Exam Notes 4.2 & 4.3

Gravitational & Electric Fields

Newton's Law of Gravitation

F = \frac{GMm}{r^2}

Force between two objects is proportional to the product of their masses and inversely proportional to the square of their separation.

Coulomb's Law

F = \frac{kQq}{r^2}

k = \frac{1}{4\pi\epsilon_0} = 9 \times 10^9

Force between two objects is proportional to the product of their charges and inversely proportional to the square of their separation.

Field Strength

Gravitational field strength:
g = \frac{GM}{r^2} (N/kg) (aka 'acceleration due to gravity')

Electric field strength:
E = \frac{kQ}{r^2} (N/C) (aka 'potential gradient')

Potential

Gravitational Potential: V = -\frac{GM}{r} (J/kg)

Electric Potential: V = \frac{kQ}{r} (J/C)

Potential is only 0 at ∞ or between unlike charges.

Potential Energy

Gravitational Potential Energy: E_p = -\frac{GMm}{r}

Electric Potential Energy: E_p = \frac{kQq}{r}

Escape Velocity

The minimum speed any object needs to be launched with in order to escape a planet's gravitational field (∞).

v = \sqrt{\frac{2GM}{r}} = \sqrt{2V}

Distance of Closest Approach

kQq/r = \frac{1}{2}mv^2
This is only true if they have like charges, and the particle is moving directly towards the other, so all E_p is converted to E_k

Satellites

A satellite in a circular orbit experiences a centripetal force that is equal to the force of gravity.

v = \sqrt{\frac{GM}{r}}

Kepler's 3rd Law

T^2 \propto r^3

Parallel Plates

For parallel plates: E = \frac{V}{d}

If levitating, the force due to the electric field must equal its weight.

Q = \frac{mgd}{V}