In a certain apparatus

PHYSICS · Multi-part · AP · Physics 2: Algebra-based · Magnetism and Electromagnetism · Magnetism and Electromagnetism

Problem

In a certain apparatus, uniform electric and magnetic fields are both directed along the positive zz-axis: E⃗=E k^,B⃗=B k^.\vec{E} = E\,\hat{k}, \qquad \vec{B} = B\,\hat{k}. At time t=0t = 0, a particle of charge qq and mass mm is released from the origin with velocity v⃗0=v0 ı^\vec{v}_0 = v_0\,\hat{\imath} (i.e. perpendicular to the common field direction). Neglect gravity. (i) Write the vector differential equation of motion for the particle. (ii) Solve the equation to obtain the components of velocity as functions of time. (iii) Hence derive the parametric equations of the particle’s trajectory and show that it is a helix whose radius is r=mv0∣q∣Br = \frac{m v_0}{|q| B} and whose pitch after one complete revolution is p=2π2qEmω2p = \frac{2\pi^2 qE}{m\omega^2}, where ω=∣q∣Bm\omega = \frac{|q|B}{m}.

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