Answer :


Given -


Kinetic energy of singly-charged potassium ions = 32 keV



Width of the magnetic region = 1.00 cm



Magnetic field’s strength, B = 0.500 T



Distance between the screen and the region = 95.5 cm



Atomic weights of the two isotopes are
m1 =39 and m2 =41


.Mass of a potassium ion = A (1.6 × 10−27) kg


For a singly-charged potassium ion K-39, separation between the points can be calculated as follows –


Mass of K-39 = 39 × 1.6 × 10−27 kg


Charge, q = 1.6 × 10−19 C


Given in the question that the narrow beam of singly-charged potassium ions is injected into a region of magnetic field.


As, given that kinetic energy K.E is



K.E = 32 keV






We know that throughout the motion, the horizontal velocity remains constant.


So, the time taken to cross the magnetic field,





Now, the acceleration in the magnetic field region-








Now, velocity in the vertical direction can be found from Newton’s 2nd law as



Since, initial velocity is 0,



Substituting the values-






Time taken by the ion to reach the screen-





Now, distance moved by the ion vertically in this given time -









Vertical distance travelled by the particle inside magnetic field


can be calculated by using 3rd equations of motion –



where,


v= final velocity


u=initial velocity


a= acceleration acting on the ion


S=distance travelled


Since initial velocity is zero,







Now, time taken t,






Now,for the potassium ion K-41






And, acceleration a is given by-





t = time taken by the ion to exit the magnetic field is given by -




From newton’s law velocity in vertical direction-









Time to reach the screen, t -



.


Now distance moved in vertical direction is-






Now,
Vertical distance travelled by the particle inside magnetic field can be found out by using 3rd equation of motion given by



where,


v= final velocity


u=initial velocity


a= acceleration acting on the ion


S=distance travelled


Since initial velocity is zero,







Net distance travelled by K-41 potassium ion






Net gap between K-39 and K-41 potassium ions is given by-



From (1) and (2)





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