Electric Potential

pot1pAni.gif (35115 bytes) The potential of the nucleus of an atom in a molecule.   The nucleus creates a positive potential.  This potential is inversely proportional to the distance, r, from the nucleus.  The potential is given by the equation:
25-26.gif (205 bytes)
which reduces to:
V = (ke q)  (1 / r)
a constant multiplied by 1/r.


The "Particle in a Box" or "Muffin-Tin" model.  When the positive nuclei of atoms are brought close together in a lattice their potentials combine.  This in physics is called "the superposition principle."  This forms a potential well or "box" where electrons become trapped.  The last frame of this animation shows the fully formed box. 

You can click eight times on the button below the picture and see how the potentials add to form the final "potential well."  The last click will trigger the entire animation.

Note: the potentials go asymptotically to infinity (because they are proportional to 1/r.)

For the more enthusiastic, two virtual reality worlds are available, on this web site, for the particle in a box.  The first (Box 1 (VRML 2.0 - 66K) ) shows the positions of the nuclei of atoms in a crystallographic plane, above the box.  The second (Box 2   (VRML 1.0 - 609 K)) is the representation of the box with potential shown by gradient of color (similar to the picture to the left). VRML Box 1 is better and takes less time to load.

An interactive Java Applet is available for this page.  Electric potential lines are an abstract concept (in that they don't really exist and can be difficult to visualize.)  We recommend that you use this applet to get a feel for electric potential lines with a number of different point charges, of different magnitudes, signs, and values.

Click here to use this applet.

Previously Asked Questions

Q:     "I have been trying to grasp the physics behind the "volt".  I am interested to what exactly is meant by "a volt".   To my understanding, a volt is a potential difference.  I tried to understand this potential difference with an analogy of gravity.  If an object is on the ground, it has no gravitational potential energy, but as work is done on lifting it, the work done on it is equal to the potential energy it gains ... right?  But how does this relate to an electrical field?  If one Joule of work is transferred to one coulomb of charge it is said to have a potential of one volt??  But in a battery, if one side is positively charged and the other negatively charged, I don't see how work was done on either side to give it this potential.  My question may seem unclear, but how is doing work on a coulomb of charge in an electric field associated with the two terminals in a battery?"

A:    PART 1

potheightgAni.gif (32623 bytes) potheighteAni.gif (32794 bytes)

Potential V is the analog of height/level/altitude/elevation h.

The concept of "volt" or "potential" in electricity is similar to the concept of "height" in gravity and the concept of "temperature" in thermodynamics. In all of these cases, a reference level is defined from which volt/height/temperature are measured. The zero for voltage is considered to be the voltage of the planet earth, called in electrical engineering "ground". The zero for gravity is considered to be the ground level (in the case of Earth). The zero for temperature is considered to be the so-called Zero Kelvin.

With respect to these reference levels, one can define potential differences (or voltage differences), as well as elevation differences or temperature differences. In the case of electricity, electric charges (for example electrons) fall from a high voltage to a low voltage. This is the electric current. In the case of gravity, material objects (for example rocks or drops of rain water) fall from a high elevation to a low elevation. In the case of thermodynamics, molecules of gas (for example air) "fall" from high temperature to low temperature. So, we have electric currents, landslides or rivers, and heat flow, all happening between points with different "characteristic elevations".

When an electric current flows through a wire, between the poles of a battery (say 5 volts potential difference), one of the poles may be grounded and the other is at an "electric elevation - called voltage - of 5 volts". The same electric current flows through the wire if one of the poles of the battery is not grounded but kept at any "electric elevation - called voltage" (for example 100 volts), and the other pole of the battery is at 105 volts. The potential difference in both cases is 5 volts.

In a similar way, rocks may fall between the top of a tall building 10 meters high to the ground, or from ground to the bottom of a pit 10 meters below the ground level. What matters is not the individual values of initial and final heights but the magnitude of the height difference.

In a similar way, heat may be transported between a hot point at 100 degrees Celsius and a cold point at 30 degrees Celsius, or from a hot point of 50 degrees Celsius to a cold point of minus 20 degrees Celsius. What is important is not the individual values of initial and final temperatures but the magnitude of the temperature difference.



Now for the concept of energy.

When a material object (a rock) is positioned at a level different from the reference level (either above or below it), it has a non-zero potential energy. This energy is exactly equal to the work needed to be done to displace the object from the reference level to the final level. The total work done, and therefore the total potential energy of the object in the final state, is equal to the weight of the object (mg) times the difference in elevation (h) between the final and the reference levels.

When an electric charge (an electron) is positioned at a potential (electric level) different from the reference level (ground level), it has a potential energy equal to the charge of the object (q) times the difference in potential (V) between the final and the reference levels.


About the concept of field.

The gravitational field exists in the direction of h (the vertical line that connects two points at different elevations, for example the top of a tree and the bottom of a pit below it). The electric field exists in the direction of V (the line that connects two points at different voltages, for example the two poles of a battery). The thermal field exists in the direction of T (the line that connects two points at different temperatures, for example a hot fire place in a room and the cold window of the room).


About the relationship between Coulomb of charge, potential in volts, and the work done in Joules. The formula that relates charge, voltage, and work is:

Work = (Charge) x (Potential difference or Voltage difference)

In the standard metric system of units, the unit for work is Joule, the unit for charge is Coulomb, and the unit for potential difference is Volt. Therefore, the equation before becomes:

1 Joule = (1 Coulomb) x (1 Volt)



Your last point referring to a battery that has one pole at a positive voltage and the other at a negative voltage is exactly similar to the case of when we consider a positive altitude (for example the top of a tree) and a negative altitude (for example a pit). Let's consider some numerical values.

Let's assume that the battery has its poles at +3 volts and -2 volts, to a total potential difference of 5 volts between the poles. An electric current flowing between the two poles of this battery is identical to the electrical current that would flow if the poles of the battery would be at + 5 volts and ground (0 volts), or at +25 volts and +20 volts, or at any other individual values of the potentials of the battery poles, with the condition that the voltage difference between the two poles is equal to 5 volts.

In the case of gravity, let's assume that the tree is 3 meters tall (as measured from the reference at the ground level) and the pit is 2 meters deep (which means that the level of the bottom of the pit has a negative 2 meter altitude, as measured from the reference at ground level). The total difference in altitude between the top of the tree and the bottom of the pit is 5 meters. The fall of a rock in this 5 meter difference of altitude is the same in the case of the three meter tall tree and the two meter deep pit, or between a 5 meter tall tree and the ground level, or between the roof of a tall building and a ledge 5 meters below, or between any two elevations that are at a difference of 5 meters with respect to each other. In all these cases, the work done by the rock in falling down is equal to the work somebody needs to do on the rock to raise it.

Q:     I've noticed that electric potential lines often resemble the lines on a topographic map.  Can you explain this?

A:    The concept of potential, V, in electricity is equivalent to the concept of "height"/"level"/"altitude", h, in the case of the gravitational field.  So, the lines of equal potential for an electric field are the equivalent of the lines of equal altitude in a topographic map.

Q:     What is the difference between electric potential and electric potential energy?

A:    Electric potential, V, and electric potential energy, U=qV, are different quantities, with different dimensions and different SI units.  Energy of any kind (electrical qV, gravitational mgh, etc.) represents the same physical quantity.  Electric potential, V, is the equivalent of "height"/"level"/"altitude", h, in the case of the gravitational field.  Therefore, electric potential, V, and electric potential energy, qV, are as different as "height", h, is different from potential gravitational energy, U = mgh.

Q:     The electric potential energy inside a sphere, E ~ r, is different from the potential energy outside of the sphere, E ~ 1/ r2.  Please explain.

A:   potsphere.gif (3256 bytes)

First we draw a diagram of the problem:

Then separate the problem into to cases:
a)   r > R   (r is outside the sphere)
b)  r < R   (r is inside the sphere)

a) r > R
        equ1.gif (502 bytes)

        So outside the sphere (r > R)  E ~ 1/r2.

b) r < R
        Assume a constant volume charge density ( equ3.gif (279 bytes)).
        equ4.gif (693 bytes).   goesto.gif (52 bytes)  equ5.gif (337 bytes)

        since:   equ2.gif (224 bytes) ,

                   equ6.gif (600 bytes)

        So inside the sphere (r < R)  E ~ r.  

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Change in the electric potential U delta.gif (839 bytes)U = Uf - Ui = -W
Electric potential energy 25-2.gif (120 bytes)
Potential difference between two points in an electric field 25-7.gif (258 bytes)
Potential at a point 25-8.gif (172 bytes)
Potential and potential difference in terms of the electric potential energy U of a particle of charge q in an electric field. V = U / q

25-6.gif (401 bytes)

Electric potential difference between any two points 25-18.gif (257 bytes)
Electric potential difference for a particular point 25-19.gif (209 bytes)
Potential due to a single point charge at a distance r from that point charge 25-26.gif (205 bytes)
The potential due to a collection of point charges 25-27.gif (369 bytes)
The electric potential of a dipole 25-30.gif (274 bytes)
Potential due to a continuous charge distribution 25-32.gif (249 bytes)
Calculating E from V

The component of E in any direction is the negative of the rate of change of the potential with distance in that direction:

25-40.gif (171 bytes)

The x, y, and z components of E may be found from

25-41.gif (430 bytes)

When E is uniform 25-40.gif (171 bytes) reduces to:

25-42.gif (171 bytes)

where s is perpendicular to an equipotential surface.  The electric field is zero in any direction parallel to an equipotential surface.

Electric potential energy of a system of point charges 25-43.gif (297 bytes)

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List of Topics

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Electrostatics Capacitance Sources of Magnetic Fields Maxwell's Equations Interference
Electric Fields Current and Resistance Magnetism in Matter Electromagnetic Waves Diffraction
Electric Flux Electrical Circuits (DC) Electromagnetic Induction Interaction of Radiation with Matter: Reflection, Refraction, Polarization