Fields

force field

• region in which a body experiences a non-contact force

• can be represented as a vector

field line

• the path which a test mass in a gravitational field field would follow

radial field

• field lines always directed towards centre for gravitational field

• straight

• converge to or diverge from a single point

properties of a radial gravitational field

• region in which a mass experiences a non contact force due to another mass

• equipotential surfaces spherical about centre of mass

• field strength has an inverse square relationship with distance form centre of mass

uniform field

• field is same strength and direction at every point

• parallel field lines

gravity

• universal attractive force acting between all matter

gravitational field strength, g

$$g=\frac{F}{m}$$

• the force per unit mass on a small test mass placed in the field at a particular position in the field where it is only acted on by gravitational force

• negative of gravitational potential gradient

• Nkg^-1

• vector

gravitational potential (V)

• the work done to move a unit mass from infinity to a particular point in a gravitational field

• zero value at infinity

• negative everywhere within the field

• energy required per unit mass to remove that mass from the gravitational field (work must be done by the mass against gravity to leave the field)

• V=W/m

• scalar

gravitational potential difference

• the work done in moving a mass m between two points in a gravitational field

• ∆W=m∆V

gravitational potential energy

• work done to move a small test mass from infinity to a particular point in a gravitational field

• also negative as 0 gpe at infinity and work must be done by gravitational force moving object from that point in field to infinity

• work is done by the field so work supplied is negative

Equipotential

• a line or surface in a field along which the electric or gravitational potential

  is constant

• no work is done when moving along an equipotential surface

• the direction of the field is perpendicular to a point on the line

Why equipotentials get more spaced apart

• at an increased distance from the mass, the gravitational field gets weaker so the gain of gpe per unit distance (height) decreases

• for an equal increase in potential, the equipotentials must become more spaced apart since that same work is done by mass against a smaller force in moving between equipotentials so a greater distance must be moved

• the closer they are, the greater the potential gradient and the stronger the field is

• not uniform field - changing potential gradient / changing field strength

the potential gradient

change of potential per unit change in distance along the field at a point in a gravitational field

• Jkg^-1m^-1 or Nkg^-1

$$\frac{\Delta V}{\Delta r}$$

graph of gravitational potential V against distance from centre of planet

• values for V all negative as gravitational potential is always negative

• $$V=-\frac{GM}{r}$$

• V is inversely proportional to -r

• g ( gravitational field strength) is the gradient at any point

newton’s law of gravitation

assumes that the force between two objects with mass is

• always attractive

• proportional to the product of the masses of the two objects

• inversely proportional to the square of their separations

$$F=\frac{Gm1m2}{r^2}$$

Gravitational constant G

• 6.67×10^-11 Nm²kg^-2

• Same for all masses

gravitational force on a test mass mt in a gravitational field between two bodies

• test mass experiences attractive forces from both objects

• point in the field between the masses at which the test mass experiences a resultant force of 0 as the forces on it will be equal and opposite

• Force from m1 = $-\frac{Gm1m3}{d^2}$

• Force from m2 =$\frac{Gm2m3}{\left(D-d\right)^2}$

• where D is the distance between the centres of both bodies and d is the distance from centre of m1 to centre of test mass

gravitational force against distance

gravitational field strength from law of gravitation

• can determine the gravitational field strength at any point in the field of a mass

• a spherical body of mass can be treated as if it was a point mass acting from a point in the centre (same field lines)

• M is the mass of the body producing the field

• $$g=\frac{GM}{r^2}$$

variation of g for spherical planet as you move a test mass beyond radius

• g drops off rapidly as it is inversely proportional to square of distance from centre

variation of g with distance from centre less than radius R ( within body of mass)

• g is 0 at centre and increases linearly with distance until it reaches radius (surface)

• g is directly proportional to r

gravitational potential from law of gravitation

$$V=-\frac{GM}{r}$$

escape velocity

• minimum velocity an object must be given to escape from a planet when projected vertically from the surface

• needs sufficient kinetic energy to overcome gravitational force of attraction towards earth

• $$v=\sqrt{\frac{2GM}{R}}$$

radial speed of orbit

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

• derived from equating centripetal force equation to gravitational force

kepler’s third law

$$\frac{r^3}{T^2}=\frac{GM_{S}}{4\pi^2}$$

linear speed of satellite in higher circular orbit

• linear speed decreases and radius of orbit increases

assumptions for calculating radial speed

• point masses

• circular orbit

geostationary satellite

• orbits in plane of earth’s equator - satellite stays above same point on earth’s equator

• 24 hr time period

• orbits in same direction as earth’s direction of rotation

• communications

• avoids dish having to track / needs to remain at same point above the earth

• same angular velocity as a point on the earth’s surface

energy of an orbiting satellite

• sub radial speed equation into kinetic energy equation $\frac{GMm}{2r}$

• gravitational potential equation x mass $-\frac{GMm}{r}$

• $$E_{_{total}}=\frac{GMm}{2r}-\frac{GMm}{r}$$

• total energy always negative

• for any object in fixed circular orbit, kinetic and potential energy constant

• elliptical orbit - total energy constant but potential and kinetic energy vary such that as one increases, other decreases

• gpe of orbiting satellite negative - energy would be required to reach 0 and leave

minimising the amount of energy required to send satellite into orbit

• equator

• highest initial speed and ke due to earth’s rotation here

field strength two masses acting on a test mass

• if their gravitational force in the same direction, add field strengths

• if gravitational force in different direction, substract field strengths

gravitational potential acting on a mass due to other masses

Vt = V1+V2

how shape of field around two masses determines relative masses

• denser field lines so a greater field strength and a greater mass

• null point is closer to the lighter mass

satellite in orbit between earth and moon. explain how the resultant force on the satellite due to the gravitational fields of the earth and the moon causes satellite to orbit with the same angular speed as the moon

• without moon’s gravitational influence, the force and gravitational fields of strength on satellite due to earth would be too large

• the required centripetal force to keep it in circular motion is smaller at the same angular velocity than for moon as it is closer to earth

• satellite angular velocity is greater than moon’s and period less than moon’s

• when moon aligns with satellite, it exerts a gravitational force along the same line as earth’s force, but in opposite direction, reducing centripetal force

• to maintain circular motion $F^{}=\frac{V^2}{r}$ radius decreases

• $T=\frac{2\pi r}{v}$ so time period increases

• $\omega=\frac{2\pi}{T}$ so angular velocity decreases to that of moon’s

Earth, moon and satellite all have same angular speed so that their centres are always in a straight line. Why centripetal motion is maintained about earth.

• earth is reference so it’s angular velocity doesn’t have an effect (since its field is radial)

• earth exerts a constant gravitational force on satellite towards its centre

• moon always exerts a constant gravitational force along the same line towards its centre

• the resultant centripetal force is the sum of these two forces and is about earth’s centre

why P cannot have a circular orbit around H

• resultant force of H and G will not be continuously centripetal - and at different locations on a circular path - magnitude of force due to G decreases as radial distance from G increases

• the circular path will not follow an equipotential so the total gpe will be different which required energy that cannot be supplied

object launched from earth’s surface. Why is gravitational force of sun more important than gravitational force due to earth on the motion of the object?

• force from earth is less than the force from sun at a similar distance due to sun’s much larger mass

• the total work done in moving a long way from sun is much greater than moving a long way from earth

electric field

region around a charged object where another charged object will experience a force

electric field lines

show the path that an isolated positive test charge would follow, if it were to be placed in that field.

electric field strength

• force per unit charge on a positive test charge placed at that point in the electric field

• vector

E = F/Q

two oppositely charged parallel plates

• uniform field

work done to move charge against a pd

• work done = Fd = VQ

uniform electric field strength

E =V/d

• V is pd

• d is distance between two plates

charged particle in uniform electric field

• constant electrostatic force

• constant acceleration

deflection of charged particles in electric fields

• gains horizontal velocity as it is accelerated by the pd at the electron gun (gains ke)

• horizontal velocity constant between plates

• vertical deflection between plates due to now under constant vertical acceleration

permittivity of free space

• electric charged try to distribute themselves equally over the surface of a metal conductor

• irregular shaped object can lead to a greater concentration of charge at some positions

• increases field strength

• E is directly proportional to Q/A

• constant of proportionality is permittivity of free space

• Fm^-1

• resistance of free space to an electric field passing through it

• charge per unit area on oppositely charged parallel plates when the electric field strength between them is 1 Vm^-1

electric potential

• the work done per unit charge to move a positive test charge from infinity to a position in the electric field

• positive test charge moving from point of higher to lower potential - work done by electric field

• positive test charge moving from lower to higher electric potential - work done by an external force

electric potential gradient

• the change in electrical potential per unit change of distance at a point in the field along the field line at that point

• $E=-\frac{\Delta V}{\Delta d}$ = — potential gradient

potential gradient in a uniform field

• constant potential gradient

• V increases linearly with d

coulomb’s law

• the force between two point charges in a vacuum

• is directly proportional to the product of their charges

• inversely proportional to the square of the distance between them

electric field strength at a distance r from a charged sphere

$$E=\frac{Q}{4\pi\char"0190 0r^2}$$

where Q is the charge creating the radial field

electric potential around a point source

$$V=\frac{Q}{4\pi\epsilon0r}$$

where Q is the charge creating the radial field

electric field strength of two charges acting on the same test charge q

• if charge pushing in the same direction add field strength

• if in opposition, subtract field strength

• vector

graph of V against distance e for a positive charge next to a negative charge