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In mathematics and
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...
, the right-hand rule is a common
mnemonic A mnemonic ( ) device, or memory device, is any learning technique that aids information retention or retrieval (remembering) in the human memory for better understanding. Mnemonics make use of elaborative encoding, retrieval cues, and image ...
for understanding orientation of axes in
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...
. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors. 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. 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 Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
,
spiral In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the American Heritage Dictionary are:electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classica ...
s,
mirror A mirror or looking glass is an object that Reflection (physics), reflects an image. Light that bounces off a mirror will show an image of whatever is in front of it, when focused through the lens of the eye or a camera. Mirrors reverse the ...
images, and enantiomers in mathematics and chemistry. The sequence is often: index finger, then middle finger, 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 finger (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 bounding a surface , 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 .


Coordinates

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.


Rotations


A rotating body

In mathematics, a rotating body is commonly represented by a pseudovector along the axis of
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
. The length of the vector gives the
speed of rotation Rotational frequency (also known as rotational speed or rate of rotation) of an object rotating around an axis is the frequency of rotation of the object. Its unit is revolution per minute (rpm), cycle per second (cps), etc. The symbol for ...
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 The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width ...
, and stars to appear to revolve westward according to the left-hand rule.


Helices and screws

A
helix A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined hel ...
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 An electromagnet is a type of magnet in which the magnetic field is produced by an electric current. Electromagnets usually consist of wire wound into a coil. A current through the wire creates a magnetic field which is concentrated in the ...
: 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 In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an ele ...
: 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''IIT Foundation Series: Physics – Class 8'', Pearson, 2009, p. 312. (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 Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
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 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 t ...
that the current created.
André-Marie Ampère André-Marie Ampère (, ; ; 20 January 177510 June 1836) was a French physicist and mathematician who was one of the founders of the science of classical electromagnetism, which he referred to as "electrodynamics". He is also the inventor of n ...
, a French physicist and mathematician, for whom the rule was named, was inspired by
Hans Christian Ørsted Hans Christian Ørsted ( , ; often rendered Oersted in English; 14 August 17779 March 1851) was a Danish physicist and chemist who discovered that electric currents create magnetic fields, which was the first connection found between electricity ...
, another physicist who experimented with magnet needles. Ørsted observed that the needles swirled when in the proximity of an
electric current An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The movin ...
-carrying wire, and concluded that electricity could create magnetic fields.


Application

This rule is used in two different applications of Ampère's circuital law: # 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 Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ...
lines around the conductor. The direction of the magnetic field (
counterclockwise Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite s ...
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. # An
electric current An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The movin ...
passes through a solenoid, resulting in a
magnetic field 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 t ...
. 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 In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and i ...
of two vectors is often taken in physics and engineering. For example, in
statics Statics is the branch of classical mechanics that is concerned with the analysis of force and torque (also called moment) acting on physical systems that do not experience an acceleration (''a''=0), but rather, are in static equilibrium with t ...
and dynamics, torque is the cross product of lever length and
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
, while
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed sy ...
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: :\mathbf = q\mathbf \times \mathbf The direction of the cross product may be found by application of the right hand rule as follows: # The index finger points in the direction of the velocity vector v. # The middle finger points in the direction of the magnetic field vector B. # 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.


Applications

The right-hand rule is in widespread use in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...
. 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 product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and i ...
s, and use the second form.) * For a rotating object, if the right-hand fingers follow the curve of a point on the object, then the thumb points along the axis of rotation in the direction of the
angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
vector. * A
torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of t ...
, the
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
that causes it, and the position of the point of application of the force. * A
magnetic field 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 t ...
, the position of the point where it is determined, and the
electric current An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The movin ...
(or change in electric flux) that causes it. * A
magnetic field 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 t ...
in a coil of wire and the
electric current An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The movin ...
in the wire. * The force of a
magnetic field 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 t ...
on a charged particle, the magnetic field itself, and the
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
of the object. * The vorticity at any point in the field of flow of a fluid * The induced current from motion in a magnetic field (known as
Fleming's right-hand rule In electromagnetism, Fleming's right-hand rule (for generators) shows the direction of induced current when a conductor attached to a circuit moves in a magnetic field. It can be used to determine the direction of current in a generator's wi ...
). * The ''x'', ''y'' and ''z'' unit vectors in a
Cartesian coordinate system 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 ...
can be chosen to follow the right-hand rule. Right-handed coordinate systems are often used in
rigid body In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external fo ...
and kinematics.


See also

*
Chirality (mathematics) In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. An object that is not chiral is said to b ...
*
Curl (mathematics) In vector calculus, the curl is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction den ...
* Fleming's left-hand rule for motors *
Improper rotation In geometry, an improper rotation,. also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendic ...
* ISO 2 * Oersted's law * Poynting vector * Pseudovector *
Reflection (mathematics) In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of r ...


References


External links


Feynman's lecture on the right-hand rule

Right and Left Hand Rules - Interactive Java Tutorial
National High Magnetic Field Laboratory *

{{Science mnemonics Electromagnetism Mechanics Orientation (geometry) Vectors (mathematics and physics) Rules of thumb Science mnemonics Hand gestures