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Step-by-Step Solution
Step 1: Understand the Concept of Pitch in a Helical Path
When a charged particle moves in a uniform magnetic field at an angle to the field, it follows a helical trajectory. The pitch of the helix is the distance the particle travels parallel to the magnetic field in one complete revolution.
Step 2: Write Down the Formula for the Pitch
The pitch $P$ of the helical path is given by:
$$
P = \frac{2\pi m}{qB} \, v \cos \theta
$$
where
$m$ is the mass of the proton,
$q$ is the charge of the proton,
$B$ is the magnetic field,
$v$ is the speed of the proton,
$\theta$ is the angle between the velocity and the magnetic field.
Step 3: Substitute the Known Values
Given values:
Mass of proton, $m = 1.67 \times 10^{-27} \, \text{kg}$
Charge of proton, $q = 1.69 \times 10^{-19} \, \text{C}$
Magnetic field, $B = 0.3 \, \text{T}$
Speed of protons, $v = 4 \times 10^5 \, \text{m s}^{-1}$
Angle, $\theta = 60^\circ$
Substitute these into the formula:
$$
P = \frac{2\pi \times 1.67 \times 10^{-27}}{(1.69 \times 10^{-19})(0.3)} \times (4 \times 10^5) \, \cos 60^\circ
$$
Step 4: Evaluate the Expression
Note that $\cos 60^\circ = \frac{1}{2}$. Performing the multiplication:
$$
P =
\frac{2 \times 3.14 \times 1.67 \times 10^{-27} \times 4 \times 10^5 \times \frac{1}{2}}{1.69 \times 10^{-19} \times 0.3}
$$
Simplify step by step to get:
$$
P \approx 0.04 \, \text{m}
$$
Step 5: Convert to Centimeters
Since $1 \, \text{m} = 100 \, \text{cm}$:
$$
P = 0.04 \, \text{m} = 4 \, \text{cm}
$$
Final Answer:
The pitch of the helical path is approximately 4 cm.
