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Electric Fields

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.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…

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# Electric Fields

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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 sources force and the charge to which that force is applied. If we express the force in terms of an electric field, the result is:

$\vec{F} = q\vec{E}$

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 Coulombs 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

Lets consider a slightly more complex situation. This is also more useful as its 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. Lets 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 fields 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 doesnt depend on the charge of the moving particle.

A field is generated by a source in the space around it. An electric field is generated by an electrically charged particle.

The movement of electrically charged particles generates a magnetic field.

Yes, it is.

Through induction, as Faraday’s law states.

An electrically charged object or a magnetic field.

## Electric Fields Quiz - Teste dein Wissen

Question

Does a charged particle create an electric field?

Yes, every charged object creates an electric field.

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Question

What does it mean that the source of an electric field is inhomogeneous?

It means that the charge of the source is not the same everywhere but changes along one or more directions.

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Question

What does the symbol ρ mean in terms of charge?

The symbol ρ represents the density of charge.

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What does the symbol ρ (x) mean in terms of charge?

The symbol ρ (x) represents a density of charge that varies with the direction x.

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Question

How does distance affect the potential?

Moving a charge away from the other will cause the potential V to become equal to U.

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Question

Is the electric field a vector or a scalar?

It's a vector.

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Question

What’s the unit of the electric field E?

The electric field E is a vector quantity measured in Newton/Coulomb or volts/m.

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Question

What is the source of an electric field?

The source of an electric field is what generates it.

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Question

How do I calculate an electric field of several charges?

By adding up the contributions of all the charges.

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Question

What does the density of charges mean?

Density of charges means that instead of a discrete number of single charges, there is a charged area in the space (e.g., an object) with a non-negligible dimension.

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Question

Why do we use a test charge to measure the electric field?

When calculating the electric field of a charge, we can refer to the formula that tells us the force between two charges, with one of them set to 1 to have no contribution to the result.

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Question

What is the electric potential?

Electric potential is the amount of energy needed to move a charge in an electric field from point A to point B.

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Is the electric potential a scalar or a vector quantity?

It's a scalar quantity.

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Question

What is the difference between U and V?

U depends on the test charge while V is an absolute reference.

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Question

Talking about potential, what does it mean to bring a point to infinite?

Since the radius is the denominator of the fraction in the formula of potential difference, bringing it to infinity will cause the quantity to decrease to an infinitesimal amount, rendering it negligible in the calculus.

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Question

What is electric potential?

It is the work required to move a point charge in an electric field.

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What is electric field strength?

It is a measure of an electric field’s density.

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How is work related to electric potential?

Electric potential is proportional to the work and charge.

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What is electric potential energy?

It is the energy required to move a point charge in an electric field.

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Question

What is the formula for electric potential difference?

E = -ΔV/Δr

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What is the unit of electric potential difference?

V

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Question

What is the direction of electric field lines?

From the positive to the negative.

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Question

What is the electric field gradient?

It shows the rate of change of electric field strength. Graphically, it is represented from the equipotential lines.

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Question

Which of the following is true?

The electric potential V decreases in the direction the test charge would normally move  due to repulsion or attraction.

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What is the difference between equipotential lines and electric field lines?

Equipotential lines express the electric potential strength and electric field lines express the electric field strength.

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Electric field lines between two oppositely charged parallel plates are?

Parallel.

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Question

Electric field strength in a uniform electric field is the same throughout the field. True of false?

True.

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Question

A test charge is referred to as a:

+ 1 C charge.

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Is electric field strength a scalar quantity or a vector quantity?

A vector quantity.

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A test charge in a uniform electric field created by two oppositely charged plates experiences a force from:

The positive terminal to the negative terminal.

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Do electric field lines point towards a negatively charged particle?

Yes, they do.

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Do electric field lines point away from a positive charge?

Yes, they do.

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Question

A charged particle produces which field around it?

Electric Field.

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Do like charges repel or attract each other?

They repel each other.

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A charged particle that enters a uniform electric field with an initial velocity moves in what kind of path?

Parabolic.

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If a charged particle enters at right angles to a uniform electric field with an initial velocity, what will its direction of acceleration be?

Parallel to the field lines.

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Question

Which of the following corresponds to an electric dipole?

Electric field lines between a positive and a negative charge.

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Which of the following is true for like particles?

They repel each other.

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Question

Which of the following quantities does the electrostatic force between two charges not depend on?

The mass of the particles.

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Question

Which of the following describes the relationship between the magnitude of the electrostatic force between two particles and the distance between them?

They are inversely proportional.

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Question

What is Coulomb’s law?

Coulomb’s law is the law that states when two or more electrically charged objects are close enough to each other, they exert a force on each other called the electrostatic force.

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Why is Coulomb’s law valid for point charges?

Coulomb’s law is only valid for point-like charges. This is due to the fact that when the two charged bodies are put together, the charge distribution does not remain uniform.

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Question

Two charges are given with the values q= (1e) and q= (-1e) and the distance between them is given as r = 6.58 ⋅ 10-11 meters. What is the value of the electrostatic force exerted on q1?

3.51 ⋅ 10-18N

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Is electrostatic force a vector or a scalar quantity?

A vector quantity.

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Question

Which principle is used when finding the net electrostatic force acting on a charge particle by multiple charges?

Superposition principle.

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Question

Why is electric force not constant?

Because when charges push or pull each other, the distance between them changes.

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Question

What is the name of the electric force of particles with a constant position?

The name of the electric force of particles with a constant position is the electrostatic force.

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Question

Which of the following laws of physics states “when two or more electrically charged objects are close enough to each other, they exert a force on each other”?

Coulomb’s Law.

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Question

Which of the following is the symbol for the charge of a particle?

q

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Question

Does the electrostatic force between two charges depend on the distance between them?

Yes.

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