Helical Path And Pitch

When a charge enters a uniform field at an angle $\theta$, its velocity is resolved into a component $v_{\parallel }=v\cos \theta$ along the field and $v_{\perp }=v\sin \theta$ across it. The magnetic force $q\vec{v}\times \vec{B}$ involves only the perpendicular component, because the parallel component is antiparallel or parallel to $\vec{B}$ and contributes nothing to the cross product. Hence $v_{\parallel }$ remains constant, carrying the particle steadily along the field, while $v_{\perp }$ produces uniform circular motion of radius $r=\frac{m{v}_{\perp }}{qB}$. The superposition of steady forward drift and circular motion is a helix, whose pitch is $p=v_{\parallel }T=\frac{2\pi mv\cos \theta }{qB}$. The option claiming the force changes the parallel component is wrong, since the force is perpendicular to velocity. The option that both components shrink contradicts the fact that magnetic force does no work and cannot change speed. The option swapping the roles of the components is geometrically incorrect. NCERT presents this helical motion explicitly. A check confirms that at $\theta =90^{\circ }$ the pitch vanishes, recovering pure circular motion. Helical motion appears throughout nature and technology: charged particles from the Sun spiral along the Earth's magnetic field lines toward the poles, producing the auroras, and the same principle guides beams in mass spectrometers and confines plasma in magnetic-bottle devices. The pitch shrinks as the entry angle approaches ninety degrees and grows as it approaches zero, smoothly interpolating between a tight circle and an almost straight drift along the field.