A 5 mg particle carrying a charge of $$5\pi \times 10^{-6}$$ C is moving with velocity of $$(3\hat{i} + 2\hat{k}) \times 10^{-2}$$ m/s in a region having magnetic field $$\vec{B} = 0.1 \hat{k}$$ Wb/m$$^2$$. It moves a distance of $$a$$ meter along $$\hat{k}$$ when it completes 5 revolutions. The value of $$a$$ is __________.

Velocity has components:

parallel to B (along k):
v∥ = 2 × 10⁻² m/s

perpendicular to B (in i direction):
v⊥ = 3 × 10⁻² m/s

motion is helical

time period:

$$T=\frac{2\pi m}{qB}$$

given:

m = 5 mg = 5 × 10⁻⁶ kg
q = 5π × 10⁻⁶ C
B = 0.1

$$T=\frac{2\pi\times5\times10^{-6}}{5\pi\times10^{-6}\times0.1}=\frac{2}{0.1}=20\text{ s}$$

time for 5 revolutions:

$$t=5T=100\text{ s}$$

distance along k:

$$a=v∥\times t=2\times10^{-2}\times100=2$$