7.0 Motion of charge particle in both electric and magnetic field

  Magnetics

    7.0 Motion of charge particle in both electric and magnetic field

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7.0 Motion of charge particle in both electric and magnetic field

Case 1 : When $\vec E || \vec B$ and particle velocity is perpendicular to both the fields.

Consider a particle of charge $q$ and mass $m$ projected from the origin with velocity $\vec v =v_0 \hat i$ into a region having electric and magnetic field i.e., $\vec E = E_0 \hat j$ and $\vec B = B_0 \hat j.$ The electric field exerts force only along y-direction and therefore particle accelerates in this direction with the acceleration ${a_y} = \frac{{{F_y}}}{m} = \frac{{q{E_0}}}{m}.$ The velocity component in $y-$direction goes on increasing with time. The magnetic field rotates the particle in a circle in $xz-$plane. The resultant path of the particle is a helical path with increasing pitch. The velocity of a particle at any time $t$ would be$\vec V = {V_x}\hat i + {V_y}\hat j + {V_y}\hat k$Here${V_y} = {a_y}t = \frac{{q{E_0}t}}{m}$And $V_x^2 + V_z^2 = V_0^2 = {\text{constant}}$Where ${{\text{V}}_x} = {V_0}\cos \theta  = {V_0}\cos \left( {\frac{{Bqt}}{m}} \right)$

${V_z} = {V_0}\sin \theta  = {V_0}\sin \left( {\frac{{Bqt}}{m}} \right)$$\theta  = \omega t = \frac{{Bq}}{m}t$$\vec V = {V_0}\cos \left[ {\frac{{Bqt}}{m}} \right]\hat i + \left[ {\frac{{q{E_0}t}}{m}} \right]\hat j + {V_0}\sin \left[ {\frac{{Bqt}}{m}} \right]\hat k$Similarly position vector of the particle$\vec r = x\hat i + y\hat j + z\hat k$ $\vec r = \left( {r\sin \theta } \right)\hat i + \left( {\frac{1}{2}ay{t^2}} \right)\hat j + \left( {r - r\cos \theta } \right)\hat k$Or$\vec r = \left[ {\frac{{m{V_0}}}{{Bq}}} \right]\sin \left[ {\frac{{Bqt}}{m}} \right]\hat i + \left[ {\frac{1}{2}\frac{{qE}}{m}{t^2}} \right]\hat j + \left[ {\frac{{m{V_0}}}{{Bq}}} \right]\left\langle {1 - \cos \left[ {\frac{{Bqt}}{m}} \right]} \right\rangle \widehat k$

Case 1 : When $\vec E || \vec B$ and particle velocity is perpendicular to both the fields.

Consider a particle of charge $q$ and mass $m$ projected from the origin with velocity $\vec v =v_0 \hat i$ into a region having electric and magnetic field i.e., $\vec E = E_0 \hat j$ and $\vec B = B_0 \hat j.$ The electric field exerts force only along y-direction and therefore particle accelerates in this direction with the acceleration ${a_y} = \frac{{{F_y}}}{m} = \frac{{q{E_0}}}{m}.$ The velocity component in $y-$direction goes on increasing with time. The magnetic field rotates the particle in a circle in $xz-$plane. The resultant path of the particle is a helical path with increasing pitch. The velocity of a particle at any time $t$ would be$\vec V = {V_x}\hat i + {V_y}\hat j + {V_y}\hat k$Here${V_y} = {a_y}t = \frac{{q{E_0}t}}{m}$And $V_x^2 + V_z^2 = V_0^2 = {\text{constant}}$Where ${{\text{V}}_x} = {V_0}\cos \theta  = {V_0}\cos \left( {\frac{{Bqt}}{m}} \right)$

${V_z} = {V_0}\sin \theta  = {V_0}\sin \left( {\frac{{Bqt}}{m}} \right)$$\theta  = \omega t = \frac{{Bq}}{m}t$$\vec V = {V_0}\cos \left[ {\frac{{Bqt}}{m}} \right]\hat i + \left[ {\frac{{q{E_0}t}}{m}} \right]\hat j + {V_0}\sin \left[ {\frac{{Bqt}}{m}} \right]\hat k$Similarly position vector of the particle$\vec r = x\hat i + y\hat j + z\hat k$ $\vec r = \left( {r\sin \theta } \right)\hat i + \left( {\frac{1}{2}ay{t^2}} \right)\hat j + \left( {r - r\cos \theta } \right)\hat k$Or$\vec r = \left[ {\frac{{m{V_0}}}{{Bq}}} \right]\sin \left[ {\frac{{Bqt}}{m}} \right]\hat i + \left[ {\frac{1}{2}\frac{{qE}}{m}{t^2}} \right]\hat j + \left[ {\frac{{m{V_0}}}{{Bq}}} \right]\left\langle {1 - \cos \left[ {\frac{{Bqt}}{m}} \right]} \right\rangle \widehat k$

Case 2: When $\overrightarrow E \bot \overrightarrow B $ and the particle is released at rest :- Consider a particle of charge $q$ and mass $m,$ is placed at the origin with zero initial velocity into a region of uniform electric and magnetic field. Let field $\vec E$ is acting along $x-$axis and field $\vec B$ is along $y-$axis, that is $\vec E= E_0\hat i$ and $\vec B=B_0\hat j$

The electric force accelerates the particle along $x-$axis and so, the particle starts gaining velocity along $x-$axis. As soon as particle starts moving, the magnetic force starts acting and bends the particle. The resulting motion of particle is in $xz-$ plane.
At any instant its velocity$\vec V = {V_x}\hat i + {V_z}\hat k$The resulatnt force is thus given by$\vec F = q\left( {\vec E + \vec V \times \vec B} \right)$$\vec F = q\left[ {{E_0}\hat i + \left( {{V_x}\hat i + {V_z}\hat k} \right) \times \overrightarrow {{B_0}} \hat j} \right]$ $z = \frac{{{E_0}}}{{{B_0}\omega }}\left( {\omega t - \sin \omega t} \right)$The above equation represents a cycloid which is defined as the path generated by the point on the circumference of a wheel rolling on the ground.