A particle with a mass of 5.0 10⁻¹² kg and a charge of 5.0 C is projected at a speed of 1.0 km s ⁻¹ into a magnetic field of magnitude 5.0 mT . The angle between the velocity and the magnetic field is ⁻¹(0.90) . Determine the diameter of the resulting helical path and its pitch.

Step-by-step solution

Given: Mass m = 5.0 10⁻¹² kg Charge q = 5.0 C = 5.0 10⁻⁶ C Speed v = 1.0 km s ⁻¹ = 1.0 10^3 m s ⁻¹ Magnetic field B = 5.0 mT = 5.0 10⁻³ T Angle = ⁻¹(0.90) = 0.90 From trigonometric identity, = 1 - ^2 = 1 - 0.90^2 = 0.19 0.436 The radius of the helical path is given by: r = mv qB Substituting the values: r = 5.0 10⁻¹² 1.0 10^3 0.90 5.0 10⁻⁶ 5.0 10⁻³ r = 4.5 10⁻⁹ 25.0 10⁻⁹ = 0.18 m = 18 cm The diameter of the helical path is: D = 2r = 2 18 cm = 36 cm The pitch of the helical path is given by: p = v T = v 2 m qB Subst