The path of a charged particle in a uniform magnetic field depends on the angle `theta` between velocity vector and magnetic field, When `theta is 0^(@) or 180^(@), F_(m) = 0` hence path of a charged particle will be linear.
When `theta = 90^(@)`, the magnetic force is perpendicular to velocity at every instant. Hence path is a circle of radius `r = (mv)/(qB)`.
The time period for circular path will be `T = (2pim)/(qB)`
When `theta` is other than `0^(@), 180^(@) and 90^(@)`, velocity can be resolved into two components, one along `vec(B)` and perpendicular to B.
`v_(|/|)=cos theta`
`v_(^)= v sin theta`
The `v_(_|_)` component gives circular path and `v_(|/|)` givestraingt line path. The resultant path is a helical path. The radius of helical path
`r=(mv sin theta)/(qB)`
ich of helix is defined as `P=v_(|/|)T`
`P=(2 i mv cos theta)`
`p=(2 pi mv cos theta)/(qB)`
Two ions having masses in the ratio 1:1 and charges 1:2 are projected from same point into a uniform magnetic field with speed in the ratio 2:3 perpendicular to field. The ratio of radii of circle along which the two particles move is :

Options

A. `4:3`
B. `2:3`
C. `3:2`
D. `3:4`

Explanation

To solve the problem, we need to find the ratio of the radii of the circular paths of two ions moving in a uniform magnetic field. We will use the formula for the radius of the circular path of a charged particle in a magnetic field, which is given by:

\[ r = \frac{mv}{qB} \]

### Step 1: Identify the given parameters
- The masses of the two ions are in the ratio \( M_1 : M_2 = 1 : 1 \).
- The charges of the two ions are in the ratio \( Q_1 : Q_2 = 1 : 2 \).
- The speeds of the two ions are in the ratio \( V_1 : V_2 = 2 : 3 \).
- The magnetic field \( B \) is uniform, so we can assume \( B_1 = B_2 \).

### Step 2: Write the formula for the radii of the two ions
Using the formula for the radius of the path in a magnetic field, we can express the radii for both ions:
- For Ion 1:
\[ r_1 = \frac{M_1 V_1}{Q_1 B} \]

- For Ion 2:
\[ r_2 = \frac{M_2 V_2}{Q_2 B} \]

### Step 3: Find the ratio of the radii \( r_1 : r_2 \)
To find the ratio \( \frac{r_1}{r_2} \), we substitute the expressions for \( r_1 \) and \( r_2 \):
\[ \frac{r_1}{r_2} = \frac{\frac{M_1 V_1}{Q_1 B}}{\frac{M_2 V_2}{Q_2 B}} = \frac{M_1 V_1 Q_2}{M_2 V_2 Q_1} \]

### Step 4: Substitute the known ratios
Now we substitute the known ratios into the equation:
- \( M_1 : M_2 = 1 : 1 \) implies \( \frac{M_1}{M_2} = 1 \)
- \( Q_1 : Q_2 = 1 : 2 \) implies \( \frac{Q_2}{Q_1} = 2 \)
- \( V_1 : V_2 = 2 : 3 \) implies \( \frac{V_1}{V_2} = \frac{2}{3} \)

Substituting these values into the ratio:
\[ \frac{r_1}{r_2} = \frac{1 \cdot V_1 \cdot 2}{1 \cdot V_2 \cdot 1} = \frac{2 V_1}{V_2} \]

### Step 5: Substitute the ratio of speeds
Now substituting \( \frac{V_1}{V_2} = \frac{2}{3} \):
\[ \frac{r_1}{r_2} = 2 \cdot \frac{2}{3} = \frac{4}{3} \]

### Conclusion
Thus, the ratio of the radii of the circular paths of the two ions is:
\[ \frac{r_1}{r_2} = \frac{4}{3} \]