To determine which of the statements is correct, let's analyze each statement step by step.
### Step 1: Analyze Statement 1
**Statement**: A charged particle enters a region of uniform magnetic field at an angle of 85 degrees to the magnetic line of force. The path of the particle is circular.
**Analysis**:
- When a charged particle enters a magnetic field at an angle, it has two components of velocity: one parallel to the magnetic field and one perpendicular to it.
- The force acting on the particle due to the magnetic field is given by the Lorentz force equation: **F = q(v × B)**.
- The component of velocity that is parallel to the magnetic field does not contribute to the magnetic force (as the angle between them is 0 degrees).
- The perpendicular component of the velocity will cause the particle to experience a magnetic force, resulting in circular motion in the plane perpendicular to the magnetic field.
- Since the particle also has a parallel component of velocity, it will not just move in a circular path but will also move along the direction of the magnetic field, resulting in a helical path.
**Conclusion**: This statement is **incorrect**.
### Step 2: Analyze Statement 2
**Statement**: An electron and a proton are moving with the same kinetic energy along the same direction, then they pass through a uniform magnetic field perpendicular to their direction of motion and describe circular paths.
**Analysis**:
- The kinetic energy (KE) of a particle is given by **KE = (1/2)mv²**. For two particles with the same kinetic energy, we can express their velocities in terms of their masses.
- Since the mass of the proton is much greater than that of the electron, for the same kinetic energy, the electron will have a higher velocity than the proton.
- When they enter the magnetic field, both will experience a magnetic force that is perpendicular to their velocity, causing them to move in circular paths.
- The radius of the circular path is given by **R = mv/(qB)**. Since the electron has a smaller mass and a higher velocity, it will have a different radius than the proton, even though they have the same kinetic energy.
**Conclusion**: This statement is **incorrect**.
### Step 3: Analyze Statement 3
**Statement**: There is no change in internal energy of a particle moving in a magnetic field although magnetic force is acting on it.
**Analysis**:
- The magnetic force acting on a charged particle is always perpendicular to the velocity of the particle.
- Since work done is defined as **W = F · d**, and the angle between the force and displacement is 90 degrees, the work done by the magnetic force is zero.
- Therefore, there is no change in kinetic energy or internal energy of the particle while it moves in the magnetic field.
**Conclusion**: This statement is **correct**.
### Step 4: Analyze Statement 4
**Statement**: Two electrons enter with the same speed but in opposite directions in a uniform transverse magnetic field, then they describe circles of the same radius.
**Analysis**:
- When two electrons enter the magnetic field in opposite directions, they will experience forces in opposite directions due to their negative charge.
- The radius of the circular path is given by **R = mv/(qB)**. Since both electrons have the same mass, charge, and speed, they will indeed describe circles of the same radius.
- However, they will move in opposite directions.
**Conclusion**: This statement is **correct**.
### Final Conclusion
The correct statements are:
- Statement 3: There is no change in internal energy of a particle moving in a magnetic field although magnetic force is acting on it.
- Statement 4: Two electrons enter with the same speed but in opposite directions in a uniform transverse magnetic field, then they describe circles of the same radius.
