-- ELECTRIC FIELDS --

Electric Fields – Lecture

15 Topics

P19-010 – Introduction to Electric Charge

P19-020 – Electric Charge

P19-030 – Electric Fields & Electric Field Lines

P19-040 – Principle of Superposition

P19-050 – Electric Field & Symmetry

P19-060 – Electric Field Created by a Point Charge

P19-070 – Electric Force

P19-080 – Electric Force between Two Point Charges: Coulomb’s Law

P19-100 – Computation: finite rod of charge

P19-110 – Computation: semicircle of charge

P19-120 – Computation: ring of charge

P19-130 – Computation: disk of charge

P19-140 – Electric Dipole in a Uniform Electric Field

P19-150 – Application: dielectric materials

P19-160 – Electric Field Created by an Electric Dipole

8B – Electric Fields Summary

Electric Fields – Problems

19 Topics

P19B2015 – Plastic Plate & Charged Ball

P19R2015 – Turning Around in E

P19R2015 – Projectile Motion Through E

P19B2015 – Plastic Balls & Fur

P19B2018 – Electric Force

P19B2018 – Five Point Charges

P19R2016 – Balancing Act

P19B2016 – Charged Plate & Point Charge

P19R2015 – Three Point Charge Distribution

P19S2014 – Infinite Sheet & Point Charge

P19S2017 – Charged Ring and Three Point Charges

P19S2017 – Four Infinite Sheets

P19B2014 – Semicircular Charged Ring

P19B2014 – Uniformly Charged Rod of Finite Length

P19B2015 – Charged Circular Ring

P19B2016 – Pair of Charged Rings

P19B2016 – Two Half Rings of Charge

P19B2018 – Charged Rod

P19L2013 – Two Electric Dipoles

-- GAUSS'S LAW --

Gauss’s Law – Lecture

10 Topics

P20-010 – Useful Definitions

P20-020 – Electric Flux

P20-030 – How Electric Flux Varies

P20-040 – Classic Computations of Electric Flux

P20-045 – Gauss’s Law

P20-050 – Application: infinite line of charge

P20-060 – Application: infinite rod of charge

P20-070 – Application: infinite sheet of charge

P20-080 – Application: solid sphere of charge

P20-090 – Electric Field at the Surface of a Conductor

8B – Gauss’s Law Summary

Gauss’s Law – Problems

15 Topics

P20B2016 – Conductor Properties

P20R2015 – About Electric Flux

P20B2017 – Infinite Wire of Unknown Charge

P20S2014 – Two Concentric Insulators

P20R2015 – Infinite Slab of Charge

P20B2014 – Three False Statements

P20B2015 – Spherical Off-Center Cavity

P20B2020 – Concentric Spheres

P20B2015 – Concentric Metal Sphere & Shell

P20R2015 – Spherical Metallic Shell

P20S2017 – Charged Sphere & Conducting Shell

P20B2018 – Concentric Conducting Shells

P20B2019 – Long Wire Inside a Metal Tube

P20B2023 – Charge Hanging from a String

P20B2018 – Charged Wire and Point Charge

-- ELECTRIC POTENTIAL --

Electric Potential – Lecture

17 Topics

P21-010 – Conceptual Introduction to Electric Potential

P21-020 – Electric Potential Created by a Charge Distribution

P21-030 – Deriving Electric Potential Energy

P21-040 – Electric Potential Difference: voltage

P21-050 – Electric Potential & Electric Potential Energy

P21-060 – Relationship Between Electric Field and Electric Potential

P21-070 – Principle of Superposition

P21-080 – Equipotentials

P21-090 – Electric Potential Created by a Point Charge

P21-100 – Electric Potential Created by a Dipole

P21-110 – Electric Potential of a Line of Charge

P21-120 – Electric Potential of a Semicircular Wire

P21-130 – Electric Potential of a Ring of Charge

P21-140 – Electric Potential of a Disk of Charge

P21-150 – Electric Potential at the Surface of a Conductor

P21-160 – Electronvolt and Conservation of Energy

P21-170 – Electrostatic Energy of a System of Point Charges

8B – Electric Potential Summary

Electric Potential – Problems

25 Topics

P21B2016 – Connecting Charged Spheres

P21B2015 – E & V of Three Charges

P21R2015 – Electrostatics of Three Point Charges

P21S2018 – Connecting Two Conducting Spheres

P21S2018 – Force from Charged Spheres

P21B2016 – Potential of an Infinite Cylinder

P21B2016 – Two Concentric Shells

P21S2018 – Electric Potential of Cylinders

P21S2018 – Electric Potential of Conducting Spheres

P21L2013 – Washer of Charge

P21B2014 – Four Point Charges

P21B2018 – Large Plate Attracting a Ball of Charge

P21B2019 – Two Charged Particles

P21B2015 – Electrostatic Analyzer

P21B2018 – Electric Field and Electric Potential

P21B2020 – Charged Plane

P21R2015 – Shooting an Electron Toward a Capacitor

P21B2021 – Potential From a Rod and a Shell

P21B2023 – Electric Potential

P21B2016 – Electrostatic Oscillator

P21B2017 – Two and Three Point Charges

P21S2014 – Electric Potential of Two Infinite Wires

P21S2014 – One Short of a Hexagon

P21S2017 – Concentric Cylindrical Shells

P21S2017 – Potential of a Spherical Shell

-- CAPACITORS --

Capacitors – Lecture

9 Topics

P22-010 – Capacitors and Capacitance

P22-020 – Parallel-Plate Capacitor

P22-030 – Cylindrical-Plate Capacitor

P22-040 – Spherical-Plate Capacitor

P22-050 – Capacitors in Parallel

P22-060 – Capacitors in Series

P22-070 – Energy Stored in a Capacitor

P22-080 – Effect of a Dielectric Between Capacitor Plates

P22-090 – Computing C by Integration

8B – Capacitors Summary

Capacitors – Problems

13 Topics

P22R2015 – Conceptual Questions About Capacitors

P22B2018 – Sequence with Parallel Plates

P22B2015 – Charge Moving in a Capacitor

P22B2017 – Miscellaneous Capacitor Questions

P22B2019 – Charge Traveling Between Parallel Plates

P22R2015 – Electrons & Protons in a Capacitor

P22B2014 – Study of a Parallel Plate Capacitor

P22B2015 – Cylindrical Capacitor

P22B2014 – Spherical Capacitor

P22S2018 – Dielectric Slab & Energy Storage

P22B2018 – Capacitor and Capacitance

P22B2015 – Storing Charge in Capacitors

P22S2013 – Equivalent Capacitance

-- DC CIRCUITS --

DC Circuits – Lecture

15 Topics

P23-010 – Electric Current

P23-020 – Current Density

P23-030 – Drift Velocity

P23-040 – Resistivity and Conductivity

P23-050 – Circuit Component: the resistor

P23-060 – Assembling Resistors in Parallel and Series

P23-070 – Circuit Component: the capacitor

P23-080 – Assembling Capacitors in Parallel and Series

P23-090 – Circuit Component: the battery

P23-100 – Generator Convention & Receptor Convention

P23-110 – Kirchhoff’s Laws

P23-120 – Electric Power

P23-130 – The RC Circuit

P23-140 – Short-Term and Long-Term Behavior

P23-150 – Time Constant of the RC Circuit

8B – DC Circuits Summary

DC Circuits – Problems

22 Topics

P23B2013 – Of Drift Velocity

P23B2014 – Resistivity & Current

P23B2023 – Properties of a Conducting Ribbon

P23B2014 – Brightness of Bulbs

P23S2014 – Three Resistor Circuit

P23S2017 – Kirchhoff’s Laws

P23B2015 – Monitoring Liquid He Levels

P23B2022 – DC Circuit with Multiple Resistors

P23B2016 – Three Resistors, Three Batteries

P23B2018 – Circuit with Bulbs

P23B2018 – Power Supplied or Consumed

P23R2015 – Equivalent Resistance

P23R2017 – Two Circuits

P23B2019 – Two Loops & a Shared Resistor

P23B2014 – Charge & Energy in Capacitors

P23B2015 – Power & Charge

P23S2017 – Resistors & Capacitors

P23B2016 – Two RC Circuits

P23B2014 – RC Circuit

P23B2014 – RC Circuit Behavior

P23B2015 – Equivalent RC Circuit

P23B2016 – Current Resistance and RC Circuit

-- MIDTERM 1 - STUDY GUIDE --

Lowhorn – 8B 2019 Spring – Midterm 1

3 Topics

P19L2019 – Square with Point Charges

P19L2019 – Semicircle with Opposite Densities

P21L2019 – E and V of an Insulating Sphere

Lowhorn – 8B 2020 Spring – Midterm 1

4 Topics

P23L2020 – Circuit with Six Resistors

P23L2020 – Power and Current

P21L2020 – E and V of an Insulating Cylinder

P21L2020 – Distant Point Charges

Lowhorn – 8B 2021 Spring – Midterm 1

4 Topics

P21L2021 – Dipole Field on Axis

P21L2021 – E and V of Concentric Cylinder and Shell

P21L2021 – E and V on Ring Axis

P23L2021 – Asymptotic Behavior of a Circuit

Lowhorn – 8B 2022 Fall – Midterm 1

4 Topics

P21L2022 – Concentric Conductor and Insulator

P21L2022 – Ring with Opposite Densities

P23L2022 – Charging Capacitors Over Time

P23L2022 – Switching in a Battery

Lowhorn – 8B 2024 Spring – Midterm 1

3 Topics

P21L2024 – Concentric Point Charge and Shells

P23L2024 – Equivalent RC Circuit

P23L2024 – Current and Equivalent Resistance

8B Masterclass – Practice Midterm 1 (graded)

8B Masterclass – Midterm 1 Review Session

-- MAGNETISM --

Magnetism – Lecture

20 Topics

P24-010 – Phenomenon

P24-020 – Magnetic Field & Magnetic Field Lines

P24-030 – Magnetic Field & Symmetry

P24-040 – Principle of Superposition

P24-050 – Reminder: cross product of two vectors

P24-060 – Magnetic Force on a Moving Point Charge

P24-070 – Magnetic Force on a Current

P24-080 – Motion of Charged Particles through B

P24-090 – The Lorentz Force

P24-100 – Biot-Savart Law

P24-110 – Biot-Savart Application: semicircular wire of current

P24-120 – Biot-Savart Application: ring of current

P24-130 – Biot-Savart Application: field on the axis of a ring of current

P24-140 – Biot-Savart Application: finite wire of current

P24-150 – Ampere’s Law

P24-160 – Ampere’s Law: infinite wire of current

P24-170 – Ampere’s Law: infinite slab of current

P24-180 – Ampere’s Law: infinite solenoid

P24-190 – Ampere’s Law: toroid with current

P24-200 – Magnetic Dipole

8B – Magnetism Summary

Magnetism – Problems

23 Topics

P24R2015 – Linear Motion Through E and B

P24B2016 – Charge Moving Between Wires

P24B2016 – Mass Spectrometer

P24B2015 – Two Vertical Wires

P24B2019 – Charged Wire and Moving Charge

P24S2013 – Traveling Through B

P24B2016 – Loop Suspended in a Magnetic Fied

P24B2016 – Magnetic Force on a Current

P24R2015 – Two Parallel Wires

P24B2017 – Forces on a Current

P24S2014 – Force on Current-Carrying Wires

P24S2018 – Wire Immersed in a Magnetic Field

P24S2021 – Parallel Rods in a Magnetic Field

P24S2014 – Arc of Current

P24B2020 – A Strange Current-Carrying Loop

P24B22021 – Two Semicircles of Current

P24B2015 – Helmholtz Pair

P24B2016 – Curved Wire of Current

P24B2013 – B-Field Created by Three Wires

P24B2014 – Coaxial Cable

P24R2015 – Hollow Cylindrical Conductor

P24S2013 – Infinite Hollow Wire

P24B2014 – Toroidal Coil

-- INDUCTION --

Induction – Lecture

19 Topics

P25-010 – Phenomenon

P25-020 – Motional EMF: moving rod in a magnetic field

P25-030 – Motional EMF: coil moving through a magnetic field

P25-040 – Magnetic Flux

P25-050 – How Magnetic Flux Varies

P25-060 – Classic Computations of Magnetic Flux

P25-070 – Lenz’s Law

P25-080 – Faraday’s Law

P25-090 – Faraday’s Law: conducting ring in a varying magnetic field

P25-100 – Faraday’s Law: conducting rod on parallel rails

P25-110 – Faraday’s Law: rotating loop in a magnetic field

P25-120 – Mutual Inductance & Self-Inductance

P25-140 – Circuit Component: inductor

P25-150 – Energy Stored in an Inductor

P25-170 – The RL Circuit

P25-180 – Short-Term & Long-Term Behavior

P25-190 – Time Constant of the RL Circuit

P25-200 – Assembling Inductors in Parallel

P25-210 – Assembling Inductors In Series

Induction – Problems

32 Topics

P25R2015 – Jumping Ring

P25B2014 – Square Loop and Induction

P25B2018 – Magnetic Field and Electromagnetic Induction

P25B2020 – Current-Carrying Sheet

P25B2015 – Tilted Coil

P25B2017 – Number of Coils of a Square Loop

P25B2018 – Rectangular Loop Straddling a Varying B

P25R2016 – Loop Traveling Through B

P25B2015 – Falling Square Loop

P25B2016 – Induction in a Vertical Moving Loop

P25B2016 – Hoop Rotating Above a Slab of Current

P25B2016 – Induction in a Zinc-Chromium Loop

P25B2016 – Electromagnetic Induction

P25B2016 – Magnetism and Electromagnetic Induction

P25B2021 – Induction from a Cylindrical Wire

P25B2016 – Flux from a Solenoid

P25B2015 – Projectile Launcher

P25B2014 – Moving Rod

P25B2023 – Rod on Inclined Rails

P25B2013 – Two Rods with Different Speeds

P25B2020 – Moving a Blue Rod on Rails

P25R2015 – Rod on Rails

P25S2013 – Moving a Rod on Rails

P25S2017 – Rod on Parallel Rails

P25B2014 – Rod on Non-Parallel Rails

P25B2014 – Current Through an RL Circuit

P25S2017 – RL Circuit Behavior

P25S2018 – Short-Term & Long-Term Currents

P25B2015 – LR Circuit

P25B2015 – Two Competing Inductors

P25R2015 – Behavior of an Inductor with Time

P25R2015 – Mutual & Self-Inductance

-- ELECTROMAGNETIC WAVES --

EM Waves – Lecture

8 Topics

P27-020 – Maxwell’s Equations

P27-030 – EM Waves

P27-040 – EM Wave Spectrum

P27-050 – Properties of EM Waves

P27-060 – Energy Density of Electric and Magnetic Fields

P27-070 – Poynting Vector and Intensity

P27-090 – Polarization of EM Waves

P27-100 – Malus’s Law

EM Waves – Problems

13 Topics

P27B2014 – Electromagnetic Wave

P27B2014 – EM Waves Questions

P27R2016 – Electromagnetic Wave

P27B2022 – Intensity of a Plane EM Wave

P27B2023 – Fields of an EM Wave

P27B2016 – Satellite Signal

P27B2015 – Light Shined Through Polarizers

P27R2015 – Two Polarizers in Series

P27R2015 – Three Polarizers in Series

P27B2020 – Plane EM Wave in Vacuum

P27B2018 – Polarized Light & Polarizers

P27B2019 – Polarized Light From a Laser

P27R2017 – Laser and Polarizers

-- OPTICS --

Optics – Lecture

28 Topics

P28-010 – Linear Propagation of Light and the Ray Model

P28-020 – Index of Refraction

P28-030 – Change in Medium

P28-040 – Dispersion

P28-050 – Law of Reflection

P28-060 – Refraction & Snell’s Law

P28-070 – Total Internal Reflection

P28-080 – Brewster’s Law

P28-090 – Object Distance, Image Distance, Real & Virtual Images

P28-100 – Magnification

P28-110 – Plane Mirrors & Ray Tracing

P28-120 – Spherical Mirrors

P28-130 – Ray Tracing Rules for Spherical Mirrors

P28-140 – Principal Rays for Concave Mirrors

P28-150 – Principal Rays for Convex Mirrors

P28-160 – Mirror Equation

P28-170 – Magnification of Spherical Mirrors

P28-180 – Thin Lenses

P28-190 – Ray Tracing Rules for Thin Lenses

P28-200 – Principal Rays for Converging Lenses

P28-210 – Principal Rays for Diverging Lenses

P28-220 – Lens Equation

P28-230 – Magnification for Thin Lenses

P28-240 – Assembling Two Lenses Together

P28-260 – Images Formed by Refracting Surfaces

P28-270 – The Lensmaker’s Equation

P28-280 – The Human Eye

P28-290 – The Magnifying Glass

Optics – Problems

46 Topics

P28B2014 – Optical Fiber

P28B2014 – Refraction Through a Prism

P28B2018 – EM Waves, Reflection, and Refraction

P28B2018 – Reflections Off of Various Surfaces

P28R2015 – Reflections Off of a Corner Mirror

P28R2015 – Fishing and Optics

P28S2013 – Reflection and Refraction in a Slab

P28S2014 – Reflection in a Sphere

P28B2015 – Cylindrical Tank

P28B2016 – CD Player Laser Beam

P28B2021 – EM Wave and Mirror Sizing

P28B2023 – Opaque Fish Tank

P28B2019 – Light Traveling Through a Prism

P28B2013 – Reflection & Refraction with Flint

P28B2014 – Transmission and Reflection of Laser Light

P28B2015 – Ice Fishing

P28B2016 – Ray Tracing

P28B2017 – In and Out of a Prism

P28S2017 – Refraction in a Crystal Ball

P28S2018 – Refraction in a Sphere

P28S2018 – Maximum and Minimum Angles

P28S2021 – Submerged Mirror

P28B2015 – Spherical Reflection

P28B2015 – Image Through a Converging Lens

P28B2016 – EM Waves and Optics

P28B2018 – Spherical Mirrors and Thin Lenses

P28B2020 – Unknown Spherical Mirror

P28B2016 – Corrective Lenses for Vision Defect

P28S2013 – Mobile Converging Lens

P28R2015 – Combining a DV Lens and a CV Lens

P28R2015 – Combining a DV Lens and a CV Mirror

P28B2014 – Afocal Telescope

P28B2015 – Eye Defect

P28B2016 – Optical Instrument

P28B2016 – Spherical Mirrors and Thin Lenses

P28B2013 – Combining Flat-Sided Lenses

P28B2014 – Image of a Candle on the Wall

P28B2020 – Two Lenses

P28S2014 – Combining Two Lenses

P28B2015 – Candle, Mirror, and Lens

P28B2016 – Unknown Optical Device

P28B2018 – Alien Eyeball

P28B2019 – Image of a Candle

P28B2016 – Image From a Microscope

P28B2017 – Candle, Mirrors, Lenses

P28R2015 – Two Lenses

-- MIDTERM 2 - STUDY GUIDE --

Lowhorn – 8B 2019 Spring – Midterm 2

4 Topics

P23L2019 – Equivalent Resistance

P23L2019 – Unknown Battery Voltage

P24L2019 – Magnetic Force on a Bent Wire

P24L2019 – Wires with Opposite Currents

Lowhorn – 8B 2020 Spring – Midterm 2

3 Topics

P24L2020 – Bent Wire of Current

P24L2020 – Pair of Infinite Wires

P25L2020 – Flux Through a Loop

Lowhorn – 8B 2021 Fall – Midterm 2

4 Topics

P28L2021 – Reflection Refraction and EM Waves

P24L2021 – Semicircles of Current

P25L2021 – Loop Falling Through B

P24L2021 – Coaxial Cable

Lowhorn – 8B 2022 Fall – Midterm 2

4 Topics

P24L2022 – Current-Carrying wire and Shell

P25L2022 – Flux and Induced Current

P28L2022 – Pool Light

P28L2022 – DV Mirror and Lens

-- INTERFERENCE & DIFFRACTION --

Interf & Diffraction – Lecture

14 Topics

P29-001 – Principle of Superposition

P29-002 – Constructive & Destructive Interference

P29-003 – Conditions for Constructive & Destructive Interference

P29-010 – Coherent Source

P29-020 – Huygens Principle

P29-030 – Double-Slit Interference: Young’s experiment

P29-040 – Phase Shift Upon Reflection

P29-050 – Thin-Film Interference

P29-060 – Diffraction Condition

P29-070 – Single-Slit Diffraction

P29-080 – Double-Slit Diffraction

P29-090 – Diffraction Grating

P29-100 – Diffraction by a Circular Aperture

P29-110 – Rayleigh’s Criterion

Interf & Diffraction – Problems

23 Topics

P29B2016 – Photon Interference. vs. Electron Diffraction

P29S2014 – Delaying Light with a Plastic Slab

P29B2016 – Shifted Interference Pattern

P29R2015 – Delaying Light

P29B2009 – Alien Skin Interference

P29R2017 – Of Thin Films

P29S2013 – Thinning Soap Film

P29S2014 – A Thin Plastic Wedge

P29S2017 – Observing a Thin Oil Layer

P29S2017 – Rings in a Convex Lens

P29B2015 – Interference with Blue and Orange Light

P29B2016 – Film Interference with Blue and Orange Light

P29S2018 – Thinning Vertical Soap Film

P29B2016 – Diffraction

P29R2017 – Double Slit Diffraction

P29S2013 – Double Slit Interference in Different Media

P29S2017 – Diffraction in Air and in Fluid

P29B2014 – Shifted Double-Slit Experiment

P29B2014 – Diffraction with Blue and Orange Light

P29B2020 – EM Waves and Diffraction

P29B2015 – Airy Disk

P29B2014 – Autopsy of a Giant Alien

P29B2015 – Resolving Traffic Lights

-- QUANTUM MECHANICS --

Quantum – Lecture

4 Topics

P31-110 – Electron Diffraction & de Broglie Wavelength

P31-120 – The Heisenberg Uncertainty Principle

P31-130 – Schrödinger’s Equation & Wavefunctions

P31-140 – Infinite Potential Well: the “particle in a box”

Quantum – Problems

10 Topics

P31B2015 – Electron Diffraction

P31R2015 – Kinetic Energy of Particles

P31S2013 – Single-Slit Diffraction with Electrons

P31B2014 – Electron in an Infinite Potential Well

P31B2015 – Infinite Potential Well with an Electron

P31B2009 – Alien Vision

P31B2014 – Finite Potential Well

P31B2020 – Excited Particle in a Potential Well

P31B2009 – Miscellaneous Quantum Questions

P31B2016 – Quantum Questions

-- RELATIVITY --

Relativity – Lecture

5 Topics

P32-010 – Invariance of Physical Laws

P32-020 – Relativity of Simultaneity

P32-030 – The Lorentz Factor

P32-040 – Time Dilation

P32-050 – Length Contraction

Relativity – Practice Problems

6 Topics

P32P2015 – Radioactive Relativistic Particle

P32P2019 – Lifetime of a Decaying Particle

P32R2015 – Detecting Muons

P32R2015 – Length of a Spaceship

P32R2015 – Moving Meter Stick

P32R2015 – Of Length Contraction

-- STUDY GUIDE - FINAL --

Lowhorn – 8B 2019 Spring – Final

5 Topics

P28L2019 – One Image or Two

P32L2019 – Contracted Denebian Starship

P29L2019 – From Bright to Dark Fringe

P21L2019 – Nested Spherical Shells

P23L2019 – Circuit with Multiple Resistors

Lowhorn – 8B 2020 Spring – Final

4 Topics

P24L2020 – Two Semicircular Wires

P32L2020 – Relativistic Rabbit

P28L2020 – Unknown Lens

P21L2020 – E and V of a Wire in a Shell

Lowhorn – 8B 2021 Fall – Final

6 Topics

P31L2021 – Four Polarizers and Photoelectrons

P29L2021 – Double Slit Diffraction Pattern

P28L2021 – Conditions on Image Formation

P25L2021 – Magnetic Field from a Circuit

P21L2021 – E and V of Concentric Sphere

P31L2021 – Quantum

P19-160 – Electric Field Created by an Electric Dipole

8B Masterclass (Lowhorn) Electric Fields – Lecture P19-160 – Electric Field Created by an Electric Dipole

Far Field Created by an Electric Dipole Along its Axis

We consider an electric dipole centered at the origin and lying on the $x$-axis as shown below.

The net electric field at a point $x\gg d$ on the $x$-axis is given by

E_{net}=E_+-E_-=\frac{kq}{{\left(x-d/2\right)}^2}-\frac{kq}{{\left(x+d/2\right)}^2}

We then simplify the above expression using $x\gg d$ and the Taylor expansion ${\left(1+\varepsilon \right)}^{\alpha }\approx 1+\alpha \varepsilon $ which is valid when $\varepsilon \ll 1$. Factoring $x$ out of the denominator of each fraction yields the following

\begin{aligned} E_{net}&=\frac{kq}{\displaystyle{{\left(x-\frac{d}{2}\right)}^2}}-\frac{kq}{\displaystyle{{\left(x+\frac{d}{2}\right)}^2}} \\ \\ &=\frac{kq}{\displaystyle{x^2{\left(1-\frac{d}{2x}\right)}^2}}-\frac{kq}{\displaystyle{x^2{\left(1+\frac{d}{2x}\right)}^2}} \\ \\ &=\frac{kq}{x^2}\left[\frac{1}{\displaystyle{{\left(1-\frac{d}{2x}\right)}^2}}-\frac{1}{\displaystyle{{\left(1+\frac{d}{2x}\right)}^2}}\right] \end{aligned}

Noticing that $d/2x\ \ll 1$, we apply the Taylor expansion above and derive the following expression

\begin{aligned} E_{net}&=\frac{kq}{x^2}\left[\frac{1}{\displaystyle{{\left(1-\frac{d}{2x}\right)}^2}}-\frac{1}{\displaystyle{{\left(1+\frac{d}{2x}\right)}^2}}\right] \\ \\ &=\frac{kq}{x^2}\left[{\left(1-\frac{d}{2x}\right)}^{-2}-{\left(1+\frac{d}{2x}\right)}^{-2}\right] \\ \\ &\approx \frac{kq}{x^2}\left[1+\frac{2d}{2x}-\left(1-\frac{2d}{2x}\right)\right] \\ \\ &\approx \frac{kq}{x^2}\cdot \frac{2d}{x} \\ \\ &\approx \frac{2kqd}{x^3} \end{aligned}

Thus, the far field generated by the electric dipole at a point $x\gg d$ along its axis is given by

\boxed{{\overrightarrow{E}}_{net}\approx\frac{2kqd}{x^3}\ \ \hat{x}\approx \frac{2kp}{x^3}\ \ \hat{x}}