While the gravitational and electric fields are very different, they share many of the same relationships. In this section, diagrams help illustrate how to translate information from our new concepts of fields and potentials back to the familiar concepts of forces between two objects and the potential energy between those objects. These diagrams are different but have very similar structure. The main difference (other than the names of the field) is that in the electric case, to translate from new to old concepts and back, we use the electric charge, while for the gravitational field we use the mass. The analogous diagram for the magnetic field is very different. That is because there is no analog of magnetic potential energy.
Potential exists for time-independent electric fields only
The magnetic field has no potential (that we can discuss in 7C), and it also depends on the velocity. The direction of the magnetic force is not in the same direction of the magnetic field; the relationships above are for the magnitudes only. Finding the direction of the magnetic field is a little bit involved, and this diagram should make more sense after studying the magnetic field later.
Relationship Between Representations
We now know three different representations of a field: using a field map, field lines or equipotentials.
- Field Map:
In the field map representation, we calculate and display the vectors for certain points at a given time. Usually these points are taken to lie on a grid. Vectors are drawn to-scale to indicate the direction and magnitude of the field. This is the most visually direct representation of the field, but when the field varies widely in magnitude, it is difficult to accurately represent the magnitude of each vector on the same scale. Creating the map also involves a lot of calculation.
This representation tries to fix some of the problems with the vector map representation. Here the field lines are joined together to make continuous field lines. The direction of the field is given by the tangent to a field line at that point. The spacing between the field lines indicates the magnitude of the field; dense lines indicate a strong field while sparse lines represent a weak field.
This representation uses lines to indicate areas of the field with equal potential. These equipotentials are always at 90° to the field lines, and no energy is gained or lost by moving along one of these lines. (Note: equipotentials do not exist for magnetic fields). We could draw any number of equipotentials, but to accurately display field magnitude using equipotentials, we adopt the convention to only draw equipotentials that are equally separated in potential. In regions where the field is stronger the equipotential lines are closer together.
The table below summarizes how to read each representation for useful information, and how to go from one representation to another.
|Field Map||Field Lines||Equipotentials|
How to read: Magnitude
|Given by the length of arrow at that point (interpolate, if necessary). Long arrows correspond to strong fields.||Indicated by density of field lines; dense lines correspond to strong fields.||Indicated by distance between neighboring equipotentials. Close equipotentials correspond to strong fields.|
How to read: Direction
|Given by direction of arrow at that point (interpolate, if necessary).||The tangent to the field line lies in the same direction as the field at that point.||Direction normal to equipotential at point, towards lower equipotential.|
|Convert to: Field Lines||Connect arrows with curves that follow the arrows' directions.||--||Take equally spaced normals along the equipotentials. Extend normals between equipotentials to connect them, making sure they follow the field.|
|Convert to: Equipotentials||Convert to Field Lines then draw equipotentials (as on the right)||Take the normals along field lines. Extend normals to connect normals with the same potential, making sure they cross field lines perpendicularly.|| |
This example should help clarify some of the points we're trying to make, and give you a little practice and insight.
Given the field map from the previous example, show the gravitational field lines and the equipotentials for two separate balls of equal mass:
Field Lines: To draw the field lines given the vector map, we try and connect the field line arrows in a smooth manner. A rough drawing is shown below:
Equipotentials: To find the equipotentials we start on a field line and draw across it perpendicular. Then we continue the field line, and keep bending it to ensure that it is perpendicular to the next field line it crosses. A rough sketch to do this is shown, where the dashed lines are the equipotentials.
How close is this to the true answer? A computer generated image shows us the actual equipotentials:
We see that our answer captured most of the information of the correct answer. One of the reasons potentials are so useful for physicists is that they are easy to calculate exactly and therefore easy to generate plots like the one above. In this course, which does not emphasize computation, there is no real advantage to constructing the equipotentials before finding the field lines or vice-versa.
Relationship Between Fields and Waves
The examples of the gravitational and electric fields shown in this chapter have all involved stationary sources, and hence the fields have not changed in time. However if we get the sources to move, then the field at a particular location starts to change in time as well; if we disturb the sources, this disturbance propagates outward like a wave pulse. By disturbing our source periodically we get waves in the field that are identical to the waves we studied earlier in 7C.
In the case of the electric and magnetic fields this sort of oscillation is what we commonly call “light”. This fact is so important that it bears repeating:
- A field can oscillate, producing a traveling wave.
We will return to this idea in more detail after we have introduced electromagnetic waves.
Up until this point we have been using a field as a convenience, but it seemed like we could always just use the direct model of forces and not have to deal with it. The fact that we can generate waves in fields that transport energy (and momentum) tells us that there is more to it than that. If we did not count the field as real, one system would lose energy, that energy would remain lost, and sometime later another system may gain energy, but this violates the conservation of energy!
If we consider the field a physical entity in its own right, rather than just something we model, the description changes. Now, a physical system can transfer energy to the field, the field transports that energy as a wave, and then the field transfers the energy to another physical system. We see that fields are more than a convenient construction, but are required if we want to preserve the conservation of energy!