A field is generated by a source in the space around it. An electrically charged particle generates an electric field. Protons and electrons have electric fields, and within a certain distance, they attract or repel other charged particles.
Definition of electric field
Forces are vectors, and so is the electric field. E is a vector quantity measured in Newton/Coulomb or volts/m:
\[\vec{E} = \frac{\vec{F}}{q}\]
Here, F and q are respectively the source’s force and the charge to which that force is applied. If we express the force in terms of an electric field, the result is:
For an isolated positive charge, the electric field around it extends radially outwards from the charge in all directions while for an isolated negative charge, the electric field around it is directed radially inwards. When it comes to forces that these charges experience, the general rule is that positive charges experience force in the direction of the electric field while negative charges experience a force that is opposite to the direction of the field.
Coulomb’s law and the field of a charge
The simplest electric field is that of a single charged particle. Using Coulomb’s law, it is possible to calculate the force between two particles q and qi at a distance of r, with ri being the vector between the particles.
\[\vec{F} = \frac{1}{4 \pi \varepsilon_0}\frac{qq_i}{r^2} r_i\]
Here, ε₀ is a constant called vacuum permittivity or absolute dielectric permittivity with a value of approximately 8.85·10-12 F/m. If we put this equation into the expression of the field and set the charge (to which the field is applied) at an intensity of 1, we obtain the field of the particle qi:
\[\vec{E} = \frac{k q_i}{r^2} r_i\]
Here, k is the constant part of the formula, including the permittivity. Its value is 9x109 kg⋅m3⋅s-2⋅C-2.
Figure 1. The electric field of a particle.
The electric field is related to the distance of the application point from the source of the field itself and to the intensity of the charge. In the case of a single particle, we locate equipotential concentric spheres where the strength of the field is the same.
The electric field of multiple charges
With proper adjustment of the terms, the formula used to describe a single charge can also be used to calculate more complex cases. In the case of multiple charges, we must consider their effect on the application point. This is calculated by adding the contribution of each charge.
Figure 2. The field of two particles at some point in the distance is equal to the sum of the field at that distance.
\[\vec{E} = k \sum_{i=1}^N \frac{q_i}{r^2} r_i\]
As you can see, there is not a big difference from the previous example. N here represents the total number of charges, but instead of calculating this once, you need to add up the result of that formula for all the charges (N). Note that it is important to respect the direction of each contribution when carrying out the vector sum.
The electric field of distributed charges
Let’s consider a slightly more complex situation. This is also more useful as it’s not very common to find spare particles moving around. Instead, consider objects with a particular shape and volume. Thanks to the superposition principle, here we consider a homogeneous density of charge ρ instead of the charge of a single particle. Calculating the electric field is a matter of doing an integral that considers the distribution of the charge inside the object:
\[\vec{E} = k \int{\frac{\rho d V}{r^2} r_i}\]
From this result, it is possible to go even further and consider, for example, a non-homogeneous density of charge. Let’s take a source whose charge varies along one or more dimensions in the space and all over the volume. To name this density, we add the dimensions on which it depends between parenthesis. For example, the case of a density of charge that varies on the x dimension is represented by ρ (x). The calculus will be more complex without any difference in the concept.
Electric potential
Sometimes you need to calculate what happens within an electric field. The movement of charge in an electric field is not the same as outside an electric field. In addition, sometimes there are other charges in the electric field’s space, and it is interesting to see what happens between them. Electric potential allows us to carry out these calculations.
From the electric field to electric potential
Electric potential is the amount of energy needed to move a charge in an electric field from point A to point B without loss or transformation of energy. To define the electric (or electrostatic) potential, we need a reference point. The first one is the source of the electric field. In the case of a single particle with a test charge immersed in its field, the potential is:
\[U = k\frac{qq_i}{r}\]
The first thing to note is that electric potential is a scalar quantity. Moreover, even if the formula is very close to that of the force, we must consider the radius, not its square power. Lastly and importantly, this quantity depends on the test charge. We need to refer to an absolute reference; thus, the definition of V is:
\[V = \frac{U}{q}\]
Consider two points and make the difference between them the potential difference ΔV:
\[V_a - V_b = kq_i \Big( \frac{1}{r_a} - \frac{1}{r_b} \Big)\]
If we bring point b very far away from point a, rb becomes greater, while 1/rb becomes smaller. The further b moves away from a, the more 1/rb approaches zero, to the point that b is so distant that we can avoid considering that term in parenthesis, and the definition of Va - Vb matches the definition of U. The unit of measure of the potential is volts.
Electric Fields (A2 Only) - Key takeaways
- A field is generated by a source in the space around it. An electric field is generated by an electrically charged particle.
- The source that generates the electric field can be discrete or continuous.
- The equipotential surfaces of a field are those where the effect of the field itself on a test charge is the same.
- Moving a charge into an electric field consumes energy, and the amount needed to go from A to B is called electric potential.
- When referring to the potential between two points, one is often described as at infinity so that it has a quantity that doesn’t depend on the charge of the moving particle.
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