JEE Main & Advanced Physics Magnetic Effects of Current / करंट का चुंबकीय प्रभाव Trajectory of a Charged Particle in a Magnetic Field

(1) Straight line : If the direction of a \[\overrightarrow{v}\]  is parallel or antiparallel to \[\overrightarrow{B},\] \[\theta =0\] or \[\theta ={{180}^{o}}\] and therefore \[F=0\]. Hence the trajectory of the particle is a straight line.

(2) Circular path : If \[\overrightarrow{v}\] is perpendicular to \[\overrightarrow{B}\] i.e. \[\theta ={{90}^{o}},\] hence particle will experience a maximum magnetic force \[{{F}_{\max }}=qvB\] which act's in a direction perpendicular to the motion of charged particle. Therefore the trajectory of the particle is a circle.

(i) In this case path of charged particle is circular and magnetic force provides the necessary centripetal force i.e. \[qvB=\frac{m{{v}^{2}}}{r}\] \[\Rightarrow \] radius of path  \[r=\frac{mv}{qB}=\frac{p}{qB}=\frac{\sqrt{2mK}}{qB}=\frac{\mathbf{1}}{B}\sqrt{\frac{\mathbf{2}mV}{q}}\]

where \[p=\] momentum of charged particle and \[K=\] kinetic energy of charged particle (gained by charged particle after accelerating through potential difference V) then \[p=mv=\sqrt{2mK}=\sqrt{2mqV}\]

(ii) If T is the time period of the particle then \[T=\frac{2\pi m}{qB}\] (i.e., time period (or frequency) is independent of speed of particle).

(3) Helical path : When the charged particle is moving at an angle to the field (other than \[{{0}^{o}},\,\,{{90}^{o}},\] or \[{{180}^{o}}\]). Particle describes a path called helix.

(i) The radius of this helical path is  \[\mathbf{r=}\frac{\mathbf{m(vsin\theta )}}{\mathbf{qB}}\]

(ii) Time period and frequency do not depend on velocity and so they are given by \[T=\frac{2\pi \,m}{qB}\] and \[\nu =\frac{qB}{2\pi \,m}\]

(iii) The pitch of the helix, (i.e., linear distance travelled in one rotation) will be given by \[p=T(v\cos \theta )=2\pi \frac{m}{qB}(v\cos \theta )\]

(iv) If pitch value is p, then number of pitches obtained in length l given as

Number of pitches\[=\frac{l}{p}\] and time required \[t=\frac{l}{v\cos \theta }\]