"Attempts to describe electromagnetic force started as early as the 18th century. Johann Tobias Mayer (1760) and Henry Cavendish (1762) were the first to suggest the force on magnetic poles and electrically charged objects obey an inverse-square law.\n",
"\n",
"## Miscalculations ;)\n",
"Sir Joseph John Thomson was first in attempting the derivation of the electromagnetic forces on a moving charged particle from Maxwell's equations (1881). In the paper he published, he gave the force as:\n",
"$\\vec{v}$ - velocity of the charged particle<br>\n",
"q - charge of the particle<br>\n",
"$\\vec{B}$ - magnetic field\n",
"\n",
"Thomson had the correct form of the equation, but due to some miscalculations and an incomplete description of the displacement current, included an incorrect scale of one-half at the front of the formula. Oliver Heaviside, who is responsible for the modern vector notation and also applied it to Maxwell's equations, fixed Thomson's mistakes and derived the correct form of the formula. For the modern formulation, Hendrik Lorentz (1892) arrived at the modern form which takes into account the total force from both the electric and magnetic fields.\n",
"It states that, to determine the direction of the magnetic force on a positive moving charge, you point the thumb of the right hand in the direction of $\\vec{v}$, the fingers in the direction of $\\vec{B}$, and a perpendicular to the palm points in the direction of $\\vec{F}.\n",
"\n",
"You can try to remember it with the following trick!<br>\n",
"<font color =\"#00cc00\">\n",
"<center>\n",
"$\\vec{v}$ Velocity is just <b>one direction</b> - so the <b>thumb</b> points it<br>\n",
"$\\vec{B}$ Magnetic field has <b>many lines</b> - so the <b>fingers</b> point them<br>\n",
"$\\vec{F}$ Force is in the direction you would <b>push</b> with your <b>palm</b><br>\n",
"</b>\n",
"<center>\n",
"</font>\n",
"<br>\n",
"<font color=\"blue\">\n",
"Try to visualize the right hand rule with the illustration below. First, guess the resultant force direction. You will see the projection of vectors on principle axises. Then, click on 'show force' to evaluate your answer.\n",
"</font>\n"
]
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"%config InlineBackend.close_figures=False \n",
"RHRSystem.FvB()"
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"# Motion of charge in uniform magnetic field\n",
"Consider a charged particle which moves perpendicular to a uniform B-field. Since the magnetic force is perpendicular to the direction of travel, a charged particle follows a curved path in a magnetic field. The particle continues to follow this curved path until it forms a complete circle. \n",
"\n",
"As the magnetic force is always perpendicular to velocity, it does no work on the charged particle. The particle’s kinetic energy and speed thus remain constant. The direction of motion is affected but not the speed.\n",
"\n",
"Since the velocity is perpendicular to the magnetic field, the magnetic force experienced by the particle is F = qvB. This supplies the centripetal force $F_{c}$ = $\\frac{mv^{2}}{2}$. Equating the two equations we obtain the radius of the circular motion as\n",
"\\begin{equation}\n",
"r = \\frac{mv}{qB}\n",
"\\end{equation}\n",
"\n",
"The time period of the circular motion is \n",
"\\begin{equation}\n",
"T = \\frac{2\\pi r}{v} = \\frac{2\\pi m}{qB}\n",
"\\end{equation}\n",
"\n",
"If the velocity is not perpendicular to the magnetic field, then we can compare each component of the velocity separately with the magnetic field. \n",
"\n",
"\\begin{equation}\n",
"v_{perpendicular} = v\\: sin \\theta\n",
"\\end{equation}\n",
"\n",
"\\begin{equation}\n",
"v_{parallel} = v\\: cos \\theta\n",
"\\end{equation}\n",
"\n",
"where $theta$ is the angle between v and B.\n",
"\n",
"The component of the velocity perpendicular to the magnetic field produces a magnetic force perpendicular to both this velocity and the field.\n",
"\n",
"The component parallel to the magnetic field creates constant motion along the same direction as the magnetic field. The result is a helical motion\n",
"\n",
"The parallel motion determines the pitch p of the helix, which is the distance between adjacent turns. This distance equals the parallel component of the velocity times the period:\n",
"\\begin{equation}\n",
"p = v\\: cos \\theta\\:T\n",
"\\end{equation}\n",
"\n",
"<font color=\"blue\">\n",
"Observe the helical motion in the illustration below\n",
"</font>\n"
]
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