Three Charge API Test

The electric field from point charges represented as a vector field.

A vector field assignes a vector to every point in some region of space. Perhaps the quintessential vector field is the electric field of an isolated charge.

Take a moment to play with the field to the left. The clearest feature is that there are vectors everywhere, not just at a specific point. This is the difference between a vector field and a vector. The electric field extends over space. But the the velocity vector of a car is defined only at the location of the car.

Spin the field around and notice that while individual arrows move, the field as a whole looks the same after any rotation in any direction. This is an example of what is called spherical symmetry. Another way to visualize spherical symmetry is that the field has the same strength at every point on a sphere centered at the origin of the field.[Considering adding a control to show such a sphere]

Taking a look at the equation for the field, we see that it is dependant on $r$ so we expect the spherical symmetry.

E = \frac{Q}{r^{2}} \hat{r}

We can also write this in terms of cartesian coordinates using

\hat{r} = \frac{x \hat{ı} + y \hat{ȷ} + z \hat{k}}{{(x^{2} + y^{2} + z^{2})}^{\frac{1}{2}}}

We substitute this into the expression for $E$ and get this longer looking expression. This is another common method for writing out vector equations, using separate equations for each of the $\hat{ı}$ , $\hat{ȷ}$ , and $\hat{k}$ components of the field.

E = \frac{Q}{{(x^{2} + y^{2} + z^{2})}^{\frac{3}{2}}} x \hat{ı} + \frac{Q}{{(x^{2} + y^{2} + z^{2})}^{\frac{3}{2}}} y \hat{ȷ} + \frac{Q}{{(x^{2} + y^{2} + z^{2})}^{\frac{3}{2}}} z \hat{k}

Comparing these two expressions for $E$ reveals that math and physics can frequently be simplified by choosing a representation that reflects the symmetry of the problem.