Problem — Two charges in circular motion

Problem

Two charges in circular motion. Charge 1 ($q_1 = 3\cdot 10^{-4}\ \text{C}$, $m = 1\ \text{kg}$) is fixed at the centre. Charge 2 ($q_2 = -2\cdot 10^{-4}\ \text{C}$, $m = 1\ \text{kg}$) orbits at distance $R = 5{,}83\ \text{m}$. The attractive Coulomb force provides the centripetal force. Find the orbital speed and period.

Solution

Using $k = 8{,}99\cdot 10^9\ \text{N·m}^2/\text{C}^2$. The attractive Coulomb force (charges of opposite sign) acts as the centripetal force:

$k\,\frac{|q_1 q_2|}{R^2} = \frac{m v^2}{R} \;\Longrightarrow\; v = \sqrt{\frac{k\,|q_1 q_2|}{m\,R}}$

With $|q_1 q_2| = (3\cdot 10^{-4})(2\cdot 10^{-4}) = 6\cdot 10^{-8}\ \text{C}^2$:

$k\,|q_1 q_2| = 8{,}99\cdot 10^9\cdot 6\cdot 10^{-8} = 539{,}4\ \text{N·m}^2$

$v = \sqrt{\frac{539{,}4}{1\cdot 5{,}83}} = \sqrt{92{,}5} \approx 9{,}62\ \text{m/s}$

Period:

$T = \frac{2\pi R}{v} = \frac{2\pi\cdot 5{,}83}{9{,}62} \approx 3{,}81\ \text{s}$

$\ev{v \approx 9{,}6\ \text{m/s},\qquad T \approx 3{,}8\ \text{s}}$

Linked atoms

Topics: Elettrostatica Concepts: Legge di Coulomb · Forza centripeta