Right-hand rule and left-hand rule

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What is the right-hand rule?

The right-hand rule is a fundamental principle in physics that helps to determine the direction of vectors. This is particularly useful for ascertaining the direction of electromagnetic phenomena, such as the magnetic field around a current-carrying conductor, the force on a moving charge carrier in a magnetic field, or the direction of the induced electric field.

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It is also referred to as the three-finger rule. In German speaking countries it is sometimes called 'UVW rule' (U=Ursache [cause], V=Vermittlung [mediation], W=Wirkung [effect]), since, for example, on the cause of the force (namely, moving charge carriers), through the mediation of a magnetic field, a force is induced as an effect. This three-finger rule is particularly useful for intuitively understanding the spatial relationships between currents, magnetic fields and forces. There are various applications of the right-hand rule that relate to specific situations, and there is also a left-hand rule for certain applications.

Mathematical background of the right-hand rule

The right-hand rule is not only central to physics but also plays an important role in mathematics, particularly in calculating the cross product of two vectors. The cross product, a fundamental concept in vector calculus, is often used to determine the direction of a new vector that is perpendicular to two given vectors. Here, the right-hand rule provides an intuitive method to visualise the direction of this resulting vector.

The cross product

The cross product of two vectors ⊽₁ and ⊽₂ is represented by the symbol × and results in a vector ⊽₁ that is perpendicular to the plane spanned by ⊽₁ and ⊽₂. The length of ⊽₃ corresponds to the area A of the parallelogram spanned by ⊽₁ and ⊽₂ (see Illustration 1).

Illustration 1: Fig. 1 Left: Representation of the cross product of two vectors ⊽₁ and ⊽₂ at an angle Θ, which corresponds to a vector ⊽₃ that is perpendicular to the plane spanned by ⊽₁ and ⊽₂, i.e. direction of the normal vector n̄, but with the length of the area A. Shown on the right, the representation of the right-hand rule for Θ = 90°.

Mathematically, the cross product is defined as follows:

⊽₃ = ⊽₁ × ⊽₂ = |⊽₁||⊽₂|sin Θ n̄

Here, \(Θ\) is the smaller angle between ⊽₁ and ⊽₂, and n̄ is a unit vector perpendicular to the plane formed by ⊽₁ and ⊽₂. The vertical lines indicate that the absolute value of the vector must be taken. The direction of n̄ (and thus the direction of ⊽₃) is determined by the right-hand rule.

Application of the right-hand rule to the cross product

To apply the right-hand rule to the cross product, hold your right hand so that the index finger points in the direction of ⊽₁ and the middle finger points in the direction of ⊽₂, with both fingers extended at a right angle to each other. The thumb, which is also extended at a right angle to the other two fingers, then points in the direction of the resulting vector ⊽₃.

This rule makes it easier to understand the orientation of the cross product in three dimensions and to ensure that the direction of the resulting vector is correctly determined. It is particularly useful in applied mathematics, physics, and engineering, where spatial visualisation and vector analysis are crucial.

In vector analysis, the right-hand rule enables an unambiguous and consistent determination of the orientation of vectors in three-dimensional space. This is important for calculating torques, analysing rotational motion, and in computer graphics for determining the visibility of surfaces. The right-hand rule thus represents an indispensable tool that combines mathematical precision with spatial intuition.

Application for magnetic phenomena

Lorentz force on a current-carrying conductor in a magnetic field

One application of the right-hand rule is to determine the direction of the Lorentz force exerted on a current-carrying conductor in an external magnetic field. Here, the thumb of the right hand points in the direction of the conventional current direction, the index finger points in the direction of the magnetic field or magnetic flux density (from north to south pole), and the bent middle finger then indicates the direction of the force acting on the conductor. This rule is also known as Fleming's rule for motor action.

Calculation of the Lorentz force

The Lorentz force acts on moving charge carriers in a magnetic field and is clearly explained by the right-hand rule. If you hold your right hand so that the thumb is pointing in the opposite direction to the direction of motion of the negative charge carriers (and thus in the direction of the conventional current direction), the index finger is pointing in the direction of the magnetic field (from the north to the south pole), and the middle finger is extended at right angles to the other two, the middle finger indicates the direction of the Lorentz force (Illustration 2). This application of the right-hand rule makes it possible to visualise the interaction between the magnetic field and the charge carriers and to determine the direction of the resulting force acting on the charge carriers and thus on the conductor itself.

Illustration 2: If the thumb of the right hand points in the conventional direction of the current from plus to minus (this is the reverse of the direction of movement of the electrons) and the index finger points in the direction of the magnetic field (from the north to the south pole), the spread middle finger indicates the direction of the Lorentz force.

Induced voltage (Faraday's law)

The right-hand rule is also used to determine the direction of the induced voltage or current by means of electromagnetic induction. If a conductor loop moves through a magnetic field or if the magnetic field changes through the loop, the thumb can be pointed in the direction of the movement or the field change, while the index finger points in the direction of the original magnetic field. The splayed middle finger would then indicate the direction of the induced EMF (electromotive force) or the current flow.

Current-carrying conductor

A variation of the right-hand rule is possible for magnetic fields of current-carrying conductors: When an electric current flows through a conductor, the right-hand rule can be used to determine the direction of the magnetic field created around the conductor. When you hold the thumb of your right hand in the direction of the conventional current direction (from plus to minus), then your fingers bend around the conductor in the direction of the magnetic field. This means that the magnetic field runs in concentric circles around the conductor.

Applying the right-hand rule in practice

In practice, the right-hand rule is an indispensable tool for engineers, physicists, and anyone working with electromagnetic systems. It facilitates the design and analysis of electric motors, generators, transformers, and other electromagnetic devices. The visual and intuitive application of the right-hand rule enables professionals to quickly determine the directions of forces and fields, which is crucial for the correct design and function of electrical and magnetic systems.

The right-hand rule is an impressive illustration of how abstract physics concepts can be made understandable and tangible through simple physical gestures, making it much easier to understand and apply electromagnetic principles.

What is the left-hand rule?

While the right-hand rule is widely used in electrodynamics and mathematics to determine the direction of vectors, such as the magnetic field around a current-carrying conductor or the resulting vector of a cross product, there are situations in which the left-hand rule is applied. This rule is used specifically in contexts that concern the direction of the force acting on moving charge carriers (such as electrons moving opposite to the conventional current direction) in a magnetic field.

Problems involving the right-hand rule can also be directly replaced with the left-hand rule if, instead of holding the thumb in the direction of the conventional current direction, which runs from plus to minus, i.e., opposite to the movement of the electrons, the thumb of the left hand is held in the direction of the movement of the electrons. Then, the direction of the force is in the direction of the extended middle finger, just as with the right-hand rule.

When is the left-hand rule and when is the right-hand rule applied?

The choice between the right-hand rule and the left-hand rule depends on the specific physics context:

For the analysis and design of motors, the left-hand rule is particularly useful because it allows one to visualise the interaction between the magnetic field and the current flowing through the motor. This helps engineers to understand how the motor’s motion is achieved.
In vector calculus and electrodynamics in physics, the right-hand rule remains a central tool for determining the direction of vector products and forces on charge carriers in magnetic fields.

Understanding both rules and their respective applications allows for deeper insight into the fundamentals of physics and engineering by providing a direct method to determine the directions of forces, fields, and other vector quantities in three-dimensional space.

Author:
Dr Franz-Josef Schmitt

Dr Franz-Josef Schmitt is a physicist and academic director of the advanced practicum in physics at Martin Luther University Halle-Wittenberg. He worked at the Technical University from 2011-2019, heading various teaching projects and the chemistry project laboratory. His research focus is time-resolved fluorescence spectroscopy in biologically active macromolecules. He is also the Managing Director of Sensoik Technologies GmbH.

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