Right-hand rule

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Finding the direction of the cross product by the right-hand rule

In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation conventions for vector cross product in three dimensions.

Most of the various left- and right-hand rules arise from the fact that the three axes of three-dimensional space have two possible orientations. This can be seen by holding your hands outward and together, palms up, with the fingers curled. If the curl of your fingers represents a movement from the first or x-axis to the second or y-axis, then the third or z-axis can point along either thumb. Left- and right-hand rules arise when dealing with coordinate axes, rotation, spirals, electromagnetic fields, mirror images, and enantiomers in mathematics and chemistry.

Electromagnetics

  • When electricity 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 you coil the wire into a right-handed (or left-handed) 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 reenter at the −Z end. The +Z end where the lines exit is defined as the north pole. If you curl the fingers of your right hand in the direction of the circular component of the current your right thumb points to the north pole.
  • Lorentz force: If a positive electric charge moves across a magnetic field it experiences a force according to the right-hand rule. If the curl of your 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 your 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

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

Ampère's right-hand grip rule[1] (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 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 .

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:

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]

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.)

See also

References

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

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
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