Vector Fields

Vector Field Equations

We defined a vector field as a function of space, which means there’s some math to be expected. Vector field equations can look many different ways — that’s because each component function, or equation corresponding to a dimension in space, will have its own separate function. Remember that a vector pointing straight along (or parallel to) an axis is one-dimensional: vectors pointing left or right only have an$x$component, while vectors pointing up or down only have a$y$component.

However, vectors that don’t point straight will have an angle, which we measure from either the$x$or$y$axis. We can split a two-dimensional vector into its horizontal and vertical components, giving us two new vectors:

A two-dimensional vector can be broken into two separate one-dimensional component vectors, StudySmarter Originals

Vector field equations aren’t too different: we have$x$and$y$components, but instead of static points, we have functions where we can plug in different values. This lets us examine a vector quantity at any point of interest, as well as create vector field graphs. Let’s put this together to write a general vector field equation:

,

where$f(x,y)$is the function for the$x$component and$g(x,y)$is the function for the$y$component. You might see this equation written with parenthesis instead of angle brackets, but both represent a vector field equation made up of component functions.

Graphing Vector Fields

Let’s go through an example using a 2D vector field equation.

Vector field functions can also be three-dimensional or even functions of time, but for now, we’ll be sticking to two-dimensional functions and graphs.

To summarize, here are the most important points to remember about vector fields.

• Vector field equations are multivariable functions of 2D or 3D space, meaning an$x$,$y$, and sometimes$z$coordinate are needed.
• Vector field equations can also be a function of time.
• Vector field equations have a separate function for each component.
• To make a vector field graph, choose evenly-spaced sample coordinates and plug your chosen$(x,y)$values into each vector component function. Plot your vectors once you have enough to get an idea about the general shape of the vector field.
• Vector field graphs can also be created with a graphing calculator by providing the vector component functions. A graphing calculator will choose evenly-spaced sample points to plot.

The Gravitational Vector Field

One of the most important physics applications of vector fields is modeling long-range forces. The gravitational vector field is the long-range force you’re probably most familiar with. We call gravity a long-range force because it’s a weak force that dominates, or appears to be the primary acting force, at larger scales.

Gravity is the universal attraction of all masses to one another, so we can model a gravitational field around any mass, whether it’s a moon, planet, or any smaller or larger object. Since the gravitational force is a vector field, we can choose many different points in space to examine how strong or weak the force of gravity would be on a smaller mass. To get a better understanding, let’s look at a couple of plots that visualize the gravitational vector field.

A portion of the 2D gravitational vector field and a 3D representation showing gravitational potential energy. The magnitude of the vectors near the central mass shows the greatest attractive force exerted, Creative Commons CC0 1.0

In this diagram, we have both a two-dimensional gravitational vector field plot and a three-dimensional gravity well. At the center of the$xy$plane, we have some mass with a gravitational field surrounding it — for example, this could be the Earth. The arrows represent the direction and magnitude of the gravitational force exerted on a smaller test mass — say a small rocky object, one that’s much smaller than the Earth. All vectors in the two-dimensional plot point directly towards the central mass: this is the attractive nature of gravity. Pay close attention to the magnitude of the vector arrows. The closer our small rocky object is to the Earth, the greater the gravitational force exerted. The strength of the force decreases as the distance between the two masses increases.

After examining the two-dimensional vector field plot, look at the visualization of the gravity well and check that this makes sense to you. Imagine that the Earth is centered at the bottom of this well — again, the gravitational force exerted on smaller masses is strongest in the space closest to the Earth. The shape of the gravity well emphasizes just how much stronger the pull of gravity is with decreased distance.

The Gravitational Vector Field Equation

The gravitational vector field equation can be written using Newton’s law of universal gravitation:

$\overrightarrow{F}\left(r\right)=\frac{G{m}_1{m}_2}{r^2}$,

where$G$is the gravitational constant,$m_1$and$m_2$are the two masses, and$r$is the distance between the centers of the two masses. This equation gives us the gravitational force exerted by an object with mass$m_1$on a second object with mass$m_2$, such as the pull of the Earth on the moon. You might also see this written as:

$\overrightarrow{F}\left(r\right)\propto \frac{GM}{r^2}$,

where$M$is the mass of the object with the gravitational vector field we are modeling. This is a way of expressing that the strength of the gravitational field is inversely proportional to the squared distance from the central mass$M$.

Types of Vector Fields

There are two types of vector fields in real two-dimensional space to consider. The first type is called a radial field.

Radial fields are rotationally symmetric, meaning the vector field will look the same after rotating the field about its center. Gravitational vector fields are an example of radial fields. All vectors point directly towards the origin, or center of mass, due to the attractive nature of gravity.

Gravitational attraction is an example of a radial vector field since all force vectors point toward the origin, Creative Commons CC0 1.0

The second type of vector field is called a rotational field or vortex field.

A rotational field can be used to model the flow of fluids or major weather events, like hurricanes.

Hurricanes, an example of a vortex field, show the swirling shape of rotational vector fields, Creative Commons CC0 1.0