Coulomb’s Law and Electric Fields (Ch. 21–23)

21 Coulomb’s Law

• Electric charge is the property of fundamental particles that can be positive or negative. Same-sign charges repel; opposite-sign charges attract.

• Neutral objects have equal numbers of positive and negative charges; excess charge means an imbalance (net charge).

• Conductors: many electrons are free to move. Insulators (nonconductors) have bound charges and little to no free movement of charge.

• Particles inside atoms: protons (positive), electrons (negative), neutrons (neutral); electrons are the conduction electrons in metals.

• Free-body/vector treatment: when multiple electrostatic forces act on a particle, the net force is the vector sum of the individual forces.

• Grounding and induction: grounding provides a conducting path to Earth so excess charge can spread away; induction can rearrange charges in a nearby object without direct contact.

• Conducting spheres: excess charge on a spherical conductor distributes uniformly on the external surface.

• Shell theorems (analogous to gravity):

  • Shell outside a uniformly charged surface acts as if all charge were at the center (attracts/repels external charges).

  • Shell inside a uniformly charged shell exerts no net force on an internal charge.

• Charge on a conductor in contact spreads to the surface; nonconductors can hold charge in the interior.

• Electric current i is the rate of charge flow through a point: $i= rac{dq}{dt}.$

• Electric charge units: the coulomb (C); a current of 1 A corresponds to 1 C moving per second.

• Coulomb’s law describes the electrostatic force between two point charges, q1 and q2, separated by distance r:

  • Vector form: $\boxed{\mathbf{F}=k\frac{q<em>1 q</em>2}{r^2}\,\hat{\mathbf{r}}}$

  • Magnitude form: $F= k\frac{|q<em>1 q</em>2|}{r^2},$ where $k=\frac{1}{4\pi \epsilon_0}=8.99\times 10^9\ \mathrm{N\,m^2/C^2}.$

  • Permittivity of free space: $\epsilon_0=8.85\times 10^{-12}\ \mathrm{C^2/(N\,m^2)}.$

• Direction of the Coulomb force aligns with the line joining the charges: opposite signs attract; same signs repel.

• For multiple charges, the net force on a particle is the vector sum of the forces from all other charges (superposition).

• Unit/quantity relations:

  • The force magnitude relates to the charge magnitudes and inverse square of distance: $F\propto\frac{q<em>1 q</em>2}{r^2}.$

  • Current and charge transport link to Coulomb’s law via the relationship between charge, current, and time.

• Example charges and problems: quantification of how charges distribute on shells and conductors; charges on spherical conductors spread uniformly on the exterior, while internal regions of conductors have zero excess charge.

• Charge quantization and elementary charge:

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

  • Any charge q can be written as $q=n e,\quad n\in\mathbb{Z}.$

  • Millikan’s oil-drop experiment established charge quantization: $q=ne\quad (n=0,\pm1,\pm2,\dots).$

• Charge conservation: the net charge of an isolated system is conserved. Examples include annihilation ($e^-+e^+\to\gamma+\gamma$) and pair production (\gamma \to e^-+e^+).

• Nucleus-scale considerations compare electromagnetic and nuclear forces, e.g., the strong nuclear force binds protons together in nuclei against the large electrostatic repulsion; the gravitational force is negligible at nuclear scales.

21-2 CHARGE IS QUANTIZED

• Charge is quantized: $q= ne$ with $e\approx 1.602\times 10^{-19}\ \mathrm{C}.$

• The elementary charge is the fundamental charge unit for electrons and protons; quarks carry fractional charges but are not isolated.

• In everyday macroscopic phenomena, the discrete nature of charge is not noticeable because of the enormous number of charges involved.

• Practical implication: charge on particles and objects comes in discrete multiples of the elementary charge, not continuous values.

21-3 CHARGE IS CONSERVED

• Net charge is conserved in isolated processes: the algebraic sum of charges before and after any process remains the same.

• Annihilation: a particle and its antiparticle annihilate to radiation, conserving charge; pair production: a photon can produce a particle–antiparticle pair, conserving charge.

• Nuclear decay exemplifies charge conservation (e.g., uranium-238 decay via emission of an alpha particle 4He, which carries 2 protons, leaving daughter with adjusted proton count but total charge conserved).

• The total charge of a closed system remains constant even if charges move internally; external charges can influence distributions but not the net enclosed charge on a closed surface.

22 Electric Fields

22-1 What Is the Electric Field?

• An electric field is a vector field set up by charges; at each point, it has a magnitude and direction and exerts a force on a test charge placed there.

• Definition via a test charge $q<em>0$: $\mathbf{E}=\frac{\mathbf{F}}{q</em>0}.$

• The field is defined by the force that would act on a positive test charge placed at that point; a positive test charge experiences a force in the same direction as the field, a negative test charge in the opposite direction.

• Electric field lines visualize the field: lines originate on positive charges and terminate on negative charges; density of lines indicates field strength; the field vector at any point is tangent to the line there.

• SI unit: $\mathrm{N/C}$ (equivalently, $\mathrm{V/m}$).

• Superposition principle applies to fields: the net field from multiple charges is the vector sum of the fields from each charge.

• For a point charge q at distance r, the field is
  $\boxed{\mathbf{E}=\frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}}$

• For a positive charge, field lines radiate outward; for a negative charge, field lines converge inward.

22-2 The Electric Field Due to a Point Charge

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

• Direction: away from a positive charge; toward a negative charge.

• The field due to several charges is the vector sum of the individual fields (superposition): $\mathbf{E}<em>{\text{net}} = \sum</em>i \mathbf{E}_i.$

• The test-charge method can be used to determine the field by placing a small test charge and measuring the force per unit charge.

22-3 THE ELECTRIC FIELD DUE TO A DIPOLE

• An electric dipole consists of two charges of equal magnitude $q$ but opposite sign, separated by distance $d$; dipole moment $\mathbf{p}=q\,\mathbf{d}$ points from negative to positive charge.

• On the axis of the dipole, for large distances (z >> d), the field is
  $\boxed{E<em>z \approx \frac{1}{4\pi\epsilon</em>0}\frac{2p}{z^3}}$ directed along the dipole moment when measured far along the axis.

• At distant points along the dipole axis, $E$ falls as $\propto \frac{1}{z^3}$, which is faster than the $1/r^2$ fall-off for a single charge, because the two charges’ fields cancel partially at large distances.

• The orientation of the field is along the dipole moment for distant points on the axis; in general, the dipole’s field pattern resembles two separated charges with nearly cancelling fields at far distances.

• Torque on a dipole in a uniform external field: $\boldsymbol{\tau}=\mathbf{p}\times\mathbf{E}$ with magnitude $|\boldsymbol{\tau}|=pE\sin\theta$. In magnitude-angle form, $\boldsymbol{\tau}=-\,pE\sin\theta\,\hat{n}$ where $ heta$ is the angle between $\mathbf{p}$ and $\mathbf{E}$.

• Potential energy of a dipole in a field: $U=-\mathbf{p}\cdot\mathbf{E}=-pE\cos\theta.$

• Work-energy relation: rotation by torque changes potential energy by $\Delta U$, and the work done by the field is $W= -\Delta U$ (external work has the opposite sign).

• Applications: dielectric behavior, microwave heating (dipoles rotate in oscillating fields, converting field energy to thermal energy).

22-4 THE ELECTRIC FIELD DUE TO A LINE OF CHARGE

• A line (rod) of charge with linear density $\lambda$: use symmetry and integration.

• Field magnitude outside a long line: $\boxed{E=\frac{\lambda}{2\pi\epsilon_0 r}}$ where $r$ is the radial distance from the line; direction is perpendicular to the line and radial.

• To handle finite lines, integrate along the line; symmetry allows cancellation of transverse components, leaving a net field along the axis perpendicular to the line.

• For a uniform line density, the total charge $Q$ along a segment of length $L$ is $Q=\lambda L$.

22-5 THE ELECTRIC FIELD DUE TO A CHARGED DISK

• On the axis of a uniformly charged disk of radius $R$ and surface charge density $\sigma$, at distance $z$ from the center:
  $\boxed{E(z)=\frac{\sigma}{2\epsilon_0}\left(1-\frac{z}{\sqrt{z^2+R^2}}\right)}.$

• Limiting cases:

  • As $R\to\infty$, the disk approaches a nonconducting infinite sheet, yielding $E=\frac{\sigma}{2\epsilon_0}.$

  • At the surface $z=0$, $E=\frac{\sigma}{2\epsilon_0}.$

• The disk field reduces to the point-charge field at large $z$, since the disk looks like a point charge of $Q=\sigma\pi R^2$ at large distances: $E \approx \frac{1}{4\pi\epsilon_0}\frac{Q}{z^2}.$

22-6 A POINT CHARGE IN AN ELECTRIC FIELD

• A charge $q$ placed in an external electric field $\mathbf{E}$ experiences a force
  $\boxed{\mathbf{F}=q\mathbf{E}}.$

• Sign rule: If $q>0$, force direction is the same as $\mathbf{E}$; if $q<0$, force is opposite to $\mathbf{E}$.

• Millikan’s experiment established the elementary charge $e$ is quantized: $q=ne,\quad n\in \mathbb{Z}.$

• The external field is the field produced by other charges; a charge does not respond to its own field.

22-7 A DIPOLE IN AN ELECTRIC FIELD

• In a uniform external field, a dipole experiences a torque that tends to align the dipole with the field:
  $\boldsymbol{\tau}=\mathbf{p}\times\mathbf{E},\qquad |\boldsymbol{\tau}|=pE\sin\theta.$

• Potential energy associated with orientation: $U=-\mathbf{p}\cdot\mathbf{E}=-pE\cos\theta.$

• Work done by the field when rotating from angle $\thetai$ to $\thetaf$ is $W=-\Delta U$.

• Practical example: dielectric rotation in external fields; orientation affects energy and torque.

22-8 REVIEW AND CONNECTIONS

• Comparison of forces and fields: force is due to the field; field is the mediator of force.

• Superposition underpins complex charge distributions and field calculations.

• Field visualization via field lines provides intuition for direction and relative magnitude.

• The concept of a test charge is central to defining and measuring fields.

23 Gauss’ Law

23-1 ELECTRIC FLUX

• Flux through a surface is the measure of how much field “pierces” the surface.

• For a patch with area element $dA$ and area vector $\hat{n}dA$, the differential flux is
  $d\Phi_E = \mathbf{E}\cdot d\mathbf{A} = \mathbf{E}\cdot \hat{n}dA = E\cos\theta\,dA.</p></li><li><p>Uniform fields and flat surfaces simplify to$\Phi_E = E\cos\theta\,A.$</p></li><li><p>Flux through a closed surface is the sum (integral) of all patch fluxes and is denoted by$\Phi_E=\oint\mathbf{E}\cdot d\mathbf{A}.$</p></li><li><p>The orientation convention: outward normal on a closed surface yields positive flux; inward flux is negative.</p></li><li><p>Gauss’ law relates the net flux through a closed surface to the enclosed charge:<br>$\boxed{\oint\mathbf{E}\cdot d\mathbf{A} = \frac{q{\text{enc}}}{\epsilon0}}.$</p></li><li><p>In vacuum, this relation holds; in materials it generalizes with the appropriate dielectric constants (Chapter 25).</p></li></ul><h5 id="0372b59b-05d7-4e6f-9eeb-a63ce4b34299" data-toc-id="0372b59b-05d7-4e6f-9eeb-a63ce4b34299" collapsed="false" seolevelmigrated="true">23-2 GAUSS’ LAW</h5><ul><li><p>Rewriting Gauss’ law using the flux definition:$\oint\mathbf{E}\cdot d\mathbf{A} = \frac{q{\text{enc}}}{\epsilon0}.

• In highly symmetric situations, Gauss’ law provides powerful shortcuts to find $E$ without performing difficult integrals.

• Examples illustrate the choice of Gaussian surfaces that exploit symmetry (spherical, cylindrical, planar).

23-3 A CHARGED ISOLATED CONDUCTOR

• Excess charge on an isolated conductor resides on its outer surface; the field inside is zero.

• For a conducting surface, just outside the surface the field is perpendicular to the surface and has magnitude $E = \sigma/\epsilon_0$ where $\sigma$ is the surface charge density.

• If a cavity exists inside a conductor and contains charge, the induced charges appear on the cavity wall and outer surface such that the net charge on the inner surface equals the enclosed charge, leaving the outer surface to carry the remainder.

• The external field near a conductor is perpendicular to the surface and proportional to the surface charge density; inside, $E=0$.

23-4 APPLYING GAUSS’ LAW: CYLINDRICAL SYMMETRY

• For an infinite line with uniform linear charge density $\lambda$, the electric field at distance $r$ is
  \boxed{E=\frac{\lambda}{2\pi\epsilon_0 r}}.

• The field is perpendicular to the line (radial) and does not depend on height along the line due to symmetry.

23-5 APPLYING GAUSS’ LAW: PLANAR SYMMETRY

• Infinite nonconducting sheet with uniform surface charge density $\sigma$ creates a uniform field perpendicular to the sheet:
  \boxed{E=\frac{\sigma}{2\epsilon_0}} (on either side, directed away from the sheet if $\sigma>0$).

• For two large parallel conducting plates, the field between them is $E=\sigma/\epsilon_0$ (with $\sigma$ the inner-surface charge density); outside fields cancel to zero due to symmetry.

23-6 APPLYING GAUSS’ LAW: SPHERICAL SYMMETRY

• Spherical symmetry lets us apply Gauss’ law with a concentric spherical Gaussian surface.

• Outside a spherical shell of radius $R$ and total charge $q$, the field is the same as if all charge were at the center:
  \boxed{E(r)=\frac{1}{4\pi\epsilon_0}\frac{q}{r^2}},\quad r>R.$</p></li><li><p>Inside the shell, the field is zero:$E(r)=0,\quad r<R.

• For a uniformly charged solid sphere of radius $R$ with total charge $q$, inside the sphere the field is radial with magnitude
  \boxed{E(r)=\frac{1}{4\pi\epsilon_0}\frac{q}{R^3}\,r},\quad r<R.

• The shell theorems (proved by Gauss’ law): inside a uniformly charged shell, $E=0$; outside, the field is as if all charge is at the center.

23-5 SUMMARY OF KEY GAUSS’ LAW RESULTS

• Planar symmetry: infinite nonconducting sheet: E=\frac{\sigma}{2\epsilon_0} perpendicular to the sheet.

• Conducting surface: outside field magnitude $E=\sigma/\epsilon_0$; inside $E=0$.

• Cylindrical symmetry (line of charge): E=\frac{\lambda}{2\pi\epsilon_0 r}$, radial.</p></li><li><p>Spherical symmetry: outside shell$E=\frac{1}{4\pi\epsilon0}\frac{q}{r^2}$, inside shell$E=0$; inside uniform sphere$E(r)=\frac{1}{4\pi\epsilon0}\frac{q}{R^3} r.$</p></li><li><p>Charge inside a Gaussian surface determines flux through that surface; charge outside does not contribute to net flux (though it affects the local field).</p></li><li><p>The conductor principle: excess charge resides on the surface; the internal field is zero. Cavities inside conductors contain no net charge unless there’s charge on surrounding surfaces that cancels it.</p></li></ul><h4 id="f50996b9-8c87-4e81-bbdd-ef197a4c764c" data-toc-id="f50996b9-8c87-4e81-bbdd-ef197a4c764c" collapsed="false" seolevelmigrated="true">CONNECTIONS AND REAL-WORLD RELEVANCE</h4><ul><li><p>Electromagnetism underpins modern electronics, telecommunications, and lighting; field concepts explain how devices operate without direct contact between charges.</p></li><li><p>Gauss’ law provides powerful analytic shortcuts in highly symmetric systems, enabling efficient calculations for charges distributed in lines, sheets, and spheres.</p></li><li><p>The concept of electric fields and flux is foundational for understanding capacitance, shielding, and electrostatics in engineering designs (e.g., cables, sensors, and integrated circuits).</p></li><li><p>Ethical/practical implications: understanding static electricity and grounding informs safety in electronics assembly, high-voltage systems, and lightning safety.</p></li></ul><h4 id="a07b8aab-fd70-475b-8d04-d033a10743ea" data-toc-id="a07b8aab-fd70-475b-8d04-d033a10743ea" collapsed="false" seolevelmigrated="true">NOTATION KEYPOINTS (recap)</h4><ul><li><p>Charge:$q$(C), elementary charge$e\approx 1.602\times 10^{-19}\ \mathrm{C}$;$q=ne$.</p></li><li><p>Coulomb constant:$k=\frac{1}{4\pi\epsilon_0}=8.99\times 10^9\ \mathrm{N\,m^2/C^2}.$</p></li><li><p>Permittivity of free space:$\epsilon_0=8.85\times 10^{-12}\ \mathrm{C^2/(N\,m^2)}.$</p></li><li><p>Dipole moment:$\mathbf{p}=q\mathbf{d},\quad p=qd.$</p></li><li><p>Electric field:$\mathbf{E}=\frac{\mathbf{F}}{q0},\quad \mathbf{F}=q0\mathbf{E}.$</p></li><li><p>Surface charge density:$\sigma,$Linear charge density:$\lambda,$Volume charge density:$\rho.$</p></li><li><p>Flux through a surface:$\Phi_E=\oint \mathbf{E}\cdot d\mathbf{A}.$</p></li><li><p>Gauss’ law:$\oint \mathbf{E}\cdot d\mathbf{A}=\frac{q{\text{enc}}}{\epsilon0}.$$