Question

Doubly-ionised helium ions are projected with a speed of 10 km s⁻¹ in a direction perpendicular to a uniform magnetic field of magnitude 1.0 T. Find (a) the force acting on an ion (b) the radius of the circle in which it circulates and (c) the time taken by an ion to complete the circle.

Answer 1

Given:
Speed of the helium ions, v = 10 km s⁻¹ = 10⁴ m/s
Uniform magnetic field, B = 1.0 T
Charge on the helium ions = 2e
Mass of the helium ions, m = $4\times 1.6\times {10}^{-27}\mathrm{Kg}$
(a) The force acting on an ion,
F = qvBsinθ
= 2 × 1.6 × 10⁻¹⁹ × 10⁴ × 1.0
= 3.2 × 10⁻¹⁵ N

(b) The radius of the circle in which it circulates,
$$\begin{gathered} r=\frac{mv}{qB} \\ =\frac{4\times 1.6\times {10}^{-27}\times {10}^4}{2\times 1.6\times {10}^{-19}\times 1} \\ =\frac{2\times {10}^{-23}}{{10}^{-19}} \\ =2\times {10}^{-4}\mathrm{m} \end{gathered}$$

(c) The time taken by an ion to complete the circle,
$$\begin{gathered} T=\frac{2\mathrm{\pi }r}{v} \\ =\frac{6.28\times 2.1\times {10}^{-4}}{{10}^4} \end{gathered}$$
= 1.31 × 10⁻⁷ s