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A proton and an α-particle, moving with the same velocity enter into uniform magnetic field, acting normal to the plane of their motion. The ratio of the radii of the circular paths described by the proton and α-particle is
  • a)
    1:2
  • b)
    1:4
  • c)
    4:1
  • d)
    1:16
Correct answer is option 'A'. Can you explain this answer?
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A proton and an α-particle, moving with the same velocity enter into u...

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A proton and an α-particle, moving with the same velocity enter into u...
Given:
- A proton and an α-particle enter a uniform magnetic field with the same velocity.
- The magnetic field acts normal to the plane of their motion.

To find:
The ratio of the radii of the circular paths described by the proton and α-particle.

Concept:
The radius of the circular path followed by a charged particle in a magnetic field is given by the equation:

r = mv / (qB)

where,
r = radius of the circular path
m = mass of the particle
v = velocity of the particle
q = charge of the particle
B = magnetic field strength

The equation shows that the radius of the circular path is directly proportional to the mass of the particle and inversely proportional to the charge and velocity of the particle.

Explanation:
Let's consider the proton and the α-particle separately.

Proton:
- The mass of a proton (m_p) is approximately 1.67 x 10^-27 kg.
- The charge of a proton (q_p) is 1.6 x 10^-19 C.
- Let the velocity of the proton be v.

α-particle:
- The mass of an α-particle (m_α) is approximately 6.64 x 10^-27 kg.
- The charge of an α-particle (q_α) is 3.2 x 10^-19 C.
- Let the velocity of the α-particle be v.

Using the equation for radius:
For the proton:
r_p = (m_p * v) / (q_p * B)

For the α-particle:
r_α = (m_α * v) / (q_α * B)

Calculating the ratio:
Taking the ratio of the radii:
r_p / r_α = [(m_p * v) / (q_p * B)] / [(m_α * v) / (q_α * B)]
= (m_p * v * q_α * B) / (m_α * v * q_p * B)
= (m_p / m_α) * (q_α / q_p)

Substituting the values:
r_p / r_α = (1.67 x 10^-27 kg) / (6.64 x 10^-27 kg) * (3.2 x 10^-19 C) / (1.6 x 10^-19 C)
= (1/4) * 2
= 1/2

Therefore, the ratio of the radii of the circular paths described by the proton and α-particle is 1:2.
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Community Answer
A proton and an α-particle, moving with the same velocity enter into u...
R= mv/qB
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