Two identical charged particles enter a uniform magnetic field with same speed but at angles `30^(@)` and `60^(@)` with field Let a,b and c be the ratio of their time periods, radii and pitches of the helical paths than .

Options

A. `abc=1`
B. `abcgt1`
C. `abclt1`
D. `a=bc`

Answer

To solve the problem, we need to find the ratios of the time periods (A), radii (B), and pitches (C) of the helical paths of two identical charged particles entering a uniform magnetic field at angles of 30° and 60°.

### Step 1: Time Period Ratio (A)
The time period (T) of a charged particle moving in a magnetic field is given by the formula:
\[ T = \frac{2\pi m}{QB} \]
where:
- \( m \) = mass of the particle
- \( Q \) = charge of the particle
- \( B \) = magnetic field strength

Since both particles have the same mass, charge, and magnetic field strength, the time periods for both particles will be equal:
\[ T_1 = T_2 \]
Thus, the ratio of their time periods (A) is:
\[ A = \frac{T_1}{T_2} = 1 \]

### Step 2: Radius Ratio (B)
The radius (r) of the helical path of a charged particle in a magnetic field is given by:
\[ r = \frac{mv \sin \theta}{QB} \]
For the two angles:
- For \( \theta_1 = 30° \)
- For \( \theta_2 = 60° \)

The ratio of the radii (B) is:
\[ B = \frac{r_1}{r_2} = \frac{mv \sin 30°}{QB} \div \frac{mv \sin 60°}{QB} = \frac{\sin 30°}{\sin 60°} \]
Calculating the sine values:
- \( \sin 30° = \frac{1}{2} \)
- \( \sin 60° = \frac{\sqrt{3}}{2} \)

Thus, the ratio becomes:
\[ B = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} \]

### Step 3: Pitch Ratio (C)
The pitch (P) of the helical path is given by:
\[ P = V \cos \theta \cdot T \]
Since the time period (T) is the same for both particles, the ratio of the pitches (C) is:
\[ C = \frac{P_1}{P_2} = \frac{V \cos 30° \cdot T}{V \cos 60° \cdot T} = \frac{\cos 30°}{\cos 60°} \]
Calculating the cosine values:
- \( \cos 30° = \frac{\sqrt{3}}{2} \)
- \( \cos 60° = \frac{1}{2} \)

Thus, the ratio becomes:
\[ C = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \]

### Final Calculation: Product of Ratios
Now, we can find the product of the ratios:
\[ A \cdot B \cdot C = 1 \cdot \frac{1}{\sqrt{3}} \cdot \sqrt{3} = 1 \]

### Conclusion
The final result for the product of the ratios A, B, and C is:
\[ A \cdot B \cdot C = 1 \]