Ncert Exercises & Solutions

Verify that the cyclotron frequency ω = eB/m has the correct dimensions of ${[T]}^{- 1}$ ${[T]}^{- 1}$ .

Hint: Use dimensional analysis.

Step 1: For a charged particle moving perpendicular to the magnetic field, the magnetic Lorentz force provides the necessary centripetal force for revolution.

∴ \frac{{mv}^{2}}{R} = qvB

$∴ \frac{{mv}^{2}}{R} = qvB$

On simplifying the terms, we have,

∴ \frac{qB}{m} = \frac{v}{R} = ω

$∴ \frac{qB}{m} = \frac{v}{R} = ω$

Step 2: Finding the dimensional formula of angular frequency,

∴ [ω] = [\frac{qB}{m}] = [\frac{v}{R}] = [T]^{- 1}

$∴ [ω] = [\frac{qB}{m}] = [\frac{v}{R}] = [T]^{- 1}$

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