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Section 11.4 The Geometry of Electric Fields

Activity 11.4.1. Drawing electric field vectors.

Using only what you know about the relationship of charges to electrostatic electric fields, namely:

\begin{equation*} \EE=\frac{1}{4\pi\epsilon_0} \, \frac{Q\hat{r}}{r^2} \end{equation*}

and the superposition principle, sketch the electric field (the vector field) for each of the following static charge configurations:

  • Four positive charges arranged in a square.

  • Two positive charges and two negative charges arranged in a square, with like charges diagonally opposite each other.

  • A line segment with constant charge density.

  • A circular loop with constant charge density.

Hint.

When working through this activity, some things that you might have needed to pay attention to are:

  1. The electric field is a vector field, not a scalar field, i.e. it is a vector at every point in space, not a scalar. Make sure to add the vectors themselves, not their magnitudes.

  2. Typically, zero potential does not correspond to zero electric field and vice versa.

  3. These examples are inherently 3-dimensional; drawing vector fields in three dimensions may be more challenging than visualizing the three dimensional fields in your head.

Solution.

The electric field due to four positive charges arranged in a square is shown below.

Figure 11.4.1. The electric field due to four (positive) charges arranged in a square. Use the check boxes to select which fields to show. (The applet can also be used to visualize the electric field due to two positive charges.)

The electric field due to four charges arranged in a square quadrupole is shown below.

Figure 11.4.2. The electric field due to four charges arranged in a square quadrupole. Use the check boxes to select which fields to show. (The applet can also be used to visualize the electric field due to a dipole.)

Activity 11.4.2. Comparing electric field vectors, electric potential, and electric field lines.

Now, you should repeat this activity in two other ways:

  1. Sketch the level curves for the electrostatic potential (Section 4.3) and then visually/geometrically take the gradient.

  2. Draw electric field lines for these charge configurations.

Hint.

There are two main approaches to drawing the electric field vectors. Which you use will depend on what starting information you have.

  • Vector addition of the known electric fields due to each source;

  • Computing the gradient of the potential, if known, for the given configuration.

Electric field lines are yet another representation of the geometry of electric fields. They satisfy the following properties:

  • Field lines start at positive charges and end at negative charges;

  • Field lines are tangent to the direction of the electric field at each point (i.e. the direction of the field lines is the same as the direction of the electric field at each point). As a consequence of this, field lines never cross.

  • The density of field lines is proportional to the strength of the electric field in that area. Be cautious in trying to represent the electric field in terms of field lines. In particular, drawing electric field lines in two dimensions will not show the correct density everywhere; it requires electric field lines in three dimensions to see the correct fall off.

It would be useful for you to stop and think about which properties of the electric field are best represented by the vector representation of the field and which by electric field lines.