Righthand rule
This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages)

In mathematics and physics, the righthand rule is a common mnemonic for understanding orientation of axes in threedimensional space. It is also a convenient method for quickly finding the direction of a crossproduct of 2 vectors.
Most of the various lefthand and righthand rules arise from the fact that the three axes of threedimensional space have two possible orientations.^{[citation needed]} One can see this by holding one's hands outward and together, palms up, with the thumbs outstretched 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. Lefthand and righthand 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 200francs banknote).
Curve orientation and normal vectors[edit]
In vector calculus, it is often necessary to relate the normal vector to a surface to the curve bounding it. For a positivelyoriented curve C bounding a surface S, the normal to the surface n̂ is defined such that the right thumb points in the direction of n̂, and the fingers curl along the orientation of the bounding curve C.
Coordinates[edit]
Axis or vector  Two fingers and thumb  Curled fingers 

x, 1, or A  First or index  Fingers extended 
y, 2, or B  Second finger or palm  Fingers curled 90° 
z, 3, or C  Thumb  Thumb 
Coordinates are usually righthanded.
For righthanded 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 counterclockwise.
For lefthanded 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[edit]
A rotating body[edit]
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 righthand 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 righthand rule. This causes the Sun, Moon, and stars to appear to revolve westward according to the lefthand rule.
Helices and screws[edit]
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 lefthanded, 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 lefthanded. The rule is this: if a screw is righthanded (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[edit]
 When electricity (conventional current) flows in a long straight wire, it creates a circular or cylindrical magnetic field around the wire according to the righthand 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 noncircular 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 reenter 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 righthand 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 righthand grip rule[edit]
Ampère's righthand grip rule^{[2]} (also called righthand screw rule, coffeemug rule or the corkscrewrule) 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.
AndréMarie 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 currentcarrying wire, and concluded that electricity could create magnetic fields.
Application[edit]
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 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.
 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[edit]
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:
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.^{[1]}
Applications[edit]
The righthand rule is in widespread use in physics. A list of physical quantities whose directions are related by the righthand rule is given below. (Some of these are related only indirectly to cross products, and use the second form.)
 For a rotating object, if the righthand 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 vector.
 A torque, the force that causes it, and the position of the point of application of the force.
 A magnetic field, the position of the point where it is determined, and the electric current (or change in electric flux) that causes it.
 A magnetic field in a coil of wire and the electric current in the wire.
 The force of a magnetic field on a charged particle, the magnetic field itself, and the velocity 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 righthand rule).
 The x, y and z unit vectors in a Cartesian coordinate system can be chosen to follow the righthand rule. Righthanded coordinate systems are often used in rigid body and kinematics.
See also[edit]
 Chirality (mathematics)
 Curl (mathematics)
 Fleming's lefthand rule for motors
 Improper rotation
 ISO 2
 Oersted's law
 Poynting vector
 Pseudovector
 Reflection (mathematics)
References[edit]
 ^ ^{a} ^{b} Watson, George (1998). "PHYS345 Introduction to the Right Hand Rule". udel.edu. University of Delaware.
 ^ IIT Foundation Series: Physics – Class 8, Pearson, 2009, p. 312.
External links[edit]
 Feynman's lecture on the righthand rule
 Right and Left Hand Rules  Interactive Java Tutorial National High Magnetic Field Laboratory
 Weisstein, Eric W. "Righthand rule". MathWorld.
 Christian Moser : righthandrule : wpftutorial.net