In a mass spectrometer, an ion of charge, q, and mass, m, moving in a path of radius, r, in a field of flux density, B, has a speed of

Analyze the question: An ion moves in a circle inside a magnetic field. We need its speed v in terms of q, m, B and r.

Concept: A charged particle moving perpendicular to a magnetic field feels a magnetic force that points toward the centre of the circle. This force supplies the centripetal force.

• Magnetic force: F = qvB
• Centripetal force: F = mv²/r

Solution:

Equate the two forces:

qvB = mv²/r

Divide both sides by v:

qB = mv/r

Make v the subject:

v = qBr/m

Check the options:

• A. qBr/m, which matches our result.
• B. mr/(qB) has the wrong arrangement (it is not dimensionally a speed).
• C and D do not contain the right combination of terms.

Unit check: q·B·r/m = C · T · m / kg. Since T = kg/(C·s), this gives (C · kg/(C·s) · m)/kg = m/s ✓.

Tip: For any "charge in a magnetic field" circular-motion question, write qvB = mv²/r immediately. From it you get v = qBr/m and r = mv/(qB). Remember that r is directly proportional to the speed and mass, and inversely proportional to the charge and field strength.