# Electric Field (+ Field Strength) Simply Explained

## Important Formula

## What do the formula symbols mean?

## Electric force

`$$ \boldsymbol{F} $$`Unit

`$$ \mathrm{N} = \frac{\mathrm{kg} \, \mathrm{m}}{\mathrm{s}^2} $$`

If you place a charge \( q \) in an electric field \( E \), this charge is accelerated along the field lines by the electric force. A positive charge is accelerated in the direction of the electric field lines and a negative charge is accelerated against the electric field lines.

## Electric charge

`$$ q $$`Unit

`$$ \mathrm{C} = \mathrm{As} $$`

The charge determines in which direction the electric force points - you can see this by its sign. For positive charges, the sign is a plus - so a force acts on this particle in the direction of the electric field lines. This is the case, for example, with protons, alpha particles, or positively charged ions, such as \( \text{Na}^+ \). For negative charges, the sign is a minus - so a force acts on this particle against the electric field lines. You can observe this for example with electrons.

An electron carries a *negative* charge \( q ~=~ -1.602 \cdot 10^{-19} \, \text{C} \). A proton, on the other hand, carries a *positive* charge: \( q ~=~ +1.602 \cdot 10^{-19} \, \text{C} \). Negative charges have a minus before the value of the charge and positive charges have a plus.

## Electric field

`$$ \class{purple}{\boldsymbol E} $$`Unit

`$$ \frac{\mathrm{V}}{\mathrm{m}} = \frac{\mathrm{N}}{\mathrm{C}} = \frac{\mathrm{kg} \, \mathrm{m}}{\mathrm{A} \, \mathrm{s}^3} $$`

*how big*the electric force is on an electric charge. The field strength is illustrated by field lines. The closer field lines are to each other, the greater is the field strength there, and the greater is the electric force acting on a charge there. The field lines can be straight like in a plate capacitor. Then the force always points in the same direction. But field lines can also be curved; then the force points in different directions, depending on where you place the charge \(q\).

Sconvert the formula according to the electric field strength: \( E ~=~ \frac{F_{ \text{E} }}{q} \), as you can see it is "*force per charge*".

During a thunderstorm, for example, there is a very large electric field between the earth and clouds; it reaches values above \( 150 \, 000 \, \frac{\text V}{\text m} \).

The electric field generated by a charged particle (e.g. an electron) can be determined by Coulomb's law.

## Basics: Electric charge

The physical quantity **electric charge** \( q \) is the property of a charge carrier, for example, a particle (electron, proton) or a large electrically charged sphere from physics class. The charge indicates how well the charge carrier can interact electrically with other charge carriers. Interaction refers to how strongly the charge carrier is repelled or attracted by other charges. Depending on whether the charge carrier is *positive* (+) or *negative* (-) charged, it will be repelled (this happens with (+)(+) and (-)(-)) or attracted (this happens with (-)(+)).

The physical unit of charge is:
`1
\[ [q] = \text{C} ~ (\text{Coulomb}) \]
`

Sometimes it is useful to express it in base SI units \( \text{C} = \text{As} \) (ampere-second) to simplify units in a calculation.

## Basics: Electric Force

The physical quantity **electric force** \( F_{\text E} \) tells you how quickly the momentum \( p \) (i.e., the force) of a body changes when this force acts on that body. In our case, the body refers to a charge carrier with an electric charge \( q \). The momentum consists of the velocity \( v \) and the mass \( m \) of the charge carrier: \( p = m\,v\). Since the mass is usually constant (you don't change your mass when accelerating in a car), only the velocity \( v \) of the charge carrier changes over time when a force acts on it. When the velocity changes over time, the charge carrier accelerates (as you know from a car).

We speak of an *electric* force when it is caused by other charge carriers and not, for example, by pushing a charge carrier with one's hand (mechanical force). So, if there is a positively charged proton near a negatively charged electron, the electron experiences an electric force that accelerates it toward the proton.

The physical unit of force is:
`2
\[ [F_{\text E}] = \text{N} ~ (\text{Newton}) \]
`
As with the unit of charge, you can break down the unit of force into SI units to simplify the units in a calculation: \( \text{N} = \frac{\text{kg} \, \text{m}}{\text{s}^2} \).

**Example of an electric force: Coulomb's Law**Coulomb's Law specifies the specific relationship by which two charges attract or repel each other with an electric force. You need to calculate the electric force concretely to determine the electric field. Coulomb's Law helps you with this.

You can easily calculate the magnitude of the electric field \( E \) (not to be confused with energy!) by observing how much force \( F \) acts per charge \( q \):

**Formula: Electric Field (Magnitude)**

`3 \[ E = \frac{F_{\text E}}{q} \]`

From equation 3

, you can immediately read the unit of the electric field. With the unit of force 2

and the unit of charge 1

, the unit of the electric field becomes:
`4
\[ [E] = \frac{\text N}{\text C} ~ (\text{Newton per Coulomb}) \]
`

**What is the difference between electric field / electric field strength?**

In general, the electric field is a three-dimensional vector \( \boldsymbol{E} \), a arrow indicating the direction of the electric field. Its length (corresponding to the magnitude \( |E| \) of the electric field) is called *electric field strength*.

**Example: Electric Field**If there is an electric field of \( E = 42 \frac{\text N}{\text C} \) at a certain location \( r \), then you know that if you place a charge carrier with \( 1 \, \text{C} \) at location \( r \), that charge carrier will experience a force of \( 42 \, \text{N} \).

Typically, electric field is expressed in the unit \( [E] = \frac{\text V}{\text m} \) (volts per meter), which you can easily obtain through unit conversion.

## Electric Field Lines

To illustrate the electric field, *electric field lines* are used. Based on the field lines, you can determine in which direction a test charge would move if placed in the electric field. Field lines *begin* by definition at a positive and *end* at a negative charge.

The electric field of a uniformly positively charged sphere radiates outward radially. The field lines of the electric field originate from the positive sphere, as mentioned, and terminate at negative charges. In this case, these negative charges are located somewhere very far away (at infinity). Therefore, when drawing the field lines of a single charge, they are not visible in the image.

The field lines of a negatively charged sphere also run radially, but they point in the opposite direction—toward the negative charge. If you place a small test charge at any location in the electric field, this test charge will move along the field lines. The field lines indicate the direction of the force and thus the direction of motion of the test charge.

An interesting feature of the radial electric field of a sphere is that the individual field lines diverge more and more as you move away from the sphere. This is an indication that the electric force on a test charge decreases as it moves away from the sphere. The electric force on a test charge decreases quadratically with distance, as prescribed by Coulomb's Law. This means that if you halve the distance between the test charge and the charged sphere, the force on the test charge quadruples.

What would the field lines look like if you bring a negatively charged sphere near the positively charged sphere? The trajectory of the field lines will completely change! In this case, you get the electric field of a so-called *electric dipole*. The field lines still start at the positive and end at the negative charge, but a test charge would move completely differently in the field, unlike in the radial field of a single charge. The way such trajectories of field lines arise can be easily understood by considering the resulting force \(\boldsymbol{F}\) on the test charge.

**Example: Force on a Test Charge**If a negative test charge is placed at any location in the electric field of an electric dipole, this test charge experiences a repulsive force \(\boldsymbol{F}_1\) from the negative charge and an attractive force \(\boldsymbol{F}_2\) from the positive charge. The resulting force \(\boldsymbol{F}\) is tangential to the field line but moves against the field lines (in the opposite direction).

## Homogeneous / inhomogeneous electric field

Depending on how the field lines of the electric field run, the respective electric field is referred to as homogeneous or inhomogeneous.

**Homogeneous**("uniform") - intuitively means that the field lines have the same distance and alignment everywhere in space. Mathematically, this means: \( E = \text{const} \), so the electric field has the same value at every point. You can find such a homogeneous electric field between two electrically charged plates (parallel plate capacitor).**Inhomogeneous**("non-uniform") - is exactly the opposite of "homogeneous". Such an inhomogeneous electric field is generated, for example, by a charged sphere. To indicate that the electric field depends on location, the position coordinate \( r \) is added: \( E(r) \). As in mathematics, \( E(r) \) is a function like \( f(x) \). Depending on the value of \( r \) you consider, the electric field (the value of the function) is different.