Electric Potentials  |  Connecting the Electric Potential and Field

Connecting the Electric Potential and the Electric Field

The electric field and potential have a close relationship. In 1-D the electric field can be thought as negative the slope of the electric potential.

$\bar{E}_x = -\frac{\Delta V}{\Delta x}$

Lessons from the above equation are:

   1) if $\frac{\Delta V}{\Delta x} = constant$ (linear), then $E_x$ is constant and uniform, this is the case near large sheets of charge and in parallel plate capacitors.

   2) The relationship between the electric field and potential is independent of the test charge $q_0$ - it's independent of other charges interacting with the field

   3) The minus sign tells us the electric field points towards decreasing electric potential - if $\frac{\Delta V}{\Delta x}$ is negative, $E_x$ is positive

In 3-D the electric field is negative the gradient of the electric potential.

$\overrightarrow{E} = - \langle \frac{\Delta V}{\Delta x}, \frac{\Delta V}{\Delta y}, \frac{\Delta V}{\Delta z} \rangle$

A gradient is a vector and can be thought of as a kind of 3-D slope that points in the direction of steepest ascent. With the minus sign the electric field points in the direction of steepest descent. Since this is the direction of the force on a positive charge we can again make the analogous connection to the gravitational field and how water is forced to run downhill, down the direction of steepest descent. This feature adds one more lesson:

   4) The electric field is perpendicular to equipotential surfaces and lines

Summary

We can now go full circle with the connections between the electric field and potential, and the electric force and energy. The electric field and potential are owned by the charge distribution and preclude any interactions with new test charges $q_0$. The electric force and potential energy require an interaction with the test charge. The concept map below helps visualize the relationship.

Key Equations and Infographics

Now, take a look at the pre-lecture reading and videos below.

Required Videos

Suggested Supplemental Videos

OpenStax Section 19.2  |  Electric Potential in a Uniform Electric Field

1.  Two plates of metal with uniform charge distribution are held a distance d apart from each other as shown in the figure below.  The metal plate on the left side is positively charged while the metal plate on the right is negatively charged.

a.  Which plate is considered to be at a higher potential ($V$)?

b.  The electron labeled $1$ is placed in-between the two sheets as shown.  Does this electron have a high or low electric potential energy?  Ignore the other charges in-between the plates.  

c.  The electron labeled $2$ is placed in-between the two sheets as shown.  Does this electron have a high or low electric potential energy?  Ignore the other charges in-between the plates.

d.  The proton labeled $1$ is placed in-between the two sheets as shown.  Does this proton have a high or low electric potential energy?  Ignore the other charges in-between the plates.

e.  The proton labeled $2$ is placed in-between the two sheets as shown.  Does this proton have a high or low electric potential energy?  Ignore the other charges in-between the plates.

2.  Two parallel plates are held $20.0 \, mm$ apart and the potential difference between the plates is $110 \, V$. 

a.  What is the magnitude of the electric field between the two parallel plates?

b.  Which direction does the electric field point if the plate on the left has a uniform negative charge distribution and the plate on the right has a uniform positive charge distribution?

c.  Sketch a physical representation of this problem.  Include a few equipotential lines between the two parallel plates.

3.  Two uniformly charged parallel plates are held some distance “$d$” apart from each other with an electric potential of $300 \, V$ between them.  An electron is placed at rest near the negative charged plate.  Calculate the speed at which the electron will hit the positive plate. 

4.  The figure below shows a charge distribution made up of $3$ point charges.  The charges are fixed in space and cannot move.  Let $q = 1 \, \mu C$ and $L = 1.0 \, cm$.

a.  Find the electric potential energy of this charge distribution.

b.  Find the electric potential at point P.

CLICK HERE for solutions.

Short foundation building questions, often used as clicker questions, can be found in the clicker questions repository for this subject.

BoxSand practice problems - Answers

BoxSand's conceptual problems

BoxSand's multiple select problems

BoxSand's quantitative problems

Recommended example practice problems 

• OpenStax and it's 4 related sections. Problems at bottom of page,
  • Electric Potential Energy, Website Link
  • Electric Potential in a Uniform Electric Field, Website Link
  • Electric Potential Due to a Point Charge, Website Link
  • Equipotential Lines, Website Link
• Physics-Prep
  • Electric Potential, Website Link
  • Electric Potential Due to Point Charges, Website Link
  • Equipotential Surfaces, Website Link

For additional practice problems and worked examples, visit the link below. If you've found example problems that you've used please help us out and submit them to the student contributed content section.

Learning Objectives

YouTube Videos

Simulations

Demos

History

Physics Fun

Other Resources

Resource Repository

The Physics Classroom has two sections for us, one on electric potential and the other on electric potential difference.

Electric Potential	Electric Potential Difference

Boston University's Page on electric potential is a neat reference with a couple of example problems

PPLATO is a complete resource with a lot of information, and several practice questions per subject. This webpage covers electric charge, the electric field, and electric potential, we've already covered charge, and the electric field, so now it's time to focus on electric potential!

Here's a link to Hyperphysics' reference for electric potential energy

 

Isaac Physics' section on the electric field is a good short resource. This page contains previously studied material on electric fields as well as information on electric potential, and the connection between electric field and electric potential.

 

 

Other Resources

This link will take you to the repository of other content related resources .

This link will take you to the repository of other content related resources.

Problem Solving Guide

Use the Tips and Tricks below to support your post-lecture study.

Assumptions

Checklist

Misconceptions & Mistakes

Pro Tips

Multiple Representations