Question

A fairly typical magnetic field strength for they Milky Way Galaxy is 5.0 × 10−10 T.
If a cosmic ray electron is moving with a speed of 3.0 × 106 m/s, what is the radius of
its spiral path (i.e., the cyclotron radius)? How long does it take such an electron to
make one full circle?

Answer 1

Answer on Question #66513, Physics / Electromagnetism

A typical magnetic field strength for the Milky Way Galaxy is $5.0 \times 10^{-10}$ T. If a cosmic ray electron is moving with a speed of $3.0 \times 10^{6}$ m/s, what is the radius of its spiral path (i.e., the cyclotron radius)? How long does it take such an electron to make one full circle?

Solution:

We use the following equation

$$r = \frac{mv}{qB}$$

Where, $r$ is radius, $m$ is the mass of a charged particle ($9.1 \times 10^{-31}$ kg), $v$ is the velocity perpendicular to the line of the magnetic field, $q$ is particle charge ($1.6 \times 10^{-19}$ C), $B$ is magnetic induction.

$$r = \frac{9.1 \cdot 10^{-31} \, \text{kg} \times 3.0 \cdot 10^{6} \, \text{m/s}}{1.6 \cdot 10^{-19} \, \text{C} \times 5.0 \cdot 10^{-10} \, \text{T}} = 3.4 \cdot 10^{4} \, \text{m}$$

Time of one complete rotation

$$\tau = 2\pi \frac{m}{qB}$$ $$\tau = 2\pi \times \frac{9.1 \cdot 10^{-31} \, \text{kg}}{1.6 \cdot 10^{-19} \, \text{C} \times 5.0 \cdot 10^{-10} \, \text{T}} = 7.14 \cdot 10^{-2} \, \text{s}$$

Answer: $3.4 \cdot 10^{4} \, \text{m}$ and $7.14 \cdot 10^{-2} \, \text{s}$

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