Midterm 1 Physics Study Guide: Gravitation, Coulomb's Law, Electric Field, and Gauss's Law

Gravitation

Newton's Law of Universal Gravitation

Gravitation describes the attractive force between any two masses. This fundamental force governs planetary motion and the structure of the universe.

• Gravitational Force: The force between two masses $m_1$ and $m_2$ separated by distance $r$ is given by: $F_g = G \frac{m_1 m_2}{r^2}$ where $G$ is the universal gravitational constant.

• Always Attractive: Gravitational force is always attractive, pulling masses together.

Shell Theorems

• Shell Theorem 1: Outside a uniform spherical shell of mass, the gravitational effect is as if all mass were concentrated at the center.

• Shell Theorem 2: Inside a uniform spherical shell of mass, the net gravitational force is zero.

Superposition Principle

• Forces Add Like Vectors: The net gravitational force on a body is the vector sum of all individual forces: $\vec{F}_{\text{net}} = \sum_{i=1}^n \vec{F}_i$

Gravity Near Earth's Surface

• Acceleration Due to Gravity: $a_g = G \frac{M_{\text{Earth}}}{r^2} \approx g$ where $g \approx 9.8\,\text{m/s}^2$ near Earth's surface.

Force Inside the Earth

• Gravitational Force at Distance $r$ from Center: $F = G \frac{M_{\text{Earth}} m}{R_{\text{Earth}}^3} r$

Gravitational Potential Energy

• Potential Energy Between Two Masses: $U = -G \frac{m_1 m_2}{r}$

Escape Velocity

• Minimum Velocity to Escape a Planet: $v_{\text{esc}} = \sqrt{\frac{2G M_{\text{planet}}}{R_{\text{planet}}}}$

Work and Energy

Work Done by a Force

Work is the energy transferred to or from an object via the application of force along a displacement.

• Definition: $W = \int \vec{F} \cdot d\vec{r}$

Energy of a System

• Energy: The ability to perform work. Types include kinetic (due to velocity) and potential (due to position or force).

• Spring Example: $U = \frac{1}{2} k x^2$

External Forces and Energy

• Work-Energy Relation: $W_{\text{ext}} = E_f - E_i = \Delta E$

Conservation of Energy

• Isolated Systems: $U_f + K_f = U_i + K_i$ Total energy remains constant if no external work is done.

Internal Forces and Energy

• Internal Work: Work done by internal forces does not change total energy, but can change potential or kinetic energy.

Key Principle

• Energy Differences Matter: Physics depends on energy differences, not absolute values. The origin of energies can be redefined as needed.

Coulomb's Law

Electric Charge and Force

Coulomb's law describes the force between two point electric charges.

• Charge Types: Electric charges come in two types: positive and negative.

• Force Between Charges: Like charges repel, opposite charges attract. $F = k \frac{q_1 q_2}{r^2}$ where $k$ is Coulomb's constant.

• Atoms: Consist of a positive nucleus and a negative electron cloud.

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

• Charging Objects: Objects can be charged by transferring charge or by induction.

• Conservation and Quantization: Electric charge is conserved and exists in discrete units (quantized).

Electric Field

Definition and Properties

The electric field describes the force per unit charge at a point in space due to electric charges.

• Electric Field Due to a Point Charge: $\vec{E} = k \frac{q_1}{r^2} \hat{r}$

• Electric Field Due to a Dipole: $E_z = k \frac{2p}{z^3}$ (along axis, far from dipole)

• Electric Field Due to a Line of Charge: $dE = k \frac{\lambda ds}{r^2}$, $E = \int dE$

• Force on a Charge in an Electric Field: $\vec{F} = q \vec{E}$

• Dipole in an Electric Field: $\vec{\tau} = \vec{p} \times \vec{E}$, $U(\theta) = -\vec{p} \cdot \vec{E} = pE \cos \theta$

Gauss's Law

Electric Flux

Electric flux quantifies the flow of the electric field through a surface.

• Definition: $\Phi = \int \vec{E} \cdot d\vec{A}$

Gauss's Law Statement

• Gauss's Law: $\Phi = \frac{q_{\text{encl}}}{\varepsilon_0}$ Relates the net electric flux through a closed surface to the enclosed charge.

Applications of Gauss's Law

• Symmetric Charge Configurations: Gauss's law is most useful for calculating electric fields in cases of high symmetry:

  • Spherical Symmetry: Point charges, spherical shells, solid spheres.

    • Outside a spherical shell: $E = k \frac{q}{r^2}$

    • Inside a spherical shell: $E = 0$

    • Inside a solid sphere: $E = k \frac{q r}{R^3}$

  • Cylindrical Symmetry: Very long charged wire with charge density $\lambda$: $E = \frac{\lambda}{2 \pi \varepsilon_0 r}$

  • Planar Symmetry: Field outside an infinite plane is uniform and orthogonal to the plane.

• Charged Isolated Conductor: Field inside a conductor is zero; outside, it is orthogonal to the surface. Net charge resides on the surface.

Summary Table: Key Equations and Concepts

Concept	Equation	Notes
Gravitational Force	$F_g = G \frac{m_1 m_2}{r^2}$	Always attractive
Gravitational Potential Energy	$U = -G \frac{m_1 m_2}{r}$	Negative sign indicates bound system
Escape Velocity	$v_{\text{esc}} = \sqrt{\frac{2G M}{R}}$	Minimum speed to escape gravity
Coulomb's Law	$F = k \frac{q_1 q_2}{r^2}$	Like charges repel, opposites attract
Electric Field (Point Charge)	$\vec{E} = k \frac{q}{r^2} \hat{r}$	Radial direction from charge
Electric Flux	$\Phi = \int \vec{E} \cdot d\vec{A}$	Flow of E-field through surface
Gauss's Law	$\Phi = \frac{q_{\text{encl}}}{\varepsilon_0}$	Relates flux to enclosed charge

Additional info:

• These notes cover topics from chapters 13, 21, 22, and 23, which correspond to Gravitation, Coulomb's Law, Electric Field, and Gauss's Law in a standard college physics curriculum.

• All equations are presented in LaTeX format for clarity and academic rigor.