Study guide: Electric Fields and Forces

📋 Course Outline

1. Electric Charge and Conservation
2. Charge Quantization and Materials
3. Charging by Induction
4. Coulomb’s Law and Forces
5. Electric Field and Superposition
6. Continuous Charge Distributions
7. Electric Field Lines
8. Motion in an Electric Field

📖 1. Electric Charge and Conservation

★ Must-know

• Electric charge has two signs, positive and negative; electrons carry negative charge and protons carry positive charge.

📌 Charges with the same sign repel one another, whereas charges with opposite signs attract one another.

📌 Electric charge is conserved in an isolated system and is transferred, not created, when objects are rubbed together.

Further detail

• When a glass rod is rubbed with silk, electrons move from the glass to the silk, leaving the silk negatively charged and an equal positive charge on the rod.

💡 Memory Hook

Like signs repel; unlike signs attract

📖 2. Charge Quantization and Materials

🔑 Key Concepts & Definitions

• Electrical conductors: Materials with some electrons not bound to atoms, allowing those electrons to move relatively freely through the material.
• Electrical insulators: Materials in which all electrons are bound to atoms and cannot move relatively freely through the material.

★ Must-know

📐 Formula - Electric charge is quantized according to $q=Ne$, where $N$ is an integer and $e$ is the fundamental unit of charge.

• The magnitude of the fundamental charge is $|e|=1.6\times10^{-19}\,\mathrm{C}$; an electron has charge $-e$ and a proton has charge $+e$.

Further detail

• Examples of good conductors are:

  • copper
  • aluminum
  • silver

• Examples of good insulators are:

  • glass
  • rubber
  • wood

• Semiconductors have electrical properties between those of conductors and insulators; silicon and germanium are examples.

💡 Memory Hook

Conductors let charge move; insulators hold it

📖 3. Charging by Induction

🔑 Key Concepts & Definitions

• Charging by induction: The charging of an object without contact with the object that induces the charge.

★ Must-know

• 🔄 A neutral sphere becomes positively charged by induction through these steps:
  1. Bring a charged rubber rod near the sphere without touching it.
  2. Ground the sphere so some electrons can leave through the ground wire.
  3. Remove the ground wire.
  4. Remove the rod; the remaining electrons redistribute and the sphere retains a net positive charge.

Further detail

• In an insulator, a process similar to induction rearranges charges within the material’s molecules.

💡 Memory Hook

Bring the rod near, ground the sphere, remove ground, then remove the rod

📖 4. Coulomb’s Law and Forces

🔑 Key Concepts & Definitions

• Point charge: A particle of zero size that carries electric charge; electrons and protons are often modeled as point charges.

★ Must-know

📐 Formula - The magnitude of the electric force between two stationary point charges is given by $F_e=k_e\frac{|q_1q_2|}{r^2}$, where $r$ is their separation.

• Coulomb’s force acts along the line joining the charges, is proportional to the product of their charge magnitudes, and is inversely proportional to the square of their separation.

📌 Coulomb’s force is attractive for opposite-sign charges and repulsive for like-sign charges, and it is a conservative force.

• For several source charges, find each force on the charge of interest and add the forces as vectors to obtain the resultant force.

Further detail

• The Coulomb constant is $k_e=8.9875\times10^9\,\mathrm{N\,m^2/C^2}=1/(4\pi\epsilon_0)$, where the permittivity of free space is $\epsilon_0=8.8542\times10^{-12}\,\mathrm{C^2/(N\,m^2)}$.

• The SI unit of electric charge is the coulomb (C), and one coulomb corresponds to approximately $6.24\times10^{18}$ electrons or protons.

📌 The electric forces between two charges obey Newton’s third law: $\vec F_{21}=-\vec F_{12}$.

• For an electron and proton in a hydrogen atom, the electrical force is $8.2\times10^8\,\mathrm{N}$, whereas the gravitational force is $3.6\times10^{-47}\,\mathrm{N}$.

📖 5. Electric Field and Superposition

🔑 Key Concepts & Definitions

• Electric field: The region around a source charge in which another charge experiences an electric force.

★ Must-know

📐 Formula - The electric field at a point is the electric force on a positive test charge divided by the test charge: $\vec E=\frac{\vec F_e}{q_0}$.

📌 A positive charge experiences force in the direction of the electric field, whereas a negative charge experiences force opposite the field.

📌 At any point, the total electric field due to multiple source charges is the vector sum of the fields produced by the individual charges: $\vec E=\sum_i\vec E_i$.

Further detail

• An electric field is a property of its source charge and exists without a test charge; the test charge detects the field.

• The SI unit of electric field is the newton per coulomb (N/C), and its direction is the direction of the force on a positive test charge.

📐 Formula - The electric field of a point source charge is $\vec E=k_e\frac{q}{r^2}\hat r$, with direction determined by the source charge and the unit vector along the radial line.

💡 Memory Hook

The source creates the field; the test charge detects it

📖 6. Continuous Charge Distributions

🔑 Key Concepts & Definitions

• Continuous charge distribution: A model of closely spaced charges as charge distributed along a line, over a surface, or throughout a volume.

★ Must-know

• 🔄 Calculate the field of a continuous distribution as follows:
  1. Divide the distribution into small charge elements.
  2. Calculate the field due to each element at the point of interest.
  3. Integrate the contributions over the distribution.

📐 Formula - For uniform charge distributions, volume, surface, and linear charge densities are $\rho=Q/V$, $\sigma=Q/A$, and $\lambda=Q/\ell$, respectively.

Further detail

📐 Formula - For a charge element $dq$ at distance $r$, its field contribution is $d\vec E=k_e\frac{dq}{r^2}\hat r$, and the total field is $\vec E=k_e\int\frac{dq}{r^2}\hat r$.

📐 Formula - The charge in a small volume, surface area, or length element is $dq=\rho\,dV$, $dq=\sigma\,dA$, or $dq=\lambda\,d\ell$, respectively.

• Coulomb’s constant calculations require charges in coulombs and distances in meters; continuous distributions require vector integration, and symmetry can simplify the calculation.

• For a uniformly charged disk, the distribution can be divided into rings of radius $r$ and surface area $2\pi r\,dr$, with the disk radius denoted by $R$.

💡 Memory Hook

Divide into elements, find each contribution, then integrate

📖 7. Electric Field Lines

🔑 Key Concepts & Definitions

• Electric field lines: A visual representation of an electric field, with the field vector tangent to each line and directed along it.

★ Must-know

📌 The number of field lines per unit area through a surface perpendicular to them is proportional to the electric-field magnitude there.

• Field lines radiate outward from a positive point charge and inward toward a negative point charge.

• The rules for drawing electric field lines are:

  • Lines begin on positive charges and terminate on negative charges.
  • When one sign of charge is in excess, some lines begin or end infinitely far away.
  • The number of lines leaving a positive charge or approaching a negative charge is proportional to the charge magnitude.
  • No two field lines cross.

Further detail

• For a dipole with equal and opposite charges, the number of field lines leaving the positive charge equals the number terminating on the negative charge.

• For equal positive charges, the same number of field lines leave each charge, and at a great distance their field is approximately that of a single charge of $2q$.

• When the positive charge has twice the magnitude of the negative charge, two field lines leave the positive charge for each line terminating on the negative charge, and the distant field is approximately that of a single charge of $+q$.

📖 8. Motion in an Electric Field

★ Must-know

📐 Formula - A charged particle in an electric field experiences force and, when this is the only force, obeys $\vec F_e=q\vec E=m\vec a$.

• In a uniform electric field, a charged particle has constant acceleration, so the kinematic equations can be used to describe its motion.

📌 A positive particle accelerates in the direction of the electric field, whereas a negative particle accelerates opposite the field.

Further detail

• An electron projected horizontally into a uniform electric field between plates accelerates opposite the field and follows a parabolic path between the plates.

💡 Memory Hook

Positive particles accelerate with the field; negative particles against it