motion of charged particle in uniform magnetic field

In the parallel case, there is no force on the particle but in the perpendicular one, there is a centripetal acceleration toward the center. TERMS AND PRIVACY POLICY, © 2017-2020 PHYSICS KEY ALL RIGHTS RESERVED. And you got, \[f = \frac{|q|B}{2\pi \, m} \tag{5} \label{5}\]. If the charge has mass $m$, the expression of the centripetal force on the charge is, Equating Equations \eqref{1} and \eqref{2}, and solving for $r$, you get, \[r = \frac{mv}{|q|B} \tag{3} \label{3}\]. A particle of charge q and mass m moves in XY plane. For example you can hold ionized gas of very high temperature such as $10^6 \text{K}$ in a magnetic bottle which can destroy any material if comes in contact with such a high temperature. The individuals who are preparing for Physics GRE Subject, AP, SAT, ACT exams in physics can make the most of this collection. From equations (2) and (3), it is evident that the angular frequency and period of rotation of the particle in the magnetic field do not depend upon (i) the velocity of the particle and (ii) radius of the circular path. Conceptual Questions At a given instant, an electron and a proton are moving with the same velocity in a constant magnetic field. Subscribe to our weekly Newsletter and stay tuned and get more freebies. So, what we got here is an expression for the radius of the circle in which the charge moves under the action of magnetic force. This is the main factor that creates a spiral or helical path. Thus, the charged particle continues to move along the field direction with a uniform motion (a motion in which speed and velocity is constant). Pitch of the helix: the distance traveled parallel to the magnetic field $B$ in one revolution is called the pitch of the helical path and is obtained as \begin{align*} p&=v_{\parallel}\,T\\&=(v\,\cos \theta)\,\left(\frac{2\pi\,m}{q\,B}\right)\end{align*} Thus, the formula for pitch of the helix is $p=\frac{2\pi\,mv\,\cos \theta}{q\,B}$. The absolute value of charge |q| is used because we are only considering the magnitude of magnetic force. So it is not strict to call only the frequency of rotation as cyclotron frequency. ELECTROMAGNETISM, ABOUT Time period of the helix is given by \begin{align*}T&=\frac{2\pi\,m}{eB}\\&=\frac{2(3.14)(9.11\times 10^{-31})}{(1.6\times 10^{-19})(0.2)}\\&=1.78\times 10^{-10}\,{\rm s}\\&=0.17\,{\rm ns}\end{align*}, Pitch of the helical motion is obtained as \begin{align*} p&=\frac{2\pi\,mv\,\cos \theta}{e\,B}\\&=\frac{2(3.14)(9.11\times 10^{-31})(1.8\times 10^{6})\cos 37^{\circ}}{(1.6\times 10^{-19})(0.2)}\\&=0.257\,{\rm mm}\end{align*}, Radius of the helical path is determined as \begin{align*}R&=\frac{mv\,\sin\theta}{eB}\\&=\frac{(9.11\times 10^{-31})(1.8\times 10^{6})\,\sin 37^{\circ}}{(1.6\times 10^{-19})(0.2)}\\&=0.193\,{\rm mm}\end{align*}. If the particle’s velocity has components parallel and perpendicular to the uniform magnetic field then it moves in a helical path. Storing charged particles (ionized gas) in a magnetic field has a huge importance. A particle of charge q and mass m moves in XY plane. So, we can change the linear speed and radii without affecting the angular speed or frequency. Applying Newton's second law of motion and balancing the centripetal force with the magnetic force we get a formula for radius of helical path as \begin{align*} F&=m\,a_r\\q\,v_{\bot}\,B&=m\,\frac{v_{\bot}^{2}}R\\\Rightarrow R&=\frac{m\,v_{\bot}}{q\,B}\\&=\frac{mv\,\sin\theta}{qB}\end{align*} Where $m$ is the mass of the charged particle. Let us consider a uniform magnetic field of induction B acting along the Z-axis. At points Q and R, the particle experiences force along QO and RO respectively. $\begingroup$ Related : Motion of charged particle in uniform magnetic field and a radially symmetric electric field. $\endgroup$ – Frobenius Nov 9 at 1:06 1 $\begingroup$ The solution is a helix-- … In the case of $\theta=90^{\circ}$, a circular motion is created. THERMODYNAMICS Every helical path has three distinct characteristics as radius, time period, and pitch. A particle of charge, At a point P, the velocity of the particle is, Magnetic field due to a circular loop carrying current, Magnetic induction due to a long solenoid carrying current, Force on a current carrying conductor placed in a magnetic field, Force between two long parallel current-carrying conductors, Torque experienced by a current loop in a uniform magnetic field, Conversion of galvanometer into an ammeter, Conversion of galvanometer into a voltmeter, The magnetic dipole moment of a revolving electron. Helical motion results if the velocity of the charged particle has a component parallel to the magnetic field as well as a component perpendicular to the magnetic field. MECHANICS Copyright © 2018-2021 BrainKart.com; All Rights Reserved. You may know that there is a difference between a moving charge and a stationary charge. In the former case, its path results in a circular path, and in the latter case, a helical path is formed. The angular speed $\omega$ is related to the linear speed $v$ and radius $r$, that is $\omega = v/r$, so the angular speed using Equation \eqref{3} is, \[\omega = \frac{|q|B}{m} \tag{4} \label{4}\], You know that the frequency $f$ of the rotation is $\omega / 2\pi$. The uniform magnetic field $B$ does not apply any force on the charged particle (say, electron) in the parallel direction that is $F_{\parallel}=q\,v_{\parallel}\,B\sin 0=0$. Helical path is the path of the motion of a charged particle when enters at an angle in a uniform magnetic field . In addition, there are hundreds of problems with detailed solutions on various physics topics. A charged particle (say, electron) can enter a region filled with uniform $B$ either with right angle $\theta=90^\circ$ or at angle $\theta$. The angular speed is also cyclotron frequency! Motion of a charged particle in a uniform magnetic field. Let's see what happens next. Radius: The normal force $F_{\bot}$ which creates a circular motion provides a centripetal force on the charged particle with a radial acceleration $a_r=\frac{m\,v_{\bot}^{2}}R$. As the charge moves the magnetic field exerts magnetic force on the charge and its direction is perpendicular to the plane containing $\vec v$ and $\vec B$. Now we want to answer this question: why do charged particles move in a helical path? There is no magnetic force for the motion parallel to the magnetic field, this parallel component remains constant and the motion of charged particle is helical, that is the charge moves in a helix as shown in figure below. But if the angle is not a right angle there is also a component of velocity vector parallel to the magnetic field. © 2015 All rights reserved. Cyclotron is a device where elementary particles are accelerated such as protons at high speeds. This period is also called cyclotron period and its frequency is the reciprocal of period with formula \[f=\frac 1T=\frac{q\,B}{2\pi\,m}\]. On the other hand, the vertical component undergoes a magnetic force of magnitude $F_{\bot}=q\,v_{\bot}\,B\sin 90^{\circ}=q\,v_{\bot}\,B$ which causes the charged particle moves uniformly around a circular path. Find the period, pitch, and radius of the helical path of the electron. Note that the magnetic field directed into the screen is represented by a collection of cross signs and those directed out of the screen towards you are represented dots (see Figure 2). This equation gives the angular frequency of the particle inside the magnetic field. Since the magnetic force is directed perpendicular to the plain containing $\vec v$ and $\vec B$, that is the magnetic force $\vec F$ is always perpendicular to $\vec v$, the charge moves in a circle of arbitrary radius $r$ (see fig). Particle with mass m, charge in a uniformed magnetic field Bz. Let us consider a uniform magnetic field of induction B acting along the Z-axis. If the charge is negative the rotation is clockwise. Physics problems and solutions aimed for high school and college students are provided. You can easily understand the proportionality of the radius to other related quantities from the above equation. So, the magnitude of the velocity remains constant and only its direction changes. The difference is that a moving charge has both electric and magnetic fields but a stationary charge has only electric field. This magnetic lorentz force provides the necessary centripetal force. In the above discussions the angle between magnetic field and velocity vector at each instant of motion of the charged particle is the right angle. Motion of a charged particle in a uniform magnetic field : Let us consider a uniform magnetic field of induction B acting along the Z -axis. SITEMAP Since the force acts perpendicular to its velocity, the force does not do any work. Here in this article we learn and study the motion of a charge moving in a magnetic field. In Figure 1 the magnetic field is directed inward into the screen (you are reading in the screen of a computer or a smart phone) represented by the cross (X) signs. A particle of charge q and mass m moves in XY plane. Those two above motions, uniform motion parallel to the field $B$ and uniform circular motion perpendicular to the field $B$, creates the actual path of a charged particle in a uniform magnetic field $B$ which is similar to a spring and is called a spiral or helical path.