Right-hand rule

This article is about three-dimensional vector geometry. For the maze-solving technique, see Maze solving algorithm § Wall follower.

In mathematics and physics, the right-hand rule is a common mnemonic for understanding notation conventions for vectors in three dimensions.

Ampère's right hand screw rules

Prediction of direction of field (B), given that the current I flows in the direction of the thumb

The right-hand rule as applied to motion produced with screw threads

Ampère's right hand screw rule (also called right-hand grip 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 version of the rule is used in two complementary applications of Ampère's circuital law:

1. 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.
2. An electric current passes through a straight wire. Grabbing the wire points the thumb in the direction of the conventional current (from positive to negative), while the fingers point in the direction of the magnetic flux lines. The direction of the magnetic field (counterclockwise instead of clockwise when viewed from the tip of the thumb) is a result of this convention and not an underlying physical phenomenon. The thumb points direction of current and fingers point direction of magnetic lines of force.

The rule is also used to determine the direction of the torque vector. When gripping the imaginary axis of rotation of the rotational force so that your fingers point in the direction of the force, the extended thumb points in the direction of the torque vector.

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 linear momentum and distance. In electricity and magnetism, the force exerted on a moving charged particle when moving in a magnetic field B is given by:

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

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 momentum 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.

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.)

• 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 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 right-hand rule)
• The x, y and z unit vectors in a Cartesian coordinate system can be chosen to follow the right-hand rule. Right-handed coordinate systems are often used in rigid body physics and kinematics.

The left-handed orientation is shown on the left, and the right-handed on the right.

Coordinate orientation

Axis or vector	Right-hand	Right-hand (alternative)
X, 1, or A	First or index	Thumb
Y, 2, or B	Second finger or palm	First or index
Z, 3, or C	Thumb	Second finger or palm

See also

• Chirality (mathematics)
• Curl (mathematics)
• Fleming's left-hand rule for motors
• Improper rotation
• ISO 2
• Pseudovector
• Reflection (mathematics)
• Vorticity

External links

• Right and Left Hand Rules - Interactive Java Tutorial National High Magnetic Field Laboratory
• A demonstration of the right-hand rule at physics.syr.edu
• Weisstein, Eric W., "Right-hand rule", MathWorld.
• Dr. Johannes Heidenhain : Right Hand Rule - Heidenhain TNC Training : heidenhain.de
• Christian Moser : right-hand-rule : wpftutorial.net