S2.12 Electrostatic forces
Preamble
Reveal

Preamble
The mathematical form of the equations in this section are very similar to those in S2.11. But here we explore what happens if we have interacting particles where the force can be attractive or repulsive. And we’ll delve slightly deeper in the world of vectors to look at the idea of a vector field.[A] About the force
Key Point 2.5
The electrostatic (Coulomb) force of interaction between two point charges q1 and q2 separated by distance r:
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with K a fundamental constant of electrostatics
- is attractive if q1, q2 are unlike
- is repulsive if q1, q2 are like

Commentary
K is a collection of constants, such that

The quantity ϵ0 is known as the Permittivity Constant of Free Space with an experimentally measured value of


And a first look at quantisation
Charge is quantised: it can only take certain values and these are integer multiples of the fundamental unit of charge, that on an electron.
So any net charge q can be written as

The quantisation of charge was first experimentally verified by Robert Millikan in 1913 by the famous Oil Drop experiment, for which combined with his work on the Photoelectric Effect, he won the Nobel prize in 1923.
The charge on an electron is very small and in many conditions we will be able to ignore the quantisation and treat charge as a continuous variable.
[B] Example problem
Estimate the ratio of the gravitational force between two electrons and the electrostatic (Coulomb) force between two electrons.

Solution
The gravitational force between two equal masses, m1=m2=m, separated by distance r is of magnitude (Key Point 2.4)
![\[ F_G =G \frac{m^2}{r^2} \]](mastermathpng-4.png)
The electrostatic (Coulomb) force between two equal charges, q1=q2=e, separated by distance r is of magnitude (Key Point 2.5)
![\[ F_E = K\frac{e^2}{r^2} \]](mastermathpng-5.png)
Both forces are of inverse-square form: their strength varies inversely as the square of the separation.
Taking the ratio of the two forces the separation r cancels to leave:
![\[ \frac{F_G}{F_E} = \frac{G}{K} \times \left( \frac{e}{m}\right)^{-2} = \frac{6.67 \times 10^{-11}}{9.0 \times 10 ^9 \times (1.76 \times 10^{11})^2} =2.4 \times 10^{-43} \hspace{0.5cm}\mbox{\rm (2sf)}\hspace{0.5cm} \]](mastermathpng-6.png)
You might want to contemplate the implications of what we have just calculated, in particular, this result seems to imply that gravity is so small as to be irrelevant and what it means for collections of like charged species in close proximity to one another, such as the protons in the nuclei that make up you, me and everything around us......
[C] Dealing with multiple charges - the principle of superposition
In the same way that we have been adding forces as vectors throughout this section, electrostatic forces are no exception and serve as an illustration of the Principle of Superposition in action.
Let’s start with the vector form of the force due on one charge from another.
If we have two point charges
q1 and q2 seperated by
a vector as shown below
then the force on q1 as aresult of q2 is a vector giving,
Key Point 2.6
Coulomb’s Law in vector form, being:

Note that this force is directed away from q1 if both charges have the same sign and towards q2 if opposite. Generalising to a series of charges, for example q1→q6
Then the force on q1 as a result of the charges q2→q6 will be a vector sum, being
Key Point 2.7

where each force is the force on q1 as a result of the
charge being given by Key Point 2.6.
[D] The Electric Field
We have just seen how to calculate the force on a charged particle due to the presence of a second charge. But if the two charges are nowhere near each other how is that force ’felt’ by the charge? How can there be this action at a distance?
We can explain this by saying that particle 2 sets up an electric field in the space all around it. Paticle 1 is affected by this field. Thus particle 2 exerts a force on particle 1 not by touching it but by the electric field its presence has created.

Scalar vs Vector fields
Imagine I asked you to measure the temperature of all points in the room you are currently in. You would divide up the space and make a measurement at each point. The collection of values is said to be a scalar field, as temperature is a scalar quantity.
But imagine I wanted you to measure the ’breeziness’ at your location and gave you something that would measure both wind speed and direction. You would return with a series of vectors at various points - values of the wind speed and direction.
An electric field is thus analagous to this ’wind-field’.
We can define the electric field at a point, P, in the vicinity of a charge by considering the electrostatic
force acting on a test positive charge of q0 at P.
Key Point 2.8

The magnitude of the field at this point is given by and the direction
is that of the force that acts on the test positive charge.
If we then extend this idea to a collection of charges as we did previously for the electrostatic force
Then we can write the force on this charged particle as,


So the Electric Field seen by charge q1 is a Linear Vector Summation of the Electric Fields due to the other charges; the principle of superposition again.
[E] Field lines
Field lines provide a convenient way for us to visualise the vector nature of an electric field. They are lines of force which show the direction of the field at a point in space, with the separation between the lines indicating the magnitude or strength.

Analysis
You can sort out the direction once and for all by considering what the field lines will look like if you have a test positive charge at some distance away from a positively charged particle.
The direction of the electrostatic force will be oriented away from the positively charged particle, along the line conneting it to the test positive charge. Likewise if your particle was negatively charged, the field lines would point towards it.
Now consider two charges of +q and -q separated by a fixed distance d as shown. This configuration is known as an Electric Dipole.

Mathematical analysis
We want to calculate the vector Electric Field at the position (0,a). There are two charges, so two contributions to the electric field, which according to the the Superposition principle will be the vector sum of the electric fields from each of the charges.
For the positive charge, located at , then at the point
(0,a) we have that the vector from the positive charge is


The Electric Field from the positive charge is thus,
![$$ \vec{E}_+ = {1\over4\pi\epsilon_0}{q\over r_+^2}\left[ - {d\over2r_+}\,\hat{\imath} + {a\over r_+}\,\hat{\jmath}\right] $$](mastermathpng-25.png)


![$$ \vec{E}_- = {1\over4\pi\epsilon_0}{-q\over r_-^2}\left[ {d\over2r_-}\,\hat{\imath} + {a\over r_-}\,\hat{\jmath}\right] $$](mastermathpng-28.png)


Key Point 2.10
The total Electric Field is given by
Considering Key Point 2.10 we see that the Electric Dipole is characterised by the vector quantity


Note the direction of , it is from the negative
towards the positive end of the dipole.
So in terms of the Electric Dipole Moment the Electric Field
a distance a perpendicular to the dipole axis is given by,
We can repeat this calculation at all point in space and form the field lines from a dipole as shown
Learning Resources
![]() | The treatment of electrostatics starts in HRW in Chapter 21, and much of the next 11 chapters is concerned with developing towards a treatment of Maxwell’s Equations. However, we will start (and remain) in the foothills, covering Chapter 21 and selected bits of Chapter 22. |
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