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Comparing Radii: Same Momentum vs Same Kinetic Energy Particles

Physics · Moving Charges And Magnetism · NEET

In a magnetic field the radius is r = mv/qB. Rewrite it two ways: for the SAME MOMENTUM (p = mv), use r = p/qB, so r depends only on charge (r is proportional to 1/q). For the SAME KINETIC ENERGY (K), use r = sqrt(2mK)/qB, so r depends on both mass and charge (r is proportional to sqrt(m)/q). Memory hook: "Momentum kills mass" - when momentum is equal, mass and speed cancel and only charge is left in the bottom.
Same charge? No. Compare radius by the equal quantityB into page (x x x)xxxxxxxprotonr = p/(qB), q = ealphaq = 2e -> r halvedSame momentum:r ~ 1/qSame energy:r ~ sqrt(m)/q
Two particles enter a magnetic field (into the page). With equal momentum the radius depends only on charge (r = p/qB), so the alpha particle (charge 2e) circles at half the proton's radius, giving ratio 2 : 1. With equal kinetic energy you switch to r = sqrt(2mK)/qB and mass returns.

Your doubts, answered

Why does the radius depend only on charge when the momentum is the same?

Start from r = mv/qB. The top part mv is exactly the momentum p. So r = p/qB. If two particles are given the SAME p, then p and B are the same for both, and only q changes. So r is proportional to 1/q. The mass and speed have already combined into p, so you no longer treat them separately. This is why a proton (charge e) and an alpha particle (charge 2e) with equal momentum give radius ratio 2 : 1.

How is the same-energy case different from the same-momentum case?

For kinetic energy K = (1/2)mv^2, you cannot write mv directly. Instead p = mv = sqrt(2mK) (from K = p^2/2m). Put this into r = p/qB to get r = sqrt(2mK)/qB. Now if K, B are the same for both particles, r is proportional to sqrt(m)/q. So mass now DOES matter. Same momentum uses r proportional to 1/q; same energy uses r proportional to sqrt(m)/q.

What if the particles are accelerated through the same voltage V?

Then the work done gives kinetic energy K = qV, which is different for each charge. Put K = qV into r = sqrt(2mK)/qB: r = sqrt(2m(qV))/qB = sqrt(2mV/q)/B. So r is proportional to sqrt(m/q). This is a THIRD case - do not mix it up with the same-energy case. Same voltage means same K only if the charges are equal.

How do I quickly get the ratio without heavy algebra?

Write down which quantity is equal, then use only the matching formula. Same momentum: r is proportional to 1/q, so just compare charges. Same kinetic energy: r is proportional to sqrt(m)/q, compare sqrt(m) over q. Same voltage: r is proportional to sqrt(m/q). Plug in the charge as multiples of e and mass in amu (proton 1, deuteron 2, alpha 4, electron ~1/1836).

⚠️ The NEET trap
✗ Using r proportional to sqrt(m)/q (the same-energy rule) for a same-momentum question, giving proton : alpha = sqrt(1)/1 : sqrt(4)/2 = 1 : 1.
✓ Same momentum means use r = p/qB, so r is proportional to 1/q only. Proton : alpha = (1/e) : (1/2e) = 2 : 1 (NEET 2019 answer).
🧠 First read WHICH quantity is equal - momentum, energy, or voltage - then pick the matching form of r. The formula changes with the given condition.

Real NEET questions

2019

Ionized hydrogen atoms and alpha-particles with same momenta enter perpendicular to a constant magnetic field B. The ratio of the radii of their paths r(H) : r(alpha) will be:

A · 2 : 1 ✓
B · 1 : 2
C · 4 : 1
D · 1 : 4
Solution: Radius in a magnetic field: r = mv/qB = p/qB. Same momentum p means r is proportional to 1/q. Ionized hydrogen H+ has charge q = e; alpha-particle has charge q = 2e. So r(H) : r(alpha) = (1/e) : (1/2e) = 2 : 1. Mass does NOT appear because it is already inside the equal momentum. Correct option: A.

Solved Moving Charges And Magnetism NEET PYQs

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Frequently asked

What are the three key forms of the radius formula?

r = mv/qB (basic), r = p/qB (use for same momentum, r proportional to 1/q), and r = sqrt(2mK)/qB (use for same kinetic energy, r proportional to sqrt(m)/q). Choose the form that matches the equal quantity in the question.

For same momentum, which particle has the bigger radius: proton or alpha?

The proton. With equal momentum r is proportional to 1/q. The proton charge is e and the alpha charge is 2e, so the proton has twice the radius. Ratio proton : alpha = 2 : 1.

For same kinetic energy, compare proton and alpha radius.

Use r proportional to sqrt(m)/q. Proton: sqrt(1)/1 = 1. Alpha: sqrt(4)/2 = 2/2 = 1. So the ratio is 1 : 1 - they have equal radii for the same kinetic energy. This is a classic NEET surprise result.

Why is remembering the condition (momentum, energy, or voltage) more important than the formula?

All three cases start from the same r = mv/qB, but the mass and speed simplify differently depending on what is held equal. Reading the condition first tells you whether mass matters (energy, voltage) or cancels out (momentum), which decides the whole answer.

Related concepts:
Radius of Circular Path: r = mv/qB Derivation and UseCircular Motion of a Charged Particle in a Magnetic FieldCyclotron Frequency and Time Period (Independent of Speed)Helical Path: Charge Moving at an Angle to BMagnetic Force F = qv x B: Magnitude, Direction and Angle Rule
Practise Physics for NEET:
Physics MCQs (free, chapter-wise)Moving Charges And Magnetism PYQs