![61. In a mass spectrometer, a deuteron with kinetic energy 17 MeV enters a uniform magnetic field of 2.4 T with its velocity perpendicular to the field. The deuteron moves in a circular path in the magnetic field. The radius of its path in the magnetic field (correct to two decimal places) is ____ cm. [mass of deuteron is 3.34 × l0-27 kg, 1MeV = 1.6 × 10-13 and e = 1.6 × 10-19 C]](https://www.letstalkacademy.com/wp-content/uploads/2026/07/51-11.jpg)
61. In a mass spectrometer, a deuteron with kinetic energy 17 MeV enters a uniform magnetic field of 2.4 T with its velocity perpendicular to the field. The deuteron moves in a circular path in the magnetic field. The radius of its path in the magnetic field (correct to two decimal places)
is ____ cm. [mass of deuteron is 3.34 × l0-27 kg, 1MeV = 1.6 × 10-13 and e = 1.6 × 10-19 C]
Radius of Circular Path of a Deuteron in a Magnetic Field – Complete Numerical Solution
Concept Used
When a charged particle enters a magnetic field perpendicular to the field lines, the magnetic force acts as the centripetal force.
Therefore,
Magnetic Force = Centripetal Force
qvB = mv²/r
Rearranging, we obtain the radius of the circular path:
r = mv/(qB)
Since the velocity is not directly given, we first calculate it using the kinetic energy.
Step 1: Convert Kinetic Energy into SI Units
Given kinetic energy,
K = 17 MeV
Since
1 MeV = 1.6 × 10−13 J
Therefore,
K = 17 × 1.6 × 10−13
K = 2.72 × 10−12 J
Step 2: Calculate the Velocity of the Deuteron
Using the kinetic energy formula,
K = ½mv²
Therefore,
v = √(2K/m)
Substituting the given values,
v = √[(2 × 2.72 × 10−12)/(3.34 × 10−27)]
v = √(1.6287 × 1015)
v ≈ 4.04 × 107 m/s
Step 3: Calculate the Radius
Using
r = mv/(qB)
Substituting the values,
r = (3.34 × 10−27 × 4.04 × 107)/(1.6 × 10−19 × 2.4)
Numerator:
3.34 × 10−27 × 4.04 × 107 = 1.349 × 10−19
Denominator:
1.6 × 10−19 × 2.4 = 3.84 × 10−19
Therefore,
r = (1.349 × 10−19)/(3.84 × 10−19)
r ≈ 0.351 m
Step 4: Convert Metres into Centimetres
Since
1 m = 100 cm
Therefore,
r = 0.351 × 100
r ≈ 35.13 cm
Why Does the Particle Move in a Circle?
The magnetic force is always perpendicular to the direction of motion of the charged particle. Because this force continuously changes only the direction of velocity and not its magnitude, the speed remains constant while the particle follows a circular path. This is exactly the condition for uniform circular motion, where the magnetic force provides the required centripetal force.
Working Principle of a Mass Spectrometer
A mass spectrometer separates charged particles according to their mass-to-charge ratio. When ions with different masses enter the same magnetic field, lighter particles move in smaller circles, while heavier particles move in larger circles. By measuring the radius of curvature, the mass of the particle can be determined accurately.
This principle is widely used in chemistry, nuclear physics, medicine, forensic science, environmental analysis, and isotope identification.
Important Formulae
Magnetic Force
F = qvB
Centripetal Force
F = mv²/r
Radius of Circular Motion
r = mv/(qB)
Kinetic Energy
K = ½mv²
Final Answer
The radius of the circular path of the deuteron in the magnetic field is
r = 35.13 cm
