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

We can use the formula for the radius of circular motion of a charged particle in a magnetic field: 
r = mv / qB 
where r is the radius of circular motion, m is the mass of the particle, v is the velocity, q is the charge, and B is the magnetic field strength. 
Since the two ions have different radii of circular motion, we can set up two equations for their radii: 
r1 = mv1 / qB 
r2 = mv2 / qB 
where the subscripts 1 and 2 denote the two ions. 
We can then take the ratio of these two equations to eliminate the charge and the magnetic field strength: 
r1 / r2 = (m/2) * (v1/v2) 
where we have used the fact that the radii differ by 5 cm or 0.05 m, which means that one radius is 0.025 m larger than the other.

We can rearrange this equation to solve for the mass difference: 
m = 2 * r1 * v2 / (v1 * (r1/r2) - 1) 
Plugging in the given values, we get: 
m = 2 * (0.025 + r2) * (5.0 * 10^6) / (5.0 * 10^6 * (0.4) * (0.025/r2) - 1) 
Simplifying and solving, we get: 
m = 3.2 x 10^-10 kg

Therefore, the difference in mass between the two ions is 3.2 x 10^-10 kg, which corresponds to choice C.

Two Single Charged Ions Moving Perpendicularly to a Uniform Magnetic