Fields

Fields (in general)

How do forces between separated objects arise?

For gravity, The Earth creates a "gravitational field" which can be represented by a sort of infinite sea of vectors, each with the direction and magnitude corresponding to the force from the Earth a "test mass" would feel at that point. This is an example olf a vector field.

Calculating Earth's gravitational field shouldn't involve the Moon in any way, so we take the Moon term out of our traditional gravitational force equation to get that the gravitational field of Earth is equal to \(\frac{GM_{earth}}{R^2}\), and \(F_{grav} = M_{moon} * \text{Gravitational field of Earth}\). The units for a gravitational field are N/kg, which is equivalent to m/s².

Electric Fields

Electric fields are the same as the electric force equation is the same. However, we have charges in the numerator instead of masses. It has units of N/C, and this is not the same as acceleration.

SIDENOTE: The electric field is represented by the variable \(E\).

Note that fields are not "mathematical tricks", they are a physical reality and can transmit energy. Every mass has a gravitational field, and every charge an electric field.

Direction

Gravity is straightforward - all masses attract other masses, making determining the direction of the field simple. However, electric charges can attract or repel so there are two possibilities: either the field has two values for each point (one if it the "test charge" is positive, and the other if it's negative) or a single-direction field but keeping in mind that the direction of \(Q_{\text{test}}\) depends on the sign \(Q_{\text{test}}\). Option 2 is what was chosen by scientists and engineers.

Positive test charge: electrical force described by field is in the same direction
Negative test charge: direction is opposite of field

No matter what, \(F = qE\).

Alternate Representations

Representing a field as rays emanating or arriving in the origin can be used to describe a field, where density of rays represents magnitude and the direction is tangent to the rays.

TODO Improve notes on alternative representations of fields: Here at ~38:00

Combined Fields

Method 1: Illustrating the combined effects on a test charge.
Method 2: Adding said forces up to show net force
Method 3: Connecting these with the line representation to show the electric field

None of these actually show repulsion, just the force on a third test charge.
SIDENOTE: This is three dimensional and better represented with a 3d diagram.

Conductors

If the charges in a conductor are static (net equilibrium), the electric field in the material must be zero (in order to be in equilibrium), and the field on the surface must be perpendicular to each point on the surface (as if there was a horizontal component to the field electrons would flow).

Example (note that due to the topology of the conductor, certain regions will feel a stronger field):

Take the example of a neutral conductor surrounding a positive charge. The conductor will be polarized - but to what extent? A balancing electric field created by the continued polarization of the conductor brings equilibrium and an \(E_{net}\) of 0 inside the conductor.