Topic 10: Fields
10.1 Describing Fields
Gravitational...
Fields

Magnitude: g = GM/r², in N kg^1.

Field lines: For a point or spherical mass M, the field is radial, with the field lines towards that mass. In the case of a planet, when very close to its surface, the planet may be considered flat and the field uniform.
Potential (Vg)

Definition: "The gravitational potential at a point P in a gravitational field is the work done per unit mass in bringing a small point mass from infinity to point P".

Vg = W/m = GM/r, in J kg^1.

Work: the work done depends only on the change of the potential, not on the path taken.

Positive work is done on the test object, increasing the gravitational potential.

Negative work is done by the test object, decreasing the gravitational potential.

Potential energy (EP)

Definition for one body: "The gravitational potential energy of one body is the work done to bring one mass from infinity to a specific point".

Definition for two bodies: "The gravitational potential energy of two bodies is the work done in bringing the bodies to their present position when they were infinitely apart".

EP = GMm/r. (negative sign implies that force is attractive and that +GMm/r must be provided to infinitely separate them).
Electric...
Fields

Magnitude: E = F/q = kQ/r².
Potential (Ve)

Definition: "The electrical potential at a point P is the work done per unit charge for a small positive test charge to be brought from infinity to that point".

May be visualized as the height of a flat surface.
Potential energy (EP)

Definition: "The electrical potential energy at a point P is the work done for a small positive test charge to be brought from infinity to that point".

Charge sign must always be taken into account.
Parallel plates

Explanation: long oppositely charge plates.

Field is uniform in the region between the plates.

Edge effect: field becomes weaker at the edges.
Equipotential surface

Explanation: consists of those points that have the same potential, i.e. which are at the same distance from the source (referred to as zero potential), and where masses or charges move without work being done on or by then.

Field lines are cut perpendicularly by the equipotential surfaces.
10.2 Fields at Work
Graphical interpretation of gravitational field strength and potential

Going upstream in the field (against) means going to a higher potential, so gain in the potential.

Going downstream in the field (in favour) means going to a lower, so loss in the potential.

Gradient of a graph of gravitational potential against distance is the gravitational field strength. g = ∆Vg/∆r
Inside a planet
Weightlessness
Feeling weightless for an astronaut in orbit around the Earth is a consequence of both ship and the astronaut "falling freely", with the same acceleration towards the center of planet, so that there is no normal force.
Orbits

Orbital speed (vorbit): sqrt(GM/r).

Orbital period (Torbit): sqrt[4π²r³/(GM)].

Polar orbit: for satellites close to the Earth's surface (100 km).

Geostationary orbit: for geosynchronous satellites, whose period is equal to 24 hours.
Total energy (ET) = kinetic energy (EK) + gravitational potential energy (EP).

ET = EK + EP = 1/2mv²  GMm/r = GMm/2r  GMm/r =  GMm/2r

Graph of the kinetic, potential and total energy of a mass in circular orbit around a planet as function of distance.

Increase in the orbit: total energy increases, potential energy increases and kinetic energy decreases.

Air friction: radius decreases, causing the total energy to decrease, potential energy to decrease and kinetic energy to increase.

Launching a body from a planet's surface cases:

If total energy is positive: object will follow a hyperbolic path and never return.

If total energy is zero: object will follow a parabolic path to infinity, where it will stop.

If total energy is negative: object will go into a circular or elliptical orbit or crash.


Escape velocity: "minimum speed of object to escape gravitational field of planet/travel to infinity, starting at the surface of a planet, without energy input".

vescape: sqrt(2GM/r).

Graphical interpretation of electric field strength and potential

Electric field strength is the force per unit charge, and thus, the area under the graph of electrical field strength against distance is the work per unit charge, i.e. the electric potential charge.
Inside a hollow conducting charged sphere

As the sphere is a conductor, all the surplus must reside on the outside of the sphere.

Charges will move until they are as far apart as possible and in equilibrium equidistant on the surface.


Inside a sphere, the force acting on a test charge are always equal in sizer and opposite in direction, and thus, cancel out: E = 0, which is the gradient ∆Ve/∆r, which means that V is constant.
Charges moving in magnetic and electric fields
Magnetic fields: force will be at right angles to velocity and magnetic field strength.

Circular path: when the charge's direction is perpendicular to magnetic field strength.

Magnetic force = centripetal force


Helical path: charge's movement when direction is not perpendicular to magnetic field strength.
Electric field produced by the uniform field in parallel plates

Only vertical acceleration, no horizontal.

Combination of magnetic and electric fields opposing each other, which may generate balance of forces and the charge may move in a horizontal path.
Inverse square law behavior

Geometric explanation: influence per unit area reduces to the power of 2.