6.2.4 Electric potential and energy

Electric potential energy
So we know that, for example, a stretched elastic band has elastic potential energy (stored energy). Well, charged particles can in fact also store energy. If you try to push like charges together the charges repel each other so you have to do work to decrease the separation between them. All the work done is stored as electric potential energy. This is recovered when you let go.

So we already know that F = Qq/4πε0r².Underneath the force-distance graph is the work done. The total work done to bring the particles from infinity to a separation r is the total area under the graph. The total work done is the same as the electric potential energy 'E':

E = Qq/4πε0r

NOTE: if one of the particles has a negative charge, the value for electric potential energy will be negative. The magnitude of E represents the external energy required to completely separate the charged particles to infinity.

Electric potential
The electric potential 'V' at a point is defined as the work one per unit charge in bringing a positive charge from infinity to that point. If the test charge is q, the equation for V can be determined by dividing the electric potential energy E by q:

V = E / q= Qq/4πε0rq = Q/4πε0r

NOTE: The units for electric potential 'V' is J C^-1.

Electric potential difference (electric p.d.)
The electric potential difference is the work done per unit charge between two points around the particle of charge Q. It is essentially the difference in the potentials at two points.

Capacitance
This is the last little bit of electric fields....

A capacitor is a device that stores charge. As isolated charged sphere of radius R also stores charge. It too is a capacitor. The capacitance 'C' of a charged sphere is the ratio of the charge it stores , Q, to the potential electric potential 'V' at its surface:

C = Q / V = 4πε0RV / V = 4πε0R

This is the capacitance of an isolated sphere (one that is very far away from other objects).

NOTE: we need to be able to derive this equation (from Q = VC and V = Q/4πε0r)

6.2.3 Uniform electric field

For parallel plates, the electric field strength (E) is uniform and related to the p.d. (V) across the plates and their separation (d). A positive charge will experience a constant force F is placed between the plates. The force is given by the equation F = EQ. The charge will gain energy as it moves from the positive plate to the negative plate:

W = Fd

VQ = EQd

Q = Ed

E = Q/d

NOTE: This equation ONLY works for parallel plates. It is useful in experiments as you only need a voltmeter and ruler to measure V and r, respectively) in order to determine electric field strength (E). As we already know, the unit for E is N C^-1. However, this equation shows us that we can also use V m^-1.

The capacitance of a parallel plate capacitor depends on the separation (d) between the plates.In a vacuum capacitance is proportional to A and inversely proportional to d:

C ∝ A/d

The constant of proportionality is the permittivity of free space (ε0):

C = ε0A/d

When an insulator other than a vacuum is used between the plates the equation for capacitance uses εr. εr is the permittivity for the insulator. This is always greater than ε0 so εr is sometimes known as relative permittivity:

ε = εr + ε0

This means that the equation for capacitance may be written as:

C = εA/d

Charged particles can be accelerated by electric fields. For example, if in between two oppositely charged horizontal plates, the charged particle will experience a constant electrostatic force because of the uniform electric field between the plates, so it has a constant acceleration. The following ideas can be used to determine the motion of a charged particle between plates:

• Electric field strength 'E'' between the plates = V/d
• force 'F' on the charged particle is given by F = EQ (Q is the charge of the particle)
• work done on the charged particle = Vq (q is the charge of the particle)

An electron travelling in the direction of the electric field from a positive to a negative plate will experience a deceleration. For charged particles moving in an electric field we see that:

• For the horizontal motion
• There is no acceleration hence horizontal velocity (VH) is constant with velocity v
• The time 't' spent in the field is given by the equation t = L/v
• For the vertical motion
• The vertical acceleration 'a' of the particle is given by the equation a = F / m = EQ / m
• The initial vertical velocity u = 0
• The final vertical component of the velocity vv as the particle exits the field is given by the equation vv = u + at = 0 + EQ/m x L/v = EQL/mv

picture credit: Kerboodle Physics OCR A textbook

6.2.2 Coulomb’s law

According to Coulomb's law, any two point charges exert an electrostatic (electrical) force on each other that is directly proportional to the product of their charges and inversely proportional to the square of the distance between them:

F ∝ Qq

F ∝ 1/r²

F = kQq/r²

The separation of the charges is 'r'. The magnitudes of the charges are Q and q. The electrostatic force experienced by each point is F. The point charges interact and will exert equal but opposite forces on each other (N3). K is the constant of proportionality:

k = 1/4πε0

NOTE: ε0 is the permittivity of free space ε0 (8.58 x 10^-12 Fm^-1). This means we can write:]

F = Qq/4πε0r²

As we know from 6.2.1, the electric field strength 'E' is equal to F/q:

E = F/q = Qq/4πε0r²q = Q/4πε0r²

From this we can see that electric field strength is directly proportional to the charge Q and is inversely proportional to the square of r. This means that a graph of E against 1/r² will produce a straight line through the origin.

Similarities and differences
Okay so we need to know some similarities and differences between electric and gravitational fields. I took the liberty of making a table:

Gravitational fields	Electric fields
Point masses produce a radial field	Point charges produce a radial field
Masses only produce an attractive field	Charges can produce an attractive or repulsive field
Gravitational field strength is the force per unit mass g = F/m = -GM/r2	Electric field strength is the force per unit positive charge E = F/q = Q/4πε0r2
F ∝ Mm	F ∝ Qq
F ∝1/r2	F ∝1/r2
F = -GMm/r2	F = Qq/4πε0r2

Different fields
It is important to remember that not just electric fields give rise to a force. We also have magnetic, and gravitational fields we cover in this spec!

6.2.1 Point and spherical charges

Electric/electrostatic fields are created by charged objects. 

Electric field strength
The electric field strength of an electric field at a point in space is defined as the force experienced per unit positive charge at that point. We can obtain it using the equation:

E = F/Q

NOTE: F is force experienced by the positive charge Q. The SI unit for electric field strength is N C^-1.

Electric field strength is a vector quantity. Its direction is the direction in which a positive charge would move when placed at that point (away from positive charges/towards negative charges).

Electric field patterns
We use electric field lines/lines of force to map electric field patterns:

• The arrow on an electric field line shows the direction of the field
• Electric field lines are always at right angles to the surface of a conductor
• Equally spaced parallel electric field lines represent a uniform field - one in which the electric field is the same everywhere
• Closer electric field lines represent greater electric field strength

We can model uniformly charged spheres as point charges at the centre. The electric field will be radial (like, sort of circular(?) - a bit like spiders legs around the point charge) and the field strength decreases with distance from the centre.

5.4.4 Gravitational potential and energy

Gravitational potential
The gravitational potential (Vg) at a point in a gravitational field is defined as the work done per unit mass to move an object to that point from infinity. It has the units J kg^-1 and infinity refers to a distance so far from the object producing the gravitational field that the gravitational field strength is zero. Gravitational potential is a scalar quantity. At infinity gravitational potential is at a maximum 0J kg^-1. Since all masses attract each other it takes energy (external work must be done) to move objects apart.

The gravitational potential at any point in a radial field around a point mass depends on:

• the distance 'r' from the point mass producing the gravitational field to that point
• the mass M of the point mass.

We can use the ideas of work done on a force and Newton's law of gravitation to form the equation:

Vg = -GM/r

A graph of Vg against 1/r will produce a straight line through the origin with a gradient equal to -GM. If a graph is drawn of the Vg around the Earth against distance r from the centre of mass of the Earth, the smallest value of r on the graph will be the Earths radius.

Gravitational potential energy
The gravitational potential energy 'E' of any object with the mass m within a gravitational field is defined as the work done to move the mass from infinity to a point in a gravitational field:

E = m Vg

In a uniform gravitational field, in order to change the gravitational potential energy of an object its height above the surface must be changed as this results in a change in gravitational potential.

In a radial field (remember Vg = -GM/r) the gravitational potential energy can be written as:

E = m Vg = -GMm/r

Force-distance graphs
We know, from Newton's law of gravitation, that the gravitational force between objects decreases with distance in an inverse-square relationship. The area under a force-distance graph is equal to the work done (remember this can be negative if the object falls as the object loses gravitational potential energy).

Escape velocity
In order to escape the gravitational field of a mass an object must be supplied with energy equal to the gain in gravitational potential energy needed to lift it out of the field:

1.5mv² = GMm/r

The minimum velocity for this condition to be met is known as the escape velocity. It is the same for all objects regardless of their mass:

v = √(2GM/r)

5.4.3 Planetary motion

Kepler had three different laws to help us with planetary motion:

• K1 - the orbit of a planet is an ellipse with the Sun at one of two foci
• In most cases orbits have low eccentricity so their orbits are modelled as circles. E.g the difference is distance from aphelion (furthest point from the Sun) and perihelion (closest point from the sun) is just 3%
• K2 - A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time
• As plants move on their elliptical orbit around the Sun their speed is not constant. When a planet is closer to the Sun it moves faster. This explains why we don't see great comets - their orbits are highly elliptical and when they get close to the sun they move fast and spend much less time on this part of their orbit. They also spend most of their time too far from the Sun to be visible.
• K3 - The square of the orbital period T of a planet is directly proportional to the cube of its average distance r from the sun.
• T² ∝ r³ ∴ T²/r³ = K

Most planets in the Solar System have almost circular orbits so we can use circular motion mathematics along with Newton's law of gravitation to relate the orbital period 'T' of a planet to it's distance 'r' from the Sun, justifying K3...

If we think about it, a planet of mass m orbiting the Sun at a distance r will have a speed of v and an orbital period of T. The mass of the sun is M (M_☉) and the centripetal force on the planet is provided by the gravitational force between it and the Sun. Therefore the gravitational force 'F' on the planet is equal to the centripetal force:

F = mv²/r = GMm/r²

v² = GM/r

The planet is moving in a circle. This means the speed of the planet can be determined by dividing the circumference of its orbit by its orbital period (v = 2πr/T). We can substitute this into the above equation:

4π²r²/T² = GM/r

T² = (4π²/GM)r³

This equation can be used to determine the mass of an object, the orbital period of a planet, or its distance from the Sun. From this we can make the following statements:

• The ratio T²/r³ is equal to 4π²/GM
• The gradient of the graph of T² against r³ is equal to 4π²/GM

It is important to note that Kepler's laws also apply to any smaller objects in orbit around a larger one (e.g satellites and moons in orbits around planets) and not just planets in our Solar System.

Geostationary satellites

Satellites orbiting the Earth obey Kepler's laws of planetary motion. Satellites always fall to Earth due to the fact that the only force acting on them is the gravitational attraction between it and the Earth. However it is travelling at such a great distance that as it falls the Earth curves away beneath it so it stays at the same height above the surface. All satellites must be at a given height and speed to keep a stable orbit. We can rearrange the above equation to form an equation for the speed 'v' of a satellite in stable orbit at a distance 'r' from the centre of the Earth:

v = √(GM/r)

NOTE: the mass 'm' of the satellite is not present meaning all satellites placed in a given orbit at a given height will travel at the same speed.Once launched they are usually above the atmosphere so they experience no air resistance to slow them down so their speed remains constant.

There are a few uses of satellites which are pretty useful (duh):

• communications: satellite phones (NOT mobile phones!!), TV, some types of satellite radio
• military uses: reconnaissance
• scientific research: looking down on Earth to monitor crops/pollution/vegetation etc, looking outwards to study the Universe
• weather and climate: predicting/monitoring the weather across the globe. Also monitoring long-term changes in climate
• global positioning

Depending on the use, different satellites have different orbits. Satellites placed in the equator can be geostationary satellites. As the height of these satellites increase, so does their period. A geostationary satellite is placed in a geostationary orbit so it remains above the same point of the Earth whilst the Earth rotates. To be a geostationary satellite the satellite must:

• be in an orbit above the Earth's equator
• rotate in the same direction as the Earth's rotation
• have an orbital period of 24 hours

5.4.2 Newton’s law of gravitation

Newton's law of gravitation can also be described as a universal law of gravity. It describes the forces between any objects that have mass. The law can be used to explain motions such as the motion of planets round a sun, and why objects near the surface of the Earth will fall to the ground.

If we think about it, two objects m and M separated by a distance r will each create it's own gravitational field. The interaction of these fields gives rise to forces between the objects. According to N3 the objects must experience an equal and opposite force. Newton's law of gravitation states the force between two point masses is:

• directly proportional to the product of the masses, F ∝ Mm
• inversely proportional to the square of their separation, F ∝ 1/r².

From these statements we can create the equation:

F ∝ Mm/r²

The constant in this equation is known as the gravitational constant 'G'. This value is 6.67408 × 10^-11 m³ kg^-1 s^-2:

F = -GMm/r²

NOTE: we must use a minus sign as this shows that gravitational force is an attractive force. The attractive force (F) decreases with distance in an inverse-square relationship (F ∝ 1/r²) so doubling the distance will decrease the force by a factor of 4 (2²).

If several objects are involved the resultant force can be determined by vector addition.