# Right-hand rule

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In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors.

## Contents

Most of the various left-hand and right-hand rules arise from the fact that the three axes of three-dimensional space have two possible orientations.[ citation needed ] One can see this by holding one's hands outward and together, palms up, with the thumbs out-stretched to the right and left, and the fingers making a curling motion from straight outward to pointing upward. (Note the picture to right is not an illustration of this.) The curling motion of the fingers represents a movement from the first (x axis) to the second (y axis); the third (z axis) can point along either thumb. Left-hand and right-hand rules arise when dealing with coordinate axes. The rule can be used to find the direction of the magnetic field, rotation, spirals, electromagnetic fields, mirror images, and enantiomers in mathematics and chemistry.

The sequence is often: index finger, then middle, then thumb. However, two other sequences also work because they preserve the cycle:

• Middle finger, then thumb, then index finger.
• Thumb, then index finger, then middle (e.g., see the ninth series of the Swiss 200-francs banknote).

## Curve orientation and normal vectors

In vector calculus, it is often necessary to relate the normal vector to a surface to the curve bounding it. For a positively-oriented curve C bounding a surface S, the normal to the surface is defined such that the right thumb points in the direction of , and the fingers curl along the orientation of the bounding curve C.

## Coordinates

For right-handed coordinates use the right hand.
For left-handed coordinates use the left hand.
Axis or vectorTwo fingers and thumbCurled fingers
x, 1, or AFirst or indexFingers extended
y, 2, or BSecond finger or palmFingers curled 90°
z, 3, or CThumbThumb

Coordinates are usually right-handed.

For right-handed coordinates the right thumb points along the z axis in the positive direction and the curling motion of the fingers of the right hand represents a motion from the first or x axis to the second or y axis. When viewed from the top or z axis the system is counter-clockwise.

For left-handed coordinates the left thumb points along the z axis in the positive direction and the curling motion of the fingers of the left hand represent a motion from the first or x axis to the second or y axis. When viewed from the top or z axis the system is clockwise.

Interchanging the labels of any two axes reverses the handedness. Reversing the direction of one axis (or of all three axes) also reverses the handedness. (If the axes do not have a positive or negative direction then handedness has no meaning.) Reversing two axes amounts to a 180° rotation around the remaining axis. [1]

## Rotations

### A rotating body

In mathematics, a rotating body is commonly represented by a pseudovector along the axis of rotation. The length of the vector gives the speed of rotation and the direction of the axis gives the direction of rotation according to the right-hand rule: right fingers curled in the direction of rotation and the right thumb pointing in the positive direction of the axis. This allows some easy calculations using the vector cross product. No part of the body is moving in the direction of the axis arrow. By coincidence, if the thumb is pointing north, Earth rotates in a prograde direction according to the right-hand rule. This causes the Sun, Moon, and stars to appear to revolve westward according to the left-hand rule.

### Helices and screws

A helix is a curved line formed by a point rotating around a center while the center moves up or down the z axis. Helices are either right- or left-handed, curled fingers giving the direction of rotation and thumb giving the direction of advance along the z axis.

The threads of a screw are a helix and therefore screws can be right- or left-handed. The rule is this: if a screw is right-handed (most screws are) point your right thumb in the direction you want the screw to go and turn the screw in the direction of your curled right fingers.

## Electromagnetism

• When electricity (conventional current) flows in a long straight wire, it creates a circular or cylindrical magnetic field around the wire according to the right-hand rule. The conventional current, which is the opposite of the actual flow of electrons, is a flow of positive charges along the positive z axis. The conventional direction of a magnetic line is given by a compass needle.
• Electromagnet: The magnetic field around a wire is quite weak. If the wire is coiled into a helix, all the field lines inside the helix point in the same direction and each successive coil reinforces the others. The advance of the helix, the non-circular part of the current, and the field lines all point in the positive z direction. Since there is no magnetic monopole, the field lines exit the +z end, loop around outside the helix, and re-enter at the −z end. The +z end where the lines exit is defined as the north pole. If the fingers of the right hand are curled in the direction of the circular component of the current, the right thumb points to the north pole.
• Lorentz force: If a positive electric charge moves across a magnetic field, it experiences a force according Lorentz force, with the direction given by the right-hand rule. If the curl of the right fingers represents a rotation from the direction the charge is moving to the direction of the magnetic field, then the force is in the direction of the right thumb. Because the charge is moving, the force causes the particle path to bend. The bending force is computed by the vector cross product. This means that the bending force increases with the velocity of the particle and the strength of the magnetic field. The force is maximum when the particle direction and magnetic fields are at right angles, is less at any other angle and is zero when the particle moves parallel to the field.

### Ampère's right-hand grip rule

Ampère's right-hand grip rule [2] (also called right-hand screw rule, coffee-mug rule or the corkscrew-rule) is used either when a vector (such as the Euler vector) must be defined to represent the rotation of a body, a magnetic field, or a fluid, or vice versa, when it is necessary to define a rotation vector to understand how rotation occurs. It reveals a connection between the current and the magnetic field lines in the magnetic field that the current created.

André-Marie Ampère, a French physicist and mathematician, for whom the rule was named, was inspired by Hans Christian Ørsted, another physicist who experimented with magnet needles. Ørsted observed that the needles swirled when in the proximity of an electric current-carrying wire, and concluded that electricity could create magnetic fields.

### Application

This rule is used in two different applications of Ampère's circuital law:

1. An electric current passes through a straight wire. When the thumb is pointed in the direction of conventional current (from positive to negative), the curled fingers will then point in the direction of the magnetic flux lines around the conductor. The direction of the magnetic field (counterclockwise rotation instead of clockwise rotation of coordinates when viewing the tip of the thumb) is a result of this convention and not an underlying physical phenomenon.
2. An electric current passes through a solenoid, resulting in a magnetic field. When wrapping the right hand around the solenoid with the fingers in the direction of the conventional current, the thumb points in the direction of the magnetic north pole.

## Cross products

The cross product of two vectors is often taken in physics and engineering. For example, in statics and dynamics, torque is the cross product of lever length and force, while angular momentum is the cross product of distance and linear momentum. In electricity and magnetism, the force exerted on a moving charged particle when moving in a magnetic field B is given by:

${\displaystyle \mathbf {F} =q\mathbf {v} \times \mathbf {B} }$

The direction of the cross product may be found by application of the right hand rule as follows:

1. The index finger points in the direction of the velocity vector v.
2. The middle finger points in the direction of the magnetic field vector B.
3. The thumb points in the direction of the cross product F.

For example, for a positively charged particle moving to the north, in a region where the magnetic field points west, the resultant force points up. [1]

### Applications

The right-hand rule is in widespread use in physics. A list of physical quantities whose directions are related by the right-hand rule is given below. (Some of these are related only indirectly to cross products, and use the second form.)

## Related Research Articles

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A Cartesian coordinate system in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference coordinate line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

In physics, a dipole is an electromagnetic phenomenon which occurs in two ways:

In physics the Lorentz force is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge q moving with a velocity v in an electric field E and a magnetic field B experiences a force of

Statics is the branch of classical mechanics that is concerned with the analysis of force and torque acting on physical systems that do not experience an acceleration (a=0), but rather, are in static equilibrium with their environment. The application of Newton's second law to a system gives:

In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment, moment of force, rotational force or turning effect, depending on the field of study. It represents the capability of a force to produce change in the rotational motion of the body. The concept originated with the studies by Archimedes of the usage of levers, which is reflected in his famous quote: "Give me a lever and a place to stand and I will move the Earth". Just as a linear force is a push or a pull, a torque can be thought of as a twist to an object around a specific axis. Torque is defined as the product of the magnitude of the force and the perpendicular distance of the line of action of a force from the axis of rotation. The law of Conservation of energy can also be used to understand Torque. The symbol for torque is typically , the lowercase Greek letter tau. When being referred to as moment of force, it is commonly denoted by M.

In mathematics, physics, and engineering, a Euclidean vector or simply a vector is a geometric object that has magnitude and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a directed line segment, or graphically as an arrow connecting an initial pointA with a terminal pointB, and denoted by .

A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to the magnetic field. A permanent magnet's magnetic field pulls on ferromagnetic materials such as iron, and attracts or repels other magnets. In addition, a magnetic field that varies with location will exert a force on a range of non-magnetic materials by affecting the motion of their outer atomic electrons. Magnetic fields surround magnetized materials, and are created by electric currents such as those used in electromagnets, and by electric fields varying in time. Since both strength and direction of a magnetic field may vary with location, it is described mathematically by a function assigning a vector to each point of space, called a vector field.

In electrodynamics, circular polarization of an electromagnetic wave is a polarization state in which, at each point, the electromagnetic field of the wave has a constant magnitude and is rotating at a constant rate in a plane perpendicular to the direction of the wave.

In physics, angular velocity or rotational velocity, also known as angular frequency vector, is a pseudovector representation of how fast the angular position or orientation of an object changes with time. The magnitude of the pseudovector represents the angular speed, the rate at which the object rotates or revolves, and its direction is normal to the instantaneous plane of rotation or angular displacement. The orientation of angular velocity is conventionally specified by the right-hand rule.

In physics, specifically electromagnetism, the Biot–Savart law is an equation describing the magnetic field generated by a constant electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the electric current. The Biot–Savart law is fundamental to magnetostatics, playing a role similar to that of Coulomb's law in electrostatics. When magnetostatics does not apply, the Biot–Savart law should be replaced by Jefimenko's equations. The law is valid in the magnetostatic approximation, and consistent with both Ampère's circuital law and Gauss's law for magnetism. It is named after Jean-Baptiste Biot and Félix Savart, who discovered this relationship in 1820.

In physics and mathematics, a pseudovector is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its opposite if the orientation of the space is changed, or an improper rigid transformation such as a reflection is applied to the whole figure. Geometrically, the direction of a reflected pseudovector is opposite to its mirror image, but with equal magnitude. In contrast, the reflection of a true vector is exactly the same as its mirror image.

A polarizer or polariser is an optical filter that lets light waves of a specific polarization pass through while blocking light waves of other polarizations. It can filter a beam of light of undefined or mixed polarization into a beam of well-defined polarization, that is polarized light. The common types of polarizers are linear polarizers and circular polarizers. Polarizers are used in many optical techniques and instruments, and polarizing filters find applications in photography and LCD technology. Polarizers can also be made for other types of electromagnetic waves besides visible light, such as radio waves, microwaves, and X-rays.

Fleming's left-hand rule for electric motors is one of a pair of visual mnemonics, the other being Fleming's right-hand rule. They were originated by John Ambrose Fleming, in the late 19th century, as a simple way of working out the direction of motion in an electric motor, or the direction of electric current in an electric generator.

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A screw axis is a line that is simultaneously the axis of rotation and the line along which translation of a body occurs. Chasles' theorem shows that each Euclidean displacement in three-dimensional space has a screw axis, and the displacement can be decomposed into a rotation about and a slide along this screw axis.

The various FBI mnemonics show the direction of the force on a conductor carrying a current in a magnetic field as predicted by Fleming's left hand rule for motors and Faraday's law of induction.

The poles of astronomical bodies are determined based on their axis of rotation in relation to the celestial poles of the celestial sphere. Astronomical bodies include stars, planets, dwarf planets and small Solar System bodies such as comets and minor planets, as well as natural satellites and minor-planet moons.

Magnets exert forces and torques on each other due to the rules of electromagnetism. The forces of attraction field of magnets are due to microscopic currents of electrically charged electrons orbiting nuclei and the intrinsic magnetism of fundamental particles that make up the material. Both of these are modeled quite well as tiny loops of current called magnetic dipoles that produce their own magnetic field and are affected by external magnetic fields. The most elementary force between magnets is the magnetic dipole–dipole interaction. If all of the magnetic dipoles that make up two magnets are known then the net force on both magnets can be determined by summing up all these interactions between the dipoles of the first magnet and that of the second.

In electromagnetism, Ørsted's law, also spelled Oersted's law, is the physical law stating that an electric current creates a magnetic field.

## References

1. Watson, George (1998). "PHYS345 Introduction to the Right Hand Rule". udel.edu. University of Delaware.
2. IIT Foundation Series: Physics – Class 8, Pearson, 2009, p. 312.