Two parallel wires in the plane of the paper are distance `X_(0)` apart. A point charge is moving with speed `u` between the wires in the same plane at a distance `X_(1)` from one of the wires. When the wires carry current of magnitude `I` in the same direction , the radius of curvature of the path of the point charge is `R_(1)`. In contrast, if the currents I in the two wires have directions opposite to each other, the radius of curvature of the path is `R_2`. if `(X_(0))/(X_(1)) = 3`, the value of `( R _(1))/( R_(2))` is

Explanation

To solve the problem, we need to analyze the forces acting on the point charge due to the magnetic fields generated by the two parallel wires carrying current.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have two parallel wires separated by a distance \( X_0 \).
- A point charge is moving with speed \( u \) at a distance \( X_1 \) from one of the wires.
- The ratio \( \frac{X_0}{X_1} = 3 \).

2. **Magnetic Field Due to a Long Straight Wire**:
- The magnetic field \( B \) at a distance \( r \) from a long straight wire carrying current \( I \) is given by:
\[ B = \frac{\mu_0 I}{2\pi r} \]
- For wire 1 (closer to the point charge), the distance is \( X_1 \).
- For wire 2 (farther from the point charge), the distance is \( X_0 - X_1 \).

3. **Calculating Magnetic Fields**:
- When the currents in both wires are in the same direction, the total magnetic field \( B_1 \) at the location of the point charge is:
\[ B_1 = B_1 + B_2 = \frac{\mu_0 I}{2\pi X_1} + \frac{\mu_0 I}{2\pi (X_0 - X_1)} \]
- When the currents are in opposite directions, the total magnetic field \( B_2 \) is:
\[ B_2 = B_1 - B_2 = \frac{\mu_0 I}{2\pi X_1} - \frac{\mu_0 I}{2\pi (X_0 - X_1)} \]

4. **Finding the Radius of Curvature**:
- The radius of curvature \( R \) of the path of the point charge in a magnetic field is given by:
\[ R = \frac{mv}{qB} \]
- Therefore, for the two cases:
- For currents in the same direction:
\[ R_1 = \frac{mu}{qB_1} \]
- For currents in opposite directions:
\[ R_2 = \frac{mu}{qB_2} \]

5. **Finding the Ratio \( \frac{R_1}{R_2} \)**:
- The ratio of the radii of curvature can be expressed as:
\[ \frac{R_1}{R_2} = \frac{B_2}{B_1} \]

6. **Substituting the Magnetic Fields**:
- Substitute \( B_1 \) and \( B_2 \) into the ratio:
\[ \frac{R_1}{R_2} = \frac{\frac{\mu_0 I}{2\pi X_1} - \frac{\mu_0 I}{2\pi (X_0 - X_1)}}{\frac{\mu_0 I}{2\pi X_1} + \frac{\mu_0 I}{2\pi (X_0 - X_1)}} \]

7. **Simplifying the Expression**:
- Factor out \( \frac{\mu_0 I}{2\pi} \):
\[ \frac{R_1}{R_2} = \frac{\frac{1}{X_1} - \frac{1}{X_0 - X_1}}{\frac{1}{X_1} + \frac{1}{X_0 - X_1}} \]
- Substitute \( X_0 = 3X_1 \):
\[ \frac{R_1}{R_2} = \frac{\frac{1}{X_1} - \frac{1}{2X_1}}{\frac{1}{X_1} + \frac{1}{2X_1}} = \frac{\frac{1}{2X_1}}{\frac{3}{2X_1}} = \frac{1}{3} \]

### Final Result:
Thus, the value of \( \frac{R_1}{R_2} \) is \( 3 \).