frac{|Q|}{r^2}$	Radially outward/inward
Uniformly Charged Insulating Sphere (inside)	$E = \frac{1}{4\pi\epsilon_0} \frac{|Q|}{R^3} r$	Radially outward/inward
Conducting Sphere (inside)	$E = 0$	None
Infinite Line of Charge	$E = \frac{1}{2\pi\epsilon_0} \frac{|\lambda|}{r}$	Perpendicular to line
Infinite Sheet of Charge	$E = \frac{|\sigma|}{2\epsilon_0}$	Perpendicular to sheet
Parallel Plates	$E = \frac{|\sigma|}{\epsilon_0}$	From positive to negative plate

Key Constants

• Elementary charge: $e = 1.602 \times 10^{-19}$ C

• Electric constant (permittivity of free space): $\epsilon_0 = 8.854 \times 10^{-12}$ C2 N-1 m-2

• Coulomb's constant: $k = \frac{1}{4\pi\epsilon_0} = 8.988 \times 10^9$ N m2 C-2

Additional info:

• These notes cover the foundational concepts of Chapters 21 and 22: Electric Charge, Electric Field, and Gauss's Law, as outlined in the review document.

• All equations are presented in LaTeX format for clarity and future use.