Two single charged ions moving perpendicularly to a uniform magnetic field (B = 0.40 T) with speeds of 5 000 km/s follow circular paths that differ in diameter by 5.0 cm. What is the difference in the mass of these two ions? A) 2.6 × 10−²⁸ kg B) 6.4 × 10−²⁸ kg C) 3.2 × 10−²⁸ kg D) 5.1 × 10−²⁸ kg E) 1.1 × 10−²⁸ kg

Question

Quizplus · Accepted Answer

The radius of the circular path of a charged particle moving in a magnetic field is given by $r = \frac{mv}{|q|B}$, where $m$ is the mass of the particle, $v$ is its speed, $|q|$ is the absolute value of its charge, and $B$ is the magnitude of the magnetic field. 
Since the two ions have different radii of circular motion, we can set up two equations for their respective radii: 
$r_1 = \frac{mv_1}{|q|B}$ 
$r_2 = \frac{mv_2}{|q|B}$ 
We are told that the difference in their radii is 5.0 cm, so: 
$r_2 - r_1 = \frac{mv_2}{|q|B} - \frac{mv_1}{|q|B} = \frac{m}{|q|B}(v_2 - v_1) = 5.0 	ext{ cm}$ 
We want to find the difference in mass, which means we need to eliminate all other variables. We can use the fact that the ions have the same charge (both are single-charged): 
$\frac{m}{|q|B}(v_2 - v_1) = 5.0 	ext{ cm}$ 
$\frac{m}{|q|} = \frac{5.0 	ext{ cm} 	imes B}{v_2 - v_1}$ 
We can now substitute in the given values and solve for $m$: 
$\frac{m}{1 \cdot 0.40} = \frac{5.0 \cdot 10^{-2} \cdot 0.40}{5.0 \cdot 10^6}$
$m = 3.2 \cdot 10^{-10}$ kg 
Since we are asked for the difference in mass, we can assume that the mass of one of the ions is $m + \Delta m$ and the mass of the other ion is $m - \Delta m$. Plugging these values into the equation above, we get: 
$\frac{m + \Delta m}{|q|} - \frac{m - \Delta m}{|q|} = \frac{2 \Delta m}{|q|} = 5.0 	ext{ cm} \cdot B/(v_2 - v_1)$ 
Substituting in the given values and solving for $\Delta m$: 
$\Delta m = \frac{5.0 \cdot 10^{-2} \cdot 0.40}{2 \cdot 1.6 \cdot 10^{-19} \cdot (5.0 \cdot 10^6)} = 1.6 \cdot 10^{-21}$ kg 
Therefore, the difference in mass is $\boxed{	extbf{(C) }3.2 \cdot 10^{-21} 	ext{ kg}}$

Two Single Charged Ions Moving Perpendicularly to a Uniform Magnetic