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# motion of charged particle in uniform magnetic field

Let us consider a uniform magnetic field of induction B acting along the Z-axis. You can easily understand the proportionality of the radius to other related quantities from the above equation. The uniform magnetic field $B$ does not apply any force on the charged particle (say, electron) in the parallel direction that is $F_{\parallel}=q\,v_{\parallel}\,B\sin 0=0$. A particle of charge q and mass m moves in X Y plane. But if the angle is not a right angle there is also a component of velocity vector parallel to the magnetic field. The above Equation \eqref{5} suggests that the frequency of rotation does not depend on the radius of the circle and speed (linear) of the charge and it is also called cyclotron frequency. On a moving charged particle in a uniform magnetic field, a magnetic force of magnitude $F_B=qv\,B\,\sin \theta$ is acted where $\theta$ is the angle of velocity $\vec{v}$ with the magnetic field $\vec{B}$. If the charge is negative the rotation is clockwise. 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). 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. Storing charged particles (ionized gas) in a magnetic field has a huge importance. 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. So, we can change the linear speed and radii without affecting the angular speed or frequency. It is evident from this equation, that the radius of the circular path is proportional to (i) mass of the particle and (ii) velocity of the particle. $\begingroup$ Related : Motion of charged particle in uniform magnetic field and a radially symmetric electric field. Note the cyclotron is just a device. The force F acting towards the point O acts as the centripetal force and makes the particle to move along a circular path. As soon as a charged particle enter a magnetic field $B$ with some angle $\theta$, one can decompose its velocity into parallel and vertical components with respect to $\vec{B}$ which are $v_{\parallel}=v\,\cos \theta$ and $v_{\bot}=v\,\sin \theta$. The individuals who are preparing for Physics GRE Subject, AP, SAT, ACT exams in physics can make the most of this collection. The frequency do not depend on the energy of the particle. Let us consider a uniform magnetic field of induction B acting along the Z-axis. At a point P, the velocity of the particle is v. (Fig 3.20). (2D case) When the charged particle is within a magnetic field, the radius of the circular motion is quite small and the frequency is huge. Copyright © 2018-2021 BrainKart.com; All Rights Reserved. In the parallel case, there is no force on the particle but in the perpendicular one, there is a centripetal acceleration toward the center. At a point P, the velocity of the particle is v. (Fig 3.20) Since the force acts … Now we want to answer this question: why do charged particles move in a helical path? The Equation \eqref{5} also suggests we can change the cyclotron frequency by simply changing the magnetic field. 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. Time period: The time needed to complete one revolution is obtained by definition of average velocity as \begin{align*} v&=\frac{\Delta x}{\Delta t}\\v_{\bot}&=\frac{2\pi\,R}{T}\\\Rightarrow T&=\frac{2\pi\,R}{v_{\bot}}\\&=\frac{2\pi}{q\,B}\,m\end{align*} where in the above we used the preceding formula for $R$ and $v_{\bot}=v\,\sin \theta$. MECHANICS 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, $r = \frac{mv}{|q|B} \tag{3} \label{3}$. (BS) Developed by Therithal info, Chennai. 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. The difference is that a moving charge has both electric and magnetic fields but a stationary charge has only electric field. 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. Let us consider a uniform magnetic field of induction B acting along the Z-axis. The result is very interesting (continue reading and you'll know what I mean by this). It is because the direction of force is always perpendicular meaning the force is always directed to the center of the circle. So, the magnetic force also provides the centripetal force to the charge. 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. In Figure 3 a charge $q$ is moving in the magnetic field $\vec B$ with speed $v$. Motion of a charged particle in a uniform magnetic field. 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's see what happens next. $\endgroup$ – Frobenius Nov 9 at 1:06 1 $\begingroup$ The solution is a helix-- … 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. 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*}. Think this way, an arrow is moving towards you and what you notice is the tip of the arrow (represented by dot), that is the same as moving outward from the screen (towards you). Radius: The normal force $F_{\bot}$ which creates a circular motion provides a centripetal force on the charged particle with a radial acceleration $a_r=\frac{m\,v_{\bot}^{2}}R$. Since the force acts perpendicular to its velocity, the force does not do any work. Cyclotron is a device where elementary particles are accelerated such as protons at high speeds. In this short tutorial, we explain the factors that cause this type of motion. Helical path is formed when a charged particle enters with an angle of $\theta$ other than $90^{\circ}$ into a uniform magnetic field. 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. You may know that there is a difference between a moving charge and a stationary charge. 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). Find the period, pitch, and radius of the helical path of the electron. Thus, the charged particle continues to move along the field direction with a uniform motion (a motion in which speed and velocity is constant). Motion of a Charged Particle in a Uniform Magnetic Field You may know that there is a difference between a moving charge and a stationary charge. Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail, Motion of a charged particle in a uniform magnetic field. This is the main factor that creates a spiral or helical path. 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 $\vec B$.