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Derivation Lorentz Force on Current-Carrying Wires

Attractive Lorentz force - two current-carrying conductors of the same direction
Level 2 (without higher mathematics)
Level 2 requires school mathematics. Suitable for pupils.
Updated by Alexander Fufaev on
Table of contents
  1. Lorentz force on a current-carrying conductor Here you will learn how to derive the Lorentz force formula for a current-carrying wire that is perpendicular or oblique to an external magnetic field.
  2. Lorentz force between two current-carrying wires Here you will learn the force with which two current-carrying conductors attract or repel each other.

In the following, we will derive a formula for Lorentz force experienced by one or two current-carrying wires in an external magnetic field.

Lorentz force on a current-carrying conductor

Current-carrying wire perpendicular to the magnetic field

Charge inside a current-carrying wire
Illustration : A current-carrying wire in an external magnetic field with field lines perpendicular to the wire.

We have the following situation:

  • An electric current \( \class{blue}{I} \) flows through a straight electric conductor (wire). It does not matter whether \(\class{blue}{I}\) is a current of positive or negative charges.

  • The wire has the length \(L\).

  • The conductor is in a homogeneous magnetic field \( \class{violet}{B} \) which is perpendicular to the current \( \class{blue}{I} \).

The electric charges flowing through the wire, experience a magnetic force (Lorentz force) \(\class{green}{F}\).

Formula: Lorentz force on the charge inside wire
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Since the velocity \(\class{blue}{v}\) is unknown, we want to use the current \(\class{blue}{I}\) instead. We can find out the current value quite easily with an ammeter. The current \(\class{blue}{I}\) is here the total amount of charge \(\class{blue}{Q}\), which flows along the distance \(L\) per time \(t\):

The individual charges in the wire move through the wire with a certain average velocity \(\class{blue}{v}\). Each individual charge experiences a Lorentz force. We can write the total Lorentz force on the wire as the Lorentz force acting on the total charge \(\class{blue}{Q}\). The total charge here is sum of the individual charges moving through the considered piece of the wire:

Definition: Current due to movement of charges
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A charged particle travels the length \(L\) of the wire within the time \(t\). "Distance per time" is exactly the definition of velocity. In this case, it is the speed with which a charge travels through the wire:

Formula: Speed is length per time
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Let us now rearrange the velocity 3 for the time \(t\): \(t ~=~ \frac{L}{\class{blue}{v}}\) and insert the time into the definition 2 of the current:

Current expressed with speed
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Then we rearrange the current 4 for the unknown velocity \(\class{blue}{v}\):

Velocity is current times length divided by charge
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Very nice, because now we can insert this relation into the Lorentz force formula 1 and thus eliminate the unknown velocity:

Lorentz force with eliminated velocity
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The unknown charge \(\class{blue}{Q}\) is thereby eliminated. The formula to be derived is thus:

Formula: Lorentz force on current-carrying wire in perpendicular B-field
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Electron current in the wire in the B-field
Illustration : A current-carrying wire in a magnetic field applied perpendicularly to it. The conductor is deflected due to the magnetic force.

Current-carrying wire oblique to the magnetic field

If the magnetic field \(\class{violet}{B}\) is oblique (at an angle not 90 degrees) to the current \(\class{blue}{I}\), then we need to make a small correction to the derived formula.

The Lorentz force \( \class{green}{F} \) on the charge \(\class{blue}{Q}\) moving not perpendicular to the homogeneous magnetic field is:

Formula: Lorentz force in the non-perpendicular B-field using velocity
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Here \(\alpha\) is the angle between the velocity direction (velocity vector) and the magnetic field direction (magnetic field vector).

Inserting the rewritten velocity 5 results in the following formula:

Formula: Lorentz force on current-carrying wire oblique to the B-field
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Lorentz force between two current-carrying wires

For a single wire, the magnetic field \(\class{violet}{B}\) was generated by some external source. If we now add a second current-carrying wire, we can use the magnetic field generated by this wire and see how another current-carrying wire behaves in this magnetic field. So we have the following setup:

Two parallel current-carrying conductors of the same direction
Illustration : A current-carrying wire that is in the magnetic field of the other wire and vice versa.
  • The first wire has the length \(L\) and a current \(\class{blue}{I_1}\) flows through it. The conductor generates a circular magnetic field \(\class{violet}{B_1}\) concentrically around the wire.

  • The second wire also has the length \(L\) and a possibly different current \(\class{blue}{I_2}\) flows through it. For example, this current may flow in the opposite direction or have a different magnitude. The second conductor also generates a circular magnetic field \(\class{violet}{B_2}\) concentrically around itself.

The magnetic field of a straight wire can be derived with the Ampere's law. We take the corresponding formula as given. The first wire produces the following magnetic field \(\class{violet}{B_1}\):

Formula: Generated magnetic field of the first wire
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Here \(\mu_0\) is the magnetic field constant and \(\pi = 3.14... \) a mathematical constant. More important here is the distance \(r\) from the wire. So the magnetic field \(\class{violet}{B_1}\) generated by the first wire depends on the magnitude of the current \(\class{blue}{I_1}\) and the distance \(r\) from the wire.

If we now place the second wire in the magnetic field \(\class{violet}{B_1}\) perpendicular to it (that is, \(\class{violet}{B_1}\) and \(\class{blue}{I_2}\) are perpendicular to each other), then we can use the previously derived formula 7 for the Lorentz force acting on a current-carrying wire:

We just need to adjust the formula a bit.

  1. The Lorentz force \(\class{green}{F}\) in this case corresponds to the Lorentz force \(\class{green}{F_2}\) on the second wire.

  2. The current \(\class{blue}{I}\) here is the current \( \class{blue}{I_2}\) flowing through the second wire.

  3. We also place the second wire at a distance \(r\) from the first wire. At this distance \(r\) the magnetic field has the value \(\class{violet}{B_1}\) generated by the first wire.

Formula: Lorentz force on the second wire due to the magnetic field of the first wire
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Now we can substitute the formula 9 for the magnetic field \(\class{violet}{B_1}\) into the Lorentz force formula 11:

Magnetic field of the first wire inserted into the Lorentz force formula
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We can analogously determine the Lorentz force \(\class{green}{F_1}\) on the first wire, which is in the magnetic field \(\class{violet}{B_2}\) of the second wire:

Formula: Lorentz force on the first wire
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As you can see, both conductors experience the same magnitude of Lorentz force: \( \class{green}{F_2} ~=~ \class{green}{F_1}\). Therefore, we can also omit the numbering of the forces and simply write \( \class{green}{F} \):

Formula: Lorentz force on a wire in the magnetic field of another wire
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Attractive Lorentz force - two current-carrying conductors of the same direction
Illustration : Two electron currents flowing in the same direction exert an attractive Lorentz force on each other.
Repulsive Lorentz force - two conductors with current flowing in opposite directions
Illustration : Two opposing electron currents exert a repulsive Lorentz force on each other.