Colorful Northern Lights can occasionally be seen in the night sky in the polar regions. Vikings used to imagine valkyries riding across the night sky in search of heroes, moonlight refracting off their silver armour.

However, the actual cause of the light spectacle lies in the solar storms and the **Lorentz force**.

## Experiment on the Lorentz force: ladder swing

But before we turn to large natural phenomena, let’s look at the effect of force on a smaller scale in an experiment. All you need for this is a **horseshoe magnet**one **ladder swing** and a **voltage source**.

**Step 1:** Connect the ladder swing to the (still switched off) voltage source and position it in the **homogeneous magnetic field** of the horseshoe magnet.

A **magnetic field** will as **homogeneous **denoted when it is the same at every point **field strength** owns. In the field line representation you draw it as **parallel arrows**which are located at regular intervals from each other and in **same direction** demonstrate:

You can find out more about this in our articles on the magnetic field or the Lorentz force in current-carrying conductors.

**Step 2:** Turn on the voltage source and see what happens. You will find that the ladder swing moves as if by itself.

Now you can try a variation of the experiment, change the orientation of the magnet and see what happens. the veYou can see the different versions of the experiment in the following table:

Attempt Attempt 1: Attempt 2: Attempt 3: DescriptionThe north side of the magnet faces upThe south side of the magnet faces upThe magnet is horizontalConstruction

Figure 3: North side facing up

Figure 4: South side faces up

Figure 5: horizontal orientation

ResultThe ladder swing moves to the right ladder swing moves to the left Die ladder swing stay calm

You would get a similar result if you positioned the ladder swing differently instead of the magnet.

If you look closely, you can see from the drawn field lines that in the first two experiments the direction of the magnetic field lines and the direction of the current are perpendicular to each other. In the last attempt, they run parallel. So in two cases there seems to be a force moving the conductor, in the third not. To explain that, you need the mathematical and physical definition of the force acting here.

## Lorentz force: formula, definition and unit

So what exactly do the Northern Lights and moving ladder swings have in common so that the same force is at work in both cases?

In both cases there is **moving loads** (the charged particles in the solar storm and the electrons in the current-carrying conductor), as well as a **magnetic field** (the earth’s magnetic field and the homogeneous magnetic field of the horseshoe magnet). The combination of these two components leads to the appearance of a force, the so-called **Lorentz force**.

The force was named after physicist Hendrik Antoon Lorentz, who discovered it in 1895.

The Lorentz force also depends on the angle between the direction of the current and the magnetic field lines. In this case, the sine can assume values between 0 and 1:

Correspondingly, the Lorentz force is maximum and minimum. The table below shows three examples of the relationship between the angle and the Lorentz force.

You can also see the angle between the direction of the magnetic field and the direction of the current in the following figure:

If the magnetic field lines and the direction of movement of the charge are perpendicular (90°) to one another, the Lorentz force is therefore greatest. At the same time, it can be explained mathematically why the ladder swing does not move in Experiment 3: when aligned parallel, the angle is 0° and the sine is therefore also zero, just like the Lorentz force.

In the case of vertical alignment, you can use a simple rule to determine the direction of the Lorentz force.

## Lorentz force: Three Finger Rule (UVW – Rule)

We can therefore state that the strength of the Lorentz force depends on the angle between the magnetic field and the direction of movement. But how can you tell in which direction the charge is deflected in the magnetic field?

For this you use the so-called **Three fingers rule **or **UVW rule**. The initials UVW stand for **cause**, **Mediation (V) **and **Effect (W)**.

For this rule you need your thumb, your index finger and your middle finger of the right hand, which you stretch out perpendicularly to each other.

Your thumb points in the direction of movement of the load – the **current direction (Cause) ** -, you position your index finger in the direction of the **magnetic field lines (mediation)**. Now your right middle finger should be pointing in the direction the ladder is moving. So towards the **Lorentz force (effect)**.

With the **Three Finger Rule (also UVW rule)** can you the **direction of the Lorentz force** to determine charges that are in a magnetic field **perpendicular** move to the field lines. To do this, your thumb points in the direction of the current flow, your index finger in the direction of the magnetic field lines and your middle finger in the direction of the Lorentz force.

Depending on the direction of the current, you use your right or your left hand. Usually from the **technical flow direction** run out (charge flows from plus to minus in the conductor) and you can use your right hand. In the** physical direction of current** (from minus to plus) you use your left hand to determine the direction of the force.

You can find out more in the article on Three Finger Rule.

## Derivation and formula for the Lorentz force on a current-carrying conductor

The Lorentz force acts on the individual **electrons** in the conductor and deflects it in a certain direction. As a result, the conductor also moves in this direction. The force on the conductor is therefore the sum of the forces on the individual electrons. In the following example you will learn how you can use this approach to derive the formula for the force on a current-carrying conductor.

If n electrons move in the conductor, the formula for the total force on the conductor applies:

But not even physicists go to the trouble of counting the electrons individually. Instead, you summarize the number of electrons n and their charge q to the total charge Q, accordingly you get the following formula:

However, since the speed of the electrons and the total charge in the conductor are unknown, the formula has to be slightly modified. You can generally write speed as distance s per time t. In the case of the ladder swing, the distance s is the length L of the ladder.

Inserted into the formula, you get the term depending on the conductor length L and the time t:

The current strength results from the quotient of the total charge Q and the time t. Both components already exist in the equation, so you can combine them into the current:

You can use the amperage, conductor size and magnetic flux density to calculate the force acting on a piece of conductor.

The one acting on a piece of ladder **power ** You calculate with the product of the **Current I**magnetic **Flux Density B** and the **Length L** of the ladder section:

You can explain the formula logically if you see the force on the conductor as the sum of the Lorentz force on each individual electron. Accordingly, it increases as more electrons flow through the conductor. You can achieve this, for example, by increasing the current.

### Lorentz force between two current-carrying conductors

So far you have used a horseshoe magnet to generate a homogeneous magnetic field in the experiment. However, a separate magnetic field also forms around a current-carrying conductor. The horseshoe magnet cannot be used here.

As you can see in the figure, the magnetic field forms in a circle around the conductor. You can determine the direction of the field with the **rule of thumb** determine. To do this, form your right hand into a fist and stretch your thumb in the direction of the current flow. Now your fingers are pointing in the direction of the **magnetic field.**

Now you extend the experiment and bring a second conductor with the same length L into the magnetic field of the first conductor. DYou connect the second conductor to a voltage source so that the current flows in the same direction in both conductors. The two conductors are now moving towards each other. You reverse the direction of the current by one of the two leaders um, they move away from each other.

This movement is triggered by the Lorentz force. The conductors are each in the magnetic field of the other conductor. The force on both conductors is the same.

For the **power** on **two current-carrying conductors of length L** is applicable:

The power to **Focus** between the conductors in **distance r** directional if the current direction is the same in both conductors. If you reverse the current direction of one of the two conductors, they move apart.

You can find out exactly how you came up with this formula in the following in-depth study:

You can determine the magnetic flux density of the first conductor by **ampere law** determine:

So you can calculate the force on the second conductor by applying Ampere’s law and plugging in the magnetic flux density of the first conductor:

Analogously, the force on the first conductor results from the magnetic flux density that the second conductor generates:

The mathematical derivation shows that the formulas for both forces are the same:

So far, the charges in the experiment have been caused by switching on the voltage, which leads to a current flow in the conductor. You can also cause the charge to move mechanically.

## The Lorentz force on moving charges

To illustrate, let’s expand our first attempt a bit. You also need a small one for this **light bulb**which you instead of **voltage source** put into the circuit. Otherwise the device remains the same.

Now you take your index finger and carefully pull the ladder swing towards you. The lamp then flickers on briefly and goes out as soon as you stop the mechanical movement. The faster you pull the ladder swing towards you, the brighter the light will shine. In this case, you create a short-term voltage with the help of the Lorentz force.

By moving the ladder you move indirectly at the same time **electrons** inside the leader. Just like the attempt before, here are a **magnetic field** and **moving charge carriers (the electrons).** This creates one **Lorentz force**which acts on the electrons and moves them in the conductor.

As a result (depending on which direction you move the conductor) there is a at one end **Plus-**on the other one **negative pole**. A voltage is briefly created between these poles, causing the lamp to light up. Here one speaks of the fact that a voltage is induced. If you stop the movement, the charge in the conductor is not separated and therefore no voltage is generated.

This attempt lies, among other things, in the **Lenz’s rule** perish.

Everything you need to know is explained in our article on Lenz’s rule.

## Lorentz force and centripetal force, simply explained

Finally, let’s look at one more attempt. Maybe you know from the…