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[[Electricity_and_Magnetism|⬅ Back to Electricity and Magnetism]]
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[[/Potentiel électrique|🌐 Version en français : Potentiel électrique]]
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<div style="border-left:5px solid #0ea5e9; padding:1em; background-color:#ecfeff; margin:0.75em 0 1em;">
<p style="margin:0; font-weight:bold;">📘 Electric Potential</p>
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= Reference Textbook =

<div style="border:1px solid #28a745; padding:0.8em; background-color:#f0fdf4; margin:1em 0;">
<p style="margin:0.2em 0;"><strong>📖 Reference Textbook:</strong><br>
[https://openstax.org/books/university-physics-volume-2/pages/7-introduction University Physics Volume 2: Chapter 7]</p>
</div>

= Theory =

== Background: The Electric Force is a Conservative Force ==

The electric force is a '''conservative force'''. That means that the work done by or against the electric force depends only on the initial and final positions, not on the path taken.

This allows us to define the change in electric potential energy:

<math>\Delta U = U_\mathrm{f}-U_\mathrm{i}=-W_\mathrm{e}</math>

where <math>W_\mathrm{e}</math> is the work done by the electric force.

Since <math>\vec{F}_\mathrm{e}=q\vec{E}</math>,

<math>W_\mathrm{e}=\int_A^B \vec{F}_\mathrm{e}\cdot d\vec{s}=\int_A^B q\vec{E}\cdot d\vec{s}</math>

and therefore

<math>\Delta U=-\int_A^B q\vec{E}\cdot d\vec{s}</math>

<div style="border-left:5px solid #ef4444; padding:0.8em 1em; background-color:#fff7f7; margin:1em 0;">
'''▶ Video: Conservative Forces and Electric Potential Energy'''

[https://www.youtube.com/watch?v=1XwxKKcn0ks Watch the video on YouTube]
</div>

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== Electric Potential and Electric Potential Energy ==

=== Electric Potential Energy ===
'''Electric potential energy''' is the energy stored in a system of charges due to their positions in an electric field. This is similar to gravitational potential energy, where the position of an object in a gravitational field determines its potential energy.

=== Electric Potential ===

The '''electric potential''' <math>V</math> at a point is the electric potential energy per unit charge:

<math>V=\frac{U}{q}</math>

Electric potential is a scalar quantity. Like potential energy, its value depends on the chosen zero reference.

=== Electric Potential Difference (Voltage) ===

The '''electric potential difference''' between points A and B is

<math>\Delta V=V_\mathrm{B}-V_\mathrm{A}=\frac{\Delta U}{q}</math>

Potential difference is commonly called '''voltage'''. Its SI unit is the volt:

<math>1~\mathrm{V}=1~\mathrm{J/C}</math>

A potential difference of one volt means that the potential energy changes by one joule for each coulomb of charge. Potential difference is often more useful than the potential at a single point because it does not depend on the choice of zero reference.

=== Finding Electric Potential Energy from Electric Potential ===

If a charge <math>q</math> moves through a potential difference <math>\Delta V</math>, its change in electric potential energy is

<math>\Delta U=q\Delta V</math>

If the external electric potential <math>V</math> is measured relative to the same zero reference as <math>U</math>, the potential energy of a charge placed at that point is

<math>U=qV</math>

The sign of the charge matters. For the same <math>\Delta V</math>, a positive and a negative charge undergo opposite changes in potential energy.

<div style="border-left:5px solid #ef4444; padding:0.8em 1em; background-color:#fff7f7; margin:1em 0;">
'''▶ Video: Electric Potential Energy'''

[https://www.youtube.com/watch?v=j3GrOKre__0 Watch the video on YouTube]
</div>

<div style="border-left:5px solid #ef4444; padding:0.8em 1em; background-color:#fff7f7; margin:1em 0;">
'''▶ Video: Electric Potential (Voltage)'''

[https://www.youtube.com/watch?v=1L5iolQj9KA Watch the video on YouTube]
</div>

<div style="border-left:5px solid #ef4444; padding:0.8em 1em; background-color:#fff7f7; margin:1em 0;">
'''▶ Video: Calculating Electric Potential Energy from Voltage'''

[https://www.youtube.com/watch?v=FMJXGoVIQ7c Watch the video on YouTube]
</div>

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== Electric Potential and the Electric Field ==

The potential difference between two points can be calculated from the electric field:

<math>\Delta V=V_\mathrm{B}-V_\mathrm{A}=-\int_A^B \vec{E}\cdot d\vec{s}</math>

Because the electrostatic force is conservative, this potential difference is independent of the path chosen between A and B.

For a uniform electric field,

<math>\Delta V=-\vec{E}\cdot\Delta\vec{s}=-E\Delta s\cos\theta</math>

where <math>\theta</math> is the angle between the electric field and the displacement. If the displacement is parallel to the field,

<math>\Delta V=-E\Delta s</math>

The electric field always points in the direction in which the electric potential decreases most rapidly.

This relationship gives another equivalent unit for electric field:

<math>1~\mathrm{N/C}=1~\mathrm{V/m}</math>

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== Electric Potential Due to Point Charges ==

=== Electric Potential Due to a Point Charge at Rest ===

Choosing the electric potential to be zero at infinity, the potential at a distance <math>r</math> from a point charge <math>Q</math> is

<math>V=k\frac{Q}{r}</math>

<div style="border-left:5px solid #ef4444; padding:0.8em 1em; background-color:#fff7f7; margin:1em 0;">
'''▶ Video: Electric Potential Due to a Point Charge'''

[https://www.youtube.com/watch?v=KzKsMZED1B4 Watch the video on YouTube]
</div>

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=== Electric Potential Due to Several Point Charges ===

The principle of superposition also applies to electric potential. Because potential is a scalar, the contributions are added algebraically:

<math>V(P)=\sum_i V_i(P)=k\sum_i\frac{q_i}{r_i}</math>

where <math>r_i</math> is the distance from charge <math>q_i</math> to point P. Positive charges make positive contributions and negative charges make negative contributions.

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== Potential Energy of a System of Charged Particles ==
The total electric potential energy of a system of charges is the sum of the potential energies between all pairs of charges in the system. For example, for three charges, <math> q_1, q_2, </math> and <math> q_3 </math>, the potential energy is:
<math> U = k_e \left( \frac{q_1 q_2}{r_{12}} + \frac{q_1 q_3}{r_{13}} + \frac{q_2 q_3}{r_{23}} \right) </math>

== Equipotential Lines and Surfaces ==

'''Equipotential lines''' connect points having the same electric potential. In three dimensions, the corresponding objects are equipotential surfaces.

* Electric field lines are always perpendicular to equipotential lines or surfaces.
* Electric field lines point from higher potential toward lower potential.
* Moving along an equipotential requires no work by the electric force because <math>\Delta V=0</math> and <math>\Delta U=q\Delta V=0</math>.
* Closely spaced equipotential lines indicate a stronger electric field because the potential changes more rapidly with distance.

Around a point charge, the equipotential surfaces are concentric spheres. In a diagram drawn in two dimensions, they appear as concentric circles.

<div style="border-left:5px solid #ef4444; padding:0.8em 1em; background-color:#fff7f7; margin:1em 0;">
'''▶ Video: Equipotential Lines'''

[https://www.youtube.com/watch?v=xwaPxhmocdQ Watch the video on YouTube]
</div>

<div style="border-left:5px solid #ef4444; padding:0.8em 1em; background-color:#fff7f7; margin:1em 0;">
'''▶ Video: Electric Potential and Electric-Field Lines'''

[https://www.youtube.com/watch?v=6F6mh4R00rI Watch the video on YouTube]
</div>

<div style="border-left:5px solid #ef4444; padding:0.8em 1em; background-color:#fff7f7; margin:1em 0;">
'''▶ Video: Finding the Electric Field from the Potential'''

[https://www.youtube.com/watch?v=vmW7h5YgZL8 Watch the video on YouTube]
</div>

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== The Electronvolt ==

The '''electronvolt''' (eV) is a unit of energy commonly used for atoms and subatomic particles. One electronvolt is the magnitude of the energy change of one elementary charge moving through a potential difference of one volt:

<math>1~\mathrm{eV}=1.602\times10^{-19}~\mathrm{J}</math>

The electronvolt is a unit of energy, not a unit of electric potential.

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== Electric Potential, Current, and Power ==

A potential difference can drive electric charges through a conductor when a conducting path is available.

'''Electric current''' <math>I</math> is the rate at which charge passes through a cross-section:

<math>I=\frac{\Delta Q}{\Delta t}</math>

The instantaneous current is <math>I=dQ/dt</math>. Its SI unit is the ampere, where <math>1~\mathrm{A}=1~\mathrm{C/s}</math>.

'''Power''' <math>P</math> is the rate at which energy is transferred or converted:

<math>P=\frac{\Delta U}{\Delta t}</math>

For a device through which a current <math>I</math> passes across a potential difference <math>\Delta V</math>, the magnitude of the electrical power is

<math>P=I\Delta V</math>

<div style="border-left:5px solid #ef4444; padding:0.8em 1em; background-color:#fff7f7; margin:1em 0;">
'''▶ Video: Electric Potential, Current, and Power'''

[https://www.youtube.com/watch?v=FViTaHwLgxo Watch the video on YouTube]
</div>

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== Charged-Particle Motion in a Uniform Electric Field ==

In a uniform electric field, a charged particle experiences the constant force <math>\vec{F}=q\vec{E}</math>. Its motion can be analyzed using constant-acceleration kinematics, in a way similar to projectile motion under gravity. It can also be analyzed using energy:

<math>\Delta K=-\Delta U=-q\Delta V</math>

<div style="border-left:5px solid #ef4444; padding:0.8em 1em; background-color:#fff7f7; margin:1em 0;">
'''▶ Example: Charged-Particle Motion in a Uniform Electric Field'''

[https://www.youtube.com/watch?v=zC7D6FtMPwk Watch the example on YouTube]
</div>

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= Electric-Potential Simulations =

Explore electric potential and equipotential lines with these simulations:

* [https://phet.colorado.edu/en/simulations/charges-and-fields PhET: Charges and Fields]
* [https://phet.colorado.edu/en/simulations/capacitor-lab-basics PhET: Capacitor Lab: Basics]

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= Additional Resources =

* [https://openstax.org/books/university-physics-volume-2/pages/7-summary OpenStax Chapter 7 Summary]
* [https://openstax.org/books/university-physics-volume-2/pages/7-key-equations OpenStax Chapter 7 Key Equations]

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<div style="float:left; font-size:90%; margin:0.25em 0 0.75em 0;">
[[The Electric Field|⬅ Previous: The Electric Field]]
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<div style="float:right; font-size:90%; margin:0.25em 0 0.75em 0;">
[[Capacitors|Next: Capacitors ➡]]
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2026-07-30T16:26:48Z

Brachers:

<div style="float:left; font-size:90%; margin:0.25em 0 0.75em 0;">
[[Electricity_and_Magnetism|⬅ Back to Electricity and Magnetism]]
</div>
<br style="clear:both;" />

<div style="float:right; font-size:90%;">
[[/Champ électrique|🌐 Version en français : Champ électrique]]
</div>


<div style="border-left:5px solid #0ea5e9; padding:1em; background-color:#ecfeff; margin:0.75em 0 1em;">
<p style="margin:0; font-weight:bold;">📘 The Electric Field</p>
</div>

= Reference Textbook =

<div style="border:1px solid #28a745; padding:0.8em; background-color:#f0fdf4; margin:1em 0;">
<p style="margin:0.2em 0;"><strong>📖 Reference Textbook:</strong><br>
[https://openstax.org/books/university-physics-volume-2/pages/5-4-electric-field University Physics Volume 2: Chapter 5.4]</p>
</div>

= Theory =

== The Electric Field ==

The electric field provides a convenient way of calculating the electric force <math>\vec{F}</math> on a test charge <math>q_\mathrm{t}</math>. It plays a role similar to that of the gravitational field <math>\vec{g}</math> when calculating gravitational forces.

Instead of recalculating the electric force directly using Coulomb's law—and adding the individual forces exerted by several source charges—we can first determine the electric field:

<math>\vec{F} = q_\mathrm{t}\vec{E}</math>

where:

* <math>\vec{F}</math> is the electric force exerted on the test charge;
* <math>q_\mathrm{t}</math> is the test charge placed in the electric field;
* <math>\vec{E}</math> is the electric field at the position of the test charge.

In the same way that the gravitational field <math>\vec{g}</math> gives the gravitational force per unit mass, the electric field <math>\vec{E}</math> gives the electric force per unit charge:

<math>\vec{E} = \frac{\vec{F}}{q_\mathrm{t}}</math>

The SI unit of electric field is the newton per coulomb:

<math>[\vec{E}] = \mathrm{N/C}</math>

<div style="border-left:5px solid #ef4444; padding:0.8em 1em; background-color:#fff7f7; margin:1em 0;">
'''▶ Video: Introduction to the Electric Field'''

[https://www.youtube.com/watch?v=qMnQQudvaE4 Watch the video on YouTube]
</div>

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=== Electric Field of a Point Charge ===

The electric field <math>\vec{E}</math> created by a stationary point charge <math>q_\mathrm{s}</math> at a distance <math>r</math> is given by:

<math>\vec{E} = k \frac{q_\mathrm{s}}{r^2}\hat{r}</math>

where:

* <math>\vec{E}</math> is the electric field created by the source charge;
* <math>k</math> is Coulomb's constant, approximately <math>8.99 \times 10^9~\mathrm{N \cdot m^2/C^2}</math>;
* <math>q_\mathrm{s}</math> is the source charge creating the electric field;
* <math>r</math> is the distance from the source charge to the point where the field is calculated;
* <math>\hat{r}</math> is a unit vector pointing from the source charge toward the point of interest.

The direction of the electric field depends on the sign of the source charge:

* The electric field points '''away from''' a positive source charge.
* The electric field points '''toward''' a negative source charge.

Combining the electric-field equation with

<math>\vec{F} = q_\mathrm{t}\vec{E}</math>

gives Coulomb's law:

<math>\vec{F}_\mathrm{ts} = k \frac{q_\mathrm{s}q_\mathrm{t}}{r^2}\hat{r}</math>

<div style="border-left:5px solid #ef4444; padding:0.8em 1em; background-color:#fff7f7; margin:1em 0;">
'''▶ Video: Electric Field of a Point Charge'''

[https://www.youtube.com/watch?v=pB-Fjrs1m94 Watch the video on YouTube]
</div>

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=== Superposition of Electric Fields ===

When several source charges are present, the total electric field at a point is the vector sum of the electric fields created by the individual charges:

<math>\vec{E}_\mathrm{net} = \sum_i \vec{E}_i</math>

For a collection of point charges:

<math>\vec{E}_\mathrm{net} = \sum_i k\frac{q_i}{r_i^2}\hat{r}_i</math>

The electric field is calculated independently for each source charge. The resulting electric-field vectors are then added using vector addition.

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== Electric Field Lines ==

'''Electric field lines''' provide a visual representation of an electric field. At any point, the electric-field vector is tangent to the field line and points in the direction of the force that would be exerted on a positive test charge.

* '''Direction''': Field lines point away from positive charges and toward negative charges.
* '''Field strength''': The density of field lines represents the magnitude of the electric field. Closely spaced lines indicate a stronger field.
* '''Positive test charge convention''': The arrows show the direction in which a positive test charge would accelerate.

=== Rules for Drawing Electric Field Lines ===

* Field lines begin on positive charges and end on negative charges or at infinity.
* The number of field lines associated with a charge is proportional to the magnitude of the charge.
* Electric field lines never cross.
* Field lines form continuous, smooth curves.
* Field lines do not form closed loops in electrostatics.
* Near the surface of a conductor in electrostatic equilibrium, field lines are perpendicular to the surface.

<div style="border-left:5px solid #ef4444; padding:0.8em 1em; background-color:#fff7f7; margin:1em 0;">
'''▶ Video: Electric Field Lines'''

[https://www.youtube.com/watch?v=pbrI1JEBAQU Watch the video on YouTube]
</div>

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== Typical Electric-Field Problems ==

Typical electric-field problems may require you to:

* calculate the electric field created by one or more point charges;
* determine the direction of the electric field at a specified point;
* add electric-field vectors using the principle of superposition;
* determine the force exerted on a test charge placed in an electric field;
* find a position where the net electric field is zero.

A useful general approach is:

# Draw a diagram showing all source charges and the point of interest.
# Determine the electric field created by each source charge.
# Establish the direction of each electric-field vector.
# Resolve the vectors into components when necessary.
# Add the components to obtain the net electric field.
# Use <math>\vec{F}=q_\mathrm{t}\vec{E}</math> if the force on a test charge is required.

<div style="border-left:5px solid #ef4444; padding:0.8em 1em; background-color:#fff7f7; margin:1em 0;">
'''▶ Video: Typical Problems with the Electric Field'''

[https://www.youtube.com/watch?v=8GzJOKjp_jA Watch the video on YouTube]
</div>

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= Demonstrations and Additional Videos =

== Faraday Cage ==

A '''Faraday cage''' is a conducting enclosure that redistributes electric charges around its exterior surface. Under electrostatic conditions, the electric field inside the conducting material and within an enclosed empty cavity is zero, provided that no charge is located inside the cavity.

<div style="border-left:5px solid #f97316; padding:0.8em 1em; background-color:#fff7ed; margin:1em 0;">
'''▶ Demonstration: MIT Physics Demo — Faraday's Cage'''

[https://youtu.be/WqvImbn9GG4 Watch the demonstration on YouTube]
</div>

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== Additional Electric-Field Videos ==

<div style="border-left:5px solid #ef4444; padding:0.8em 1em; background-color:#fff7f7; margin:1em 0;">
'''▶ Additional Video 1'''

[https://www.youtube.com/watch?v=M1XHjl_6HtM Watch the video on YouTube]
</div>

<div style="border-left:5px solid #ef4444; padding:0.8em 1em; background-color:#fff7f7; margin:1em 0;">
'''▶ Additional Video 2'''

[https://www.youtube.com/watch?v=Zi4kXgDBFhw Watch the video on YouTube]
</div>

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= Electric-Field Simulations =

Explore electric fields with these simulations:

* [https://phet.colorado.edu/en/simulations/efield PhET: Electric Field of Dreams]
* [https://phet.colorado.edu/en/simulations/charges-and-fields PhET: Charges and Fields]
* [https://phet.colorado.edu/en/simulations/electric-hockey PhET: Electric Field Hockey]

Additional simulations involving time-varying electric fields and electromagnetic waves:

* [https://phet.colorado.edu/en/simulations/radio-waves PhET: Radio Waves and Electromagnetic Fields]
* [https://phet.colorado.edu/en/simulations/microwaves PhET: Microwaves]

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= Additional Resources =

* [https://iwant2study.org/lookangejss/05electricitynmagnetism_11efield/ejss_model_electricfieldwee/electricfieldwee_Simulation.xhtml Electric Field Vectors]
* [https://www.falstad.com/vector3de/ Three-Dimensional Electric-Field Visualization]
* [http://ffden-2.phys.uaf.edu/212_fall2003.web.dir/kristina_smith/description.html Lightning Applet]

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<div style="float:left; font-size:90%; margin:0.25em 0 0.75em 0;">
[[Electrostatics|⬅ Previous: Electrostatics]]
</div>

<div style="float:right; font-size:90%; margin:0.25em 0 0.75em 0;">
[[Electric Potential|Next: Electric Potential ➡]]
</div>

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2025-12-15T11:08:55Z

Brachers: /* General Resources */