A beam of protons with a velocity `4 xx 10^(5)` m/sec enters a uniform magnetic field of 0.3 Testa at an angle of `60^(@)` to the magnetic field. Find the radius of the helical path taken by the proton beam. Also find the pitch of the helix, which is the distance travelled by a proton in the beam parallel to the magnetic field during one period of rotation
[Mass of proton `= 1.67 xx 10^(-27)` kg. charge on proton `= 1.6 xx 10^(19)C`]

Answer

To solve the problem step-by-step, we will first calculate the radius of the helical path taken by the proton beam and then find the pitch of the helix.

### Step 1: Identify the Given Values
- Velocity of protons, \( v = 4 \times 10^5 \, \text{m/s} \)
- Magnetic field strength, \( B = 0.3 \, \text{T} \)
- Angle with the magnetic field, \( \theta = 60^\circ \)
- Mass of proton, \( m = 1.67 \times 10^{-27} \, \text{kg} \)
- Charge of proton, \( q = 1.6 \times 10^{-19} \, \text{C} \)

### Step 2: Calculate the Components of Velocity
The velocity of the proton can be resolved into two components:
- Component parallel to the magnetic field:
\[ v_{\parallel} = v \cos \theta = 4 \times 10^5 \cos(60^\circ) = 4 \times 10^5 \times \frac{1}{2} = 2 \times 10^5 \, \text{m/s} \]

- Component perpendicular to the magnetic field:
\[ v_{\perpendicular} = v \sin \theta = 4 \times 10^5 \sin(60^\circ) = 4 \times 10^5 \times \frac{\sqrt{3}}{2} = 2 \sqrt{3} \times 10^5 \, \text{m/s} \]

### Step 3: Calculate the Radius of the Helical Path
The radius \( r \) of the circular motion (helical path) can be calculated using the formula:
\[ r = \frac{mv_{\perpendicular}}{qB} \]
Substituting the values:
\[ r = \frac{(1.67 \times 10^{-27} \, \text{kg}) \cdot (2 \sqrt{3} \times 10^5 \, \text{m/s})}{(1.6 \times 10^{-19} \, \text{C}) \cdot (0.3 \, \text{T})} \]

Calculating the numerator:
\[ = 1.67 \times 10^{-27} \cdot 2 \sqrt{3} \times 10^5 \approx 1.67 \times 10^{-27} \cdot 3.464 \times 10^5 \approx 5.78 \times 10^{-22} \]

Calculating the denominator:
\[ = (1.6 \times 10^{-19}) \cdot (0.3) = 4.8 \times 10^{-20} \]

Now, calculating the radius:
\[ r = \frac{5.78 \times 10^{-22}}{4.8 \times 10^{-20}} \approx 1.204 \times 10^{-2} \, \text{m} \approx 1.2 \times 10^{-2} \, \text{m} \]

### Step 4: Calculate the Pitch of the Helix
The pitch \( P \) of the helix is the distance traveled by the proton in the direction of the magnetic field during one complete revolution. The formula for pitch is:
\[ P = v_{\parallel} \cdot T \]
where \( T \) is the time period of one complete revolution, given by:
\[ T = \frac{2\pi m}{qB} \]

Calculating \( T \):
\[ T = \frac{2\pi (1.67 \times 10^{-27})}{(1.6 \times 10^{-19})(0.3)} \approx \frac{3.34 \times 10^{-27}}{4.8 \times 10^{-20}} \approx 6.95 \times 10^{-8} \, \text{s} \]

Now, substituting to find the pitch:
\[ P = (2 \times 10^5) \cdot (6.95 \times 10^{-8}) \approx 1.39 \times 10^{-2} \, \text{m} \approx 1.4 \times 10^{-2} \, \text{m} \]

### Final Results
- Radius of the helical path, \( r \approx 1.2 \times 10^{-2} \, \text{m} \)
- Pitch of the helix, \( P \approx 1.4 \times 10^{-2} \, \text{m} \)