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.