22.13:
Electric Field Lines
Consider the two-dimensional projection of a positive point charge's electric field. The vectors point away, and their magnitudes decrease with distance.
For a dipole, this field distribution becomes challenging to visualize. For a large number of charges, it becomes intractable.
An alternative visualization via electric field lines solves this problem.
They are defined as lines with direction, such that at any point, the tangent to the electric field line gives the direction of the electric field.
The magnitude of the electric field is given by the density of the electric field lines, the number of field lines per unit cross-sectional area perpendicular to the field.
This representation does not require tracing the lengths of the vector arrows but still traces the electric field at any point uniquely.
For example, the electric field lines of a dipole indicate that the field points away from the positive charge and into the negative charge.
If the magnitude of the positive charge is larger than the negative charge, the field lines become denser near the former.
22.13:
Electric Field Lines
The three-dimensional representation of the electric field of a positive point charge requires tracing the electric field vectors, whose lengths decrease as the square of their distance from the charge and which point away from the charge at each point. This vector field is no doubt challenging to visualize. The visualization of electric fields becomes quickly intractable as the number of charges increases.
The solution to this problem is to use electric field lines, which are not vectors but describe a vector field.
They are defined such that the magnitude of the electric field at any point is given by the density of the electric field lines around that point. Since the density varies with distance from the charges and is defined uniquely at each point in space, it uniquely describes the field vector's magnitude. That is, there is a one-to-one correspondence between the magnitude of the electric field and the density of the electric field lines, thus justifying the definition.
If the electric field lines are close together, the field's magnitude is large at that point. The magnitude is small if the field lines are far apart at the cross-section.
The direction of the electric field is also uniquely defined by the electric field lines. At any point, the tangent to the electric field line determines the electric field's direction. Thus, the definition necessitates that electric field lines do not crisscross each other.
It is important to note that a single electric field line does not trace the magnitude of the electric field, nor does it represent a specific value of the field. Instead, the significant physical quantity is the density of the field lines.
Although the direction and relative intensity of the electric field can be deduced from a set of field lines, the lines can also be misleading. For example, the field lines drawn to represent the electric field in a region must, by necessity, be discrete. However, the actual electric field in that region exists at every point in space.
Suggested Reading
- OpenStax. (2019). University Physics Vol. 2. [Web version]. Retrieved from openstax.org/books/university-physics-volume-2/pages/5-6-electric-field-lines; section 5.6; page 213.