Draft:Right-hand rule

Right-hand rule in Calculus and Electromagnetism

The right-hand rule is a common convention used in mathematics to define the direction of the cross product vector in three-dimensional space and in physics to determine the direction of the force on a current-carrying conductor in a magnetic field.

Right-hand rule in Calculus

The right-hand rule dates back to the 19th century when it was implemented as a way for identifying the positive direction of coordinate axes in three dimensions. William Rowan Hamilton, recognized for his development of quaternions, a mathematical system for representing three-dimensional rotations, is often attributed with the introduction of this convention.

Right-hand rule for cross product

In the context of quaternions, the Hamiltonian product of two vector quaternions yields a quaternion comprising both scalar and vector components.Josiah Willard Gibbs recognized that treating these components separately, as dot and cross product, simplifies vector formalism. Following a substantial debate, the mainstream adoption shifted from Hamilton's quaternionic system to Gibbs' three-vectors system. This transition led to the right-hand rule being commonly used today. The cross product of vectors ${\vec {a}}$ and ${\vec {b}}$ is a vector perpendicular to the plane spanned by ${\vec {a}}$ and ${\vec {b}}$ with the direction given by the right-hand rule: If you put the index of your right hand on ${\vec {a}}$ and the middle finger on ${\vec {b}}$, then the thumb points in the direction of ${\vec {a}}\times {\vec {b}}$ .

Right-hand rule in Electromagnetism

Fleming's right-hand rule

The right-hand rule in physics was introduced in the late 19th century by John Fleming in his book Magnets and Electric Currents. Fleming described the direction of the induced electromotive force with reference to the motion of the conductor and the direction of the magnetic field as follows:

“If a conductor, represented by the middle finger, be moved in a field of magnetic flux, the direction of which is represented by the direction of the forefinger, the direction of this motion, being in the direction of the thumb, then the electromotive force set up in it will be indicated by the direction in which the middle finger points."

Right-hand rule for ${\vec {F}}=q{\vec {v}}\times {\vec {B}}$

When considering the magnetic force on a current-carrying wire, it is to be recognized as the sum of the forces on the individual moving charges. The force on each particle represented as ${\vec {F}}=q{\vec {v}}\times {\vec {B}}$, results from the charge $q$ times the cross product of its velocity ${\vec {v}}$ and the magnetic field ${\vec {B}}$. According to the right-hand rule the force ${\vec {F}}$ aligns with the direction of ${\vec {v}}$ $\times$ ${\vec {B}}$ pointing at the thumb if $q$ is positive and in the opposite the direction of the thumb if $q$ is negative. Regardless of the sign of the charge, the magnetic force acting on a charged particle moving with a velocity ${\vec {v}}$ in a magnetic field ${\vec {B}}$ is always perpendicular to both ${\vec {v}}$ and ${\vec {B}}$.

Right-hand palm rule for ${\vec {F}}=q{\vec {v}}\times {\vec {B}}$

An alternative convention, beneficial in dealing with fields, involves holding the right hand flat with the thumb pointing in the direction of the velocity of the charged particle ${\vec {v}}$ and the four fingers pointing in the direction of the magnetic field ${\vec {B}}$. The direction of the force ${\vec {F}}$ is then identified as outward from the palm, in the direction one would push.

Right-hand palm rule for ${\vec {F}}=i{\vec {L}}\times {\vec {B}}$

For a straight wire carrying current $i$ within a magnetic field ${\vec {B}}$, the resulting magnetic force is determined by the cross product ${\vec {F}}=i{\vec {L}}\times {\vec {B}}$, where ${\vec {L}}$ represents a length vector aligned with the conventional current direction. If the fingers of the right hand point in the direction of the magnetic field ${\vec {B}}$ and the thumb posts in the direction of ${\vec {L}}$ or the direction positive charges move, then the magnetic force ${\vec {F}}$ is outward from the palm and can be identified by the direction one would push as indicated by the right-hand rule.

Right-hand rule for ${\vec {\tau }}={\vec {\mu }}\times {\vec {B}}$

Right-hand grip rule

When a rectangular current-carrying loop is placed in a magnetic field, the magnetic forces induce a torque that causes rotation. The loop's orientation is linked to its magnetic dipole moment or magnetic moment ${\vec {\mu }}$, perpendicular to the loop's plane. By pointing or curling the fingers of the right hand in the direction of the current $i$ at any point on the loop, the extended thumb points towards ${\vec {\mu }}$. The torque is defined to express a sense of rotation through a single vector, given by ${\vec {\tau }}={\vec {\mu }}\times {\vec {B}}$ . The right-hand rule connects this rotational sense to the appropriate direction for the torque vector ${\vec {\tau }}$. The torque acts to align the direction of the magnetic moment ${\vec {\mu }}$ with the direction of the magnetic field ${\vec {B}}$. In the context of a motor, when the magnetic moment ${\vec {\mu }}$ starts aligning with the direction of ${\vec {B}}$, the current in the coil is reversed, so that a torque continues to rotate the coil.

Right-hand grip rule

An alternative version of the right-hand rule, referred to as the right-hand grip rule or curl right-hand rule, helps determine the magnetic field's direction due to a current in a section of a long, straight wire. When grasping the wire section, the extended thumb points in the direction of the current $i$, and the fingers naturally curl around in the direction of the magnetic field ${\vec {B}}$ produced around the wire, forming closed concentric circles.