Suppose the electric field between the electric plates in th

Suppose the electric field between the electric plates in the mass spectrometer of the figure below is 2.48 104 V/m and the magnetic fields B = B \' = 0.50 T. The source contains carbon isotopes of mass numbers 12, 13, and 14 from a long-dead piece of a tree. (To estimate atomic masses, multiply by 1.66 10-27 kg.)

(a) How far apart are the lines formed by the singly charged ions of each type on the photographic film?

______mm

(b) What if the ions were doubly charged?

______mm

Solution

the electric field between the electric plates in the mass spectrometer E = 2.48 10⁴ V/m

the magnetic fields B = B \' = 0.50 T

Mass of carbon 12 is m = 12 x1.67 x10 ^-27 kg

Mass of carbon 13 is m \' = 13 x1.67 x10 ^-27 kg

Mass of carbon 14 is m \" = 14 x1.67 x10 ^-27 kg

(a). For carbon -12 :

Charge of singly charged ion q = 1.6 x10 ^-19 C

Velocity of the ion in velocity selector v = E / B

                                                        =(2.48 x10 ⁴)/(0.5)

                                                        = 4.96 x10 ⁴ m/s

Charge particle in magnetic field B \' ,

B\' v q = mv ²/ r

   B \' q = mv / r

        r = mv / B \' q

          = (12 x1.67x10 ^-27 )(4.96 x10 ⁴ ) /(0.5)(1.6 x10 ^-19 )

          = 12.4248x10 ^-3 m

So, 2r = 2(12.4248 x10 ^-3 m)

          = 24.8496x10 ^-3 m

For carbon -13 :

Charge of singly charged ion q = 1.6 x10 ^-19 C

Velocity of the ion in velocity selector v = E / B

                                                        =(2.48 x10 ⁴)/(0.5)

                                                        = 4.96 x10 ⁴ m/s

Charge particle in magnetic field B \' ,

B\' v q = m \' v ²/ r \'

   B \' q = m \' v / r \'

        r \' = m \' v / B \' q

          = (13 x1.67x10 ^-27 )(4.96 x10 ⁴ ) /(0.5)(1.6 x10 ^-19 )

          = 13.4602x10 ^-3 m

So, 2r \' = 2(13.4602x10 ^-3 m)

          = 26.9204x10 ^-3 m

For carbon -14 :

Charge of singly charged ion q = 1.6 x10 ^-19 C

Velocity of the ion in velocity selector v = E / B

                                                        =(2.48 x10 ⁴)/(0.5)

                                                        = 4.96 x10 ⁴ m/s

Charge particle in magnetic field B \' ,

B\' v q = m\"v ²/ r \"

   B \' q = m\"v / r\"

      r \" = m\"v / B \' q

          = (14 x1.67x10 ^-27 )(4.96 x10 ⁴ ) /(0.5)(1.6 x10 ^-19 )

          = 14.4956x10 ^-3 m

So, 2r \" = 2(14.4956 x10 ^-3 m)

          = 28.9912x10 ^-3 m

Required separation = (2 r \' - 2r ) , ( 2 r \" - 2r \')

                             = (26.9204 mm -24.8496 mm) , (28.9912 mm - 26.9204 mm)

                             = 2.0708 mm , 2.0708 mm

For doubly charged ion :

For carbon -12 :

Charge of doubly charged ion q \' = 2x1.6 x10 ^-19 C

Velocity of the ion in velocity selector v = E / B

                                                        =(2.48 x10 ⁴)/(0.5)

                                                        = 4.96 x10 ⁴ m/s

Charge particle in magnetic field B \' ,

B\' v q \' = mv ²/ r

   B \' q \' = mv / r

        r = mv / B \' q \'

          = (12 x1.67x10 ^-27 )(4.96 x10 ⁴ ) /(0.5)(2x1.6 x10 ^-19 )

          = 6.2124x10 ^-3 m

So, 2r = 2(6.2124x10 ^-3 m)

          = 12.4248x10 ^-3 m

For carbon -13 :

Charge of singly charged ion q = 2 x1.6 x10 ^-19 C

Velocity of the ion in velocity selector v = E / B

                                                        =(2.48 x10 ⁴)/(0.5)

                                                        = 4.96 x10 ⁴ m/s

Charge particle in magnetic field B \' ,

B\' v q = m \' v ²/ r \'

   B \' q = m \' v / r \'

        r \' = m \' v / B \' q

          = (13 x1.67x10 ^-27 )(4.96 x10 ⁴ ) /(0.5)(2x1.6 x10 ^-19 )

          = 6.7301x10 ^-3 m

So, 2r \' = 2(6.7301x10 ^-3 m)

          = 13.4602x10 ^-3 m

For carbon -14 :

Charge of singly charged ion q = 2x1.6 x10 ^-19 C

Velocity of the ion in velocity selector v = E / B

                                                        =(2.48 x10 ⁴)/(0.5)

                                                        = 4.96 x10 ⁴ m/s

Charge particle in magnetic field B \' ,

B\' v q = m\"v ²/ r \"

   B \' q = m\"v / r\"

      r \" = m\"v / B \' q

          = (14 x1.67x10 ^-27 )(4.96 x10 ⁴ ) /(0.5)(2x1.6 x10 ^-19 )

          = 7.2478x10 ^-3 m

So, 2r \" = 2(7.2478 x10 ^-3 m)

          = 14.4956x10 ^-3 m

Required separation = (2 r \' - 2r ) , ( 2 r \" - 2r \')

                             = (13.4602 mm -12.4248 mm) , (14.4956 mm - 13.4602 mm)

                             = 1.0354 mm , 1.0354 mm