SOLUTION Let\'s use our equations for motion in a magnetic field to help us desi
ID: 1772920 • Letter: S
Question
SOLUTION Let's use our equations for motion in a magnetic field to help us design a compact mass spectrometer for a spacecraft. Scientists want to include a compact mass spectrometer on future missions to other planets, such as Mars. Among other things, the instrument will search for signs of life by measuring the relative abundances of the carbon isotopes 12C and 13C. Suppose the instrument is designed as shown in (Figure 1), has a magnetic field of 0.0180 T, and selects carbon ions that have a speed of 3.65x103 m/s and are singly ionized (i.e., have a charge of +e). Each ion emerging from the velocity selector travels through a quarter circle before striking the detector plate oriented at 45° to the direction of the velocity selector). (a) What are the radii RK12 and R13 of the orbits of 12C and 13C ions in this spectrometer? These ions have masses of 1.99x10-26 kg and 2.16x1026 kg , respectively (b) How far apart are the spots these ions produce on the detector plate? SET UP AND SOLVE Part (a): To find the radii of the orbits, we use the cyclotron equation with charge +e = 1.60 × 10-19C (1.99×10-26kg)(3.65×103m/s) = 0 0252m (1.60x10-19)(0.018oT) ma 13u-(2.16×10- 26kr)(3.65x10'm/s) )-0.0274m 13 (1.60x1019C)(0.0180T) Part (b): As the geometry of the figure shows, the ions strike the detector plate at spots separated by a distance of v2 V2(0.0274m-0.0252m) 3.05×10-3m=3.05mm (R13-R12) = = REFLECT Notice that the radius is quite small, just a few centimeters. Even though the complete mass spectrometer will be larger than this, it will still fit easily on a robotic rover. The quarter-circle design also helps reduce size and weight. Part A - Practice Problem: If the isotope 14C is also present in the sample, what will its radius be at the detection plate? The mass of 4C is 2.33x10 26 kg Express your answer in centimeters to three significant figures. Figure 1 of 1 ma = 1.99 × 10-26 kg ma = 2.16 × 10-26 kg 0 Separation of sotopes at- detector slightly exaggerated 12 My Answers Give Up 13 Incorrect; Try Again; 9 attempts remaining Detector ContinueExplanation / Answer
R14 = M14*V/eB
= (2.33 *10-26 *3.65 * 1030)/(1.6 * 10-19*0.018)
=0.0295m
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