Right-hand rule

Left-handed coordinates on the left,
right-handed coordinates on the right.

For right-handed coordinates, if the thumb of a person's right hand points along the z-axis in the positive direction (third coordinate vector), then the fingers curl from the positive x-axis (first coordinate vector) toward the positive y-axis (second coordinate vector). When viewed at a position along the positive z-axis, the ¼ turn from the positive x- to the positive y-axis is counter-clockwise.

For left-handed coordinates, the above description of the axes is the same, except using the left hand; and the ¼ turn is clockwise.

Interchanging the labels of any two axes reverses the handedness. Reversing the direction of one axis (or three axes) also reverses the handedness. Reversing two axes amounts to a 180° rotation around the remaining axis, also preserving the handedness. These operations can be composed to give repeated changes of handedness.[2] (If the axes do not have a positive or negative direction, then handedness has no meaning.)

In vector calculus, it is necessary to relate a normal vector of a surface to the boundary curve of the surface. Given a surface S with a specified normal direction n̂ (a choice of "upward direction" with respect to S), the boundary curve C around S is defined to be positively oriented provided that the right thumb points in the direction of n̂ and the fingers curl along the orientation of the bounding curve C.

Right-hand rule for curve orientation.

• When electricity flows (with direction given by conventional current) in a long straight wire, it creates a cylindrical magnetic field around the wire according to the right-hand rule. The conventional direction of a magnetic line is given by a compass needle.
• Electromagnet: The magnetic field around a wire is relatively 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 an electric charge moves across a magnetic field, it experiences a force according to Lorentz force, with the direction given by the right-hand rule. If the index finger represents the direction of magnetic field in 3 dimensional space and the middle finger represents the direction of flow of charge (i.e current), the direction of the force on the charge is represented by the 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

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

Finding direction of magnetic field (B) for an electrical coil

Ampère's right-hand grip rule,[3] also called the 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. Ampère was inspired by fellow physicist Hans Christian Ørsted, who observed that needles swirled when in the proximity of an electric current-carrying wire and concluded that electricity could create magnetic fields.

Illustration of the right-hand rule on the ninth series of the Swiss 200-francs banknote.

The cross product of two vectors is often taken in physics and engineering. For example, as discussed above, the force exerted on a moving charged particle when moving in a magnetic field B is given by the magnetic term of Lorentz force:

${\displaystyle \mathbf {F} =q\mathbf {v} \times \mathbf {B} }$ (vector cross product)

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.[2]

Unlike most mathematical concepts, the meaning of a right-handed coordinate system cannot be expressed in terms of any mathematical axioms. Rather, the definition depends on chiral phenomena in the physical world, for example the culturally transmitted meaning of right and left hands, a majority human population with dominant right hand, or certain phenomena involving the weak force.

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

Wikimedia Commons has media related to Right-hand rule.

• Feynman's lecture on the right-hand rule
• Right and Left Hand Rules - Interactive Java Tutorial National High Magnetic Field Laboratory
• Weisstein, Eric W. "Right-hand rule". MathWorld.
• Christian Moser : right-hand-rule : wpftutorial.net