Gravitational and Electric Fields
Electric fields
can be attractive or repulsive. Any object with charge has an electric field around it - the region where it can attract or repel other charges. If a charged object is placed in an electric field, then it will experience a force.
Charged particles in a uniform electric field
A uniform electric field can be used to determine whether a particle is charged or not. The path of a charged particle moving through an electric field (at an angle to the field lines) will bend - the direction depends on the charge. A charged particle that enters an electric field at right angles to the field feels a constant force parallel to the electric field lines. If the particle is positively charged, the force acts on it in the same direction as the field lines (attracts). If it's negatively charged, the force is in the opposite direction to the field lines (repels). This causes the particle to accelerate at right angles to the particle's original motion, and so it follows a curved path (see figure on page 91). In a 3D situation, the motion is the same (curved) as there are no other significant forces acting on the charged particle.
Electric field strength in uniform fields
A uniform field can be produced by connecting two parallel plates to the opposite poles of a battery. The field strength E is the same at all points between the two plates and is given by: E=change in V/d Unit for E [Vm^-1 or NC^-1]
Electric potential
All points in an electric field have an absolute electric potential V. This is the electric potential energy that a unit positive charge (+1C) would have at that point. It depends on how far it is from the charge creating the electric field and the size of that charge. Negative V = attractive Positive V = repulsive The absolute magnitude of V is greatest on the surface of the charge, and decreases as the distance from the charge increases - V will be zero at an infinite distance from the charge.
Point mass
An object which behaves as if all its mass is concentrated at the centre. In some questions, you have to assume that the earth is a point mass so r is the distance of the object from the earth's centre.
Kinetic and potential energy of satellites
An orbiting satellite has kinetic and potential energy - its total energy (i.e. kinetic and potential) is always constant. In a circular orbit, a satellite's orbital speed and distance above the mass it's orbiting are constant. In an elliptical orbit, a satellite will speed up as its orbital radius decreases. This means that its kinetic energy increases as its potential energy decreases and vice versa. So the total energy remains CONSTANT.
What is a gravitational field?
Any object with mass will experience an attractive force if you put it in the gravitational field of another object. A gravitational field is a force field (a region in which a body experiences a non-contact force)
Geostationary satellites
Are examples of synchronous orbit - they're always above the same point on Earth. To achieve this, their orbit must be in the plane of the equator, so they will always be directly above the equator. Geostationary satellites travel at the same angular speed as the earth turns below them and in the same direction (west to east). Their orbit takes approx. 24 hours and their orbital radius is about 42000km, which is about 36000km above the surface of the earth.
Equipotentials
Are lines (in 2D) and surfaces (in 3D) that join together all of the points with the same gravitational potential V. This means that as you travel along an equipotential, your potential doesn't change - you don't lose or gain energy. So as you're moving along the equipotential, the gravitational potential difference is zero: change in V = 0. As change in work done = m x change in V, this means that the amount of work done is also zero. e.g. A satellite travelling in a circular orbit is an example of an object which travels along an equipotential surface. It stays at the same distance from the earth's centre, and so no work is done in keeping it in orbit.
Escape velocity derivation
At an infinite distance --- gpe = 0 At the surface --- gpe = negative The increase in gpe comes from the inital ke of the object. If the object is given just enough ke to escape the gravitational field of the planet, you know from conservation of energy that all of its initial ke will be converted into gpe. So ke lost = gpe gained and so 0.5mv^2 = GMm/r Simplifying and rearranging for v gives v = square root of 2GM/r NOTE: the escape velocity equation is not dependent on the mass of the object, just the mass M of the planet.
Measuring electric field lines
Conducting paper - used to map out the field lines of a 2D electric field. A positive charge is on one edge of the paper and a negative charge is on the opposite edge. A voltmeter is then used to measure the potential difference at different points on the paper. Points with the same voltage can be joined up to show equipotential lines (which are always perpendicular to field lines). Electrolytic tank - tank of water with positive and negative ions dissolved in it. Electrodes are put in the water to create a positive charge on one side of the tank and a negative charge on the other side. A voltmeter is then used to find points within the water where the potential difference is the same. From this, both equipotentials and field lines can be mapped out.
Derivation of T^2 is proportional to r^3
F=mv^2/r F=GMm/r^2 Set equal to each other and rearrange to make v the subject (You will find v is proportional to 1/r^-1/2) The time taken for a satellite to make one orbit is called the orbital period T. Speed=distance/time and the distance for a circular orbit is 2pi x r. So v=2pi.r/T Rearrange to make T the subject and sub in v. You will find that T^2 is proportional to r^3.
Use of geostationary satellites
For sending TV and telephone signals - the satellite is stationary relative to the Earth's surface, so you don't have to alter the angle of your receiver (or transmitter) to keep up.
Representation of gravitational fields
Force fields can be represented as vectors, showing the direction of the force they would exert on an object placed in that field. Gravitational field lines, or "lines of force", are arrows showing the direction of the force that masses would feel in a gravitational field.
How do force fields arise?
From interactions between objects or particles e.g. between static or moving charges, or in the case of gravity, between masses. Only objects with a large mass, such as stars and planets, have gravitational fields that produce a significant effect. Smaller objects do still have gravitational fields that attract other masses, but the effect is too weak to detect without specialised equipment.
Combining gravitational field strengths
Gravitational fields are vector fields, which means you can add them up to find the combined effect of more than one object. Remember vector fields means the direction matters.
Difference between gravitational and electric fields
Gravitational forces are always attractive but electrostatic forces can be attractive or repulsive.
Question where particle is not moving
If a charged particle is stationary in a uniform field and the top plate has no charge but the bottom plate has the same charge as the particle, the solution is to understand the upwards repelling force must be equal to the downwards gravitational force.
Electric potential graph
If repelling, V initially starts positive and decreases to zero with distance from the charge. If attracting, V initially starts negative and increases to zero with distance from the charge. The gradient of a tangent to either graph gives the field strength E at that point.
Electric potential difference
If two points in an electric field have a different absolute electric potential, ten there is an electric potential difference between them. This is the energy needed to move a unit charge between those points. The electric potential difference in a radial field can be found from the area under a graph of E against r.
Earth's gravitational field
If you put a small mass m anywhere in the Earth's gravitational field, it will always be attracted towards the earth. The earth's gravitational field is radial - the lines of force meet at the centre of the earth. If you move mass m further away from the earth - where the lines of force are further apart - the force it experiences decreases. The lines can be used to show the strength of the field at each point, where a higher line density shows a stronger gravitational field. Close to the earth's surface, the field is uniform - the field lines are parallel and equally spaced.
Electric field strength graph
Inverse square law. Graph is same shape as gravitational field strength graph. If the charge isn't a point charge e.g. a charged metal sphere, then the electric field strength inside the object doesn't have the same 'E is proportional to 1/r^2' relation. Point charge - exponential Charged sphere - same as g graph: starts by showing E at the surface of the charged sphere
Gravitational Field Strength (g)
Is the force per unit mass. Its value depends on where you are in the field. g is a vector quantity - it has a magnitude and a direction. It's always pointing towards the centre of mass of the object whose field you're describing. Since the gravitational field is almost uniform at the Earth's surface, you can assume g is constant near the earth's surface. g can also be seen as the acceleration of a mass in a gravitational field. It is often called the acceleration due to gravity. Unit [Nkg^-1] Point masses have radial gravitational fields. The magnitude of g depends on the distance r from the point mass M.
Escape velocity
Is the minimum speed an unpowered object needs in order to leave the gravitational field of a planet and not fall back towards the planet due to gravitational attraction. The direction of the escape velocity is not important. Although it's called a velocity, it's actually a speed.
Coulomb's Law
It gives the force of attraction or repulsion between two point charges Q1 and Q2, in a vacuum. The force on Q1 is always equal and opposite to the force on Q2 - the direction depends on the charges.
Satellites
Planets and satellites are kept in orbit by gravitational forces. A satellite is just any smaller mass which orbits a much larger mass - the Moon is a satellite of the Earth, etc. They are kept in orbit by the gravitational 'pull' of the mass they're orbiting.
Gravitational potential graph
See V-r graph on page 78. If you find the gradient of this graph at a particular point, you get the value of -g at that point. In other words: g = -change in V/change in r The area under this graph gives you change in V, the change in gravitational potential between two radial distances.
Similarities and differences between gravitational and electrostatic forces:
Similarities: Both have inverse-square force laws that have many characteristics in common e.g. use of field lines, use of potential concept, equipotential surfaces etc Differences: masses always attract, but charges may attract or repel
Graphs of orbital period
Sketching T against r is usually done on a logarithmic scale, otherwise the numbers involved are so big and can vary a lot between planets.
Forces between objects consideration
Sometimes we only consider the force acting on the smaller object because that's the one that experiences a greater acceleration - a = F/m. This is why we don't notice Earth's acceleration towards us when we're falling to the ground.
Equipotentials - uniform spherical mass
The equipotentials are spherical surfaces. The equipotentials and the field lines are always perpendicular. See figure on page 81.
Newton's law of gravitation
The force experienced by an object in a gravitational field is always attractive. It depends on the masses involved and the distances between them.
Coulomb's Law - inverse square law
The further apart the charges, the weaker the force between them. If the point charges aren't in a vacuum, then the size of the force F also depends on the permittivity epsilon of the material between them. If in a vacuum or in the air, use the epsilon constant (the permittivity of free space).
T^2 is proportional to r^3 meaning
The greater the radius of a satellite's orbit, the slower it will travel and the longer it will take to complete one orbit.
Inverse square laws
The law of gravitation is an inverse square law: F is proportional to 1/r^2 This means, if the distance r between the masses increases, then the force F will decrease. Because it's r^2 and not just r, if the distance doubles, the force will only be a quarter the strength of the original force. The law of gravitation is an inverse square law because it is radial. The force on any point a distance r from the mass will be the same - if you draw an imaginary sphere with radius r, the force will be the same at any point on its surface. If you double the distance from a mass, the area on the surface of the imaginary sphere of equal force covered by a particular group of radial field lines will be four times greater. The spread of field lines indicates the strength of a force felt in that field, so here the force will be four times weaker at a distance 2r than at r for a given mass.
Gravitational Field Strength (g) - graph
This is another case of the inverse square law - as r doubles, g decreases to a quarter of its original value. If you plot a graph of g against r for the earth, you get a curve that looks exponential. It starts at g=9.81 and radius is R (no y-intercept). It shows that g is greatest at the surface of the earth (R), but decreases rapidly as r increases and you move further from the centre of the earth. The area under this curve can be used to find the gravitational potential V.
Work done in an electric field
To move a charge across a potential difference (i.e. from one electric potential to another), you need to use energy. The amount of energy you need (or the work done) depends on the size of the charge you're moving and the size of the potential difference you want to move it across. Derived by setting the two E equations equal to each other. Rearrange to find Fd=Q x change in V Fd = work done so change in work done = Q x change in V Work done is always positive.
Gravitational potential V
Unit [Jkg^-1] The gravitational potential at a point is the gravitational potential energy that a unit mass at that point would have e.g. if a 1kg mass has -10J of potential energy at a point Z, then the gravitational potential at Z is -10Jkg^-1. Gravitational potential is negative on the surface of the mass and increases with distance from the mass. You can think of this negative energy as being caused by you having to do work AGAINST the gravitational field to move an object out of it. This means that the gravitational potential at an infinite distance from the mass will be zero. Gravitational potential energy is also negative because it points 'downwards' towards the centre of the planet - potential energy becomes less negative as the object moves upwards. ZERO GRAVITATIONAL POTENTIAL IS AT INFINITY.
Doing work meaning
Using a force to transfer energy from one type to another e.g. if you drop a ball from a height, gravitational potential energy is converted into kinetic energy.
Forces at subatomic levels
When you get down to the subatomic level of electrons, protons and neutrons, the distances between particles become tiny. As both the gravitational and electrostatic forces have an inverse square relationship with distance, you'd expect these forces to be huge. However, gravity at this level can be ignored because although they're close together, all of the particles have incredibly small masses - the gravitational force at these distances is much weaker than the electrostatic force. The nucleus doesn't break apart from all of the electrostatic repulsion because there are other forces at work.
Orbital period and speed
You can use the equations of circular motion to investigate planets' orbital speed and period. Objects undergoing circular motion are kept in their paths by a centripetal force. In the case of satellites, the centripetal force is the gravitational force due to it being caused by gravitational attraction.
Electric field lines
are drawn to show the direction of the force that would act on a positive charge. Point charges have a radial field. For a positive point charge, the field lines point away from the point charge and for a negative point charge they point towards it. For parallel plates, the field lines point from the plate with the more positive voltage to the plate with the less positive voltage.
Low orbiting satellites
defined as any satellites which orbit between 180-2000km above the earth e.g. the International Space Station. Satellites designed for low Earth orbits are cheaper to launch and require less powerful transmitters as they're closer, which makes them useful for communications. However, their proximity to Earth and high orbital speed (in comparison to Earth's) means you need multiple satellites working together to maintain constant coverage. Low orbital satellites are close enough to see the earth's surface in a high level of detail. Imaging satellites are usually placed in this type of orbit and are used for things like imaging (e.g. mapping) and monitoring the weather. Their orbits usually lie in a plane that includes the north and south pole. As the planet and the satellite rotate at different angular speeds, the satellite doesn't stay over the same part of the earth, and so the whole of the surface can be scanned.
Electric field strength (E)
defined as the force per unit positive charge. It's the force that a charge of +1C would experience if it was placed in an electric field. E is a vector pointing in the direction that a positive charge would move. Unit [NC^-1]
Similarities between gravitational and electric fields
g is force per unit mass E is force per unit positive charge Newton's law of gravitation is an inverse square law Coulomb's law for the electric force is an inverse square law Gravitational field lines for a spherical/point mass are the same (direction wise etc) to the electric field lines for a NEGATIVE spherical/point charge V in both are potential energies per unit mass/per unit positive charge and is zero at infinity Equipotentials for a uniform spherical mass/point charge form spherical surfaces The work done equations are similar.
Gravitational potential difference
is the energy needed to move a unit mass. Two points at different distances from a mass will have different gravitational potentials (because gravitational potential increases with distance), thus there is a difference between these two points. When you move an object, you do work against gravity. Change in work done = mass x change in V Change in gravitational potential energy(Ep) = m x change in V Therefore, Ep = -GMm/r
Synchronous orbit
is when an orbiting object has an orbital period equal to the rotational period of the object it is orbiting e.g. a synchronous orbit of Earth would take approximately 24 hours.