A proton and an electron are projected into a region of uniform magnetic field in a direction perpendicular to the field. If they have the same initial velocities then

Options

A. They move in circular paths of same radii
B. They experience equal force
C. The trajectory of the electron is more curved
D. Their velocity remain equal to each other through out the motion

Answer

To solve the problem step by step, we need to analyze the motion of a proton and an electron projected into a uniform magnetic field with the same initial velocities.

### Step 1: Understanding the Motion in a Magnetic Field
When a charged particle moves in a magnetic field, it experiences a magnetic force given by the equation:
\[ F = q(v \times B) \]
where:
- \( F \) is the magnetic force,
- \( q \) is the charge of the particle,
- \( v \) is the velocity of the particle,
- \( B \) is the magnetic field.

### Step 2: Analyzing the Forces on Proton and Electron
Both the proton and the electron are subjected to the same magnetic field \( B \) and have the same initial velocity \( v \). The charges of the proton and electron are equal in magnitude but opposite in sign:
- Charge of proton, \( q_p = +e \)
- Charge of electron, \( q_e = -e \)

Since the velocity is perpendicular to the magnetic field, the magnetic force acting on both particles can be expressed as:
\[ F = qvB \]

### Step 3: Calculating the Radius of Circular Motion
The radius \( r \) of the circular path of a charged particle in a magnetic field is given by:
\[ r = \frac{mv}{qB} \]
where:
- \( m \) is the mass of the particle.

For the proton and electron:
- For proton: \( r_p = \frac{m_p v}{q_p B} \)
- For electron: \( r_e = \frac{m_e v}{q_e B} \)

### Step 4: Comparing the Radii
Since the mass of the proton (\( m_p \)) is much greater than the mass of the electron (\( m_e \)), and both have the same charge magnitude and are in the same magnetic field:
- \( r_p > r_e \)

This means the radius of the circular path of the proton is greater than that of the electron.

### Step 5: Analyzing the Force
Since both particles have the same charge magnitude and experience the same magnetic field, they will experience equal magnitudes of force:
\[ F_p = q_p v B \]
\[ F_e = |q_e| v B \]
Thus, the forces are equal in magnitude but opposite in direction due to the opposite charges.

### Step 6: Trajectories of the Particles
The trajectory of the electron will be more curved than that of the proton because the radius of curvature for the electron is smaller. Therefore, the electron will move in a tighter circular path compared to the proton.

### Step 7: Velocity vs. Speed
While both particles have the same initial speed, their velocities (which are vector quantities) will change due to the circular motion. The direction of the velocity vector changes, but the speed (the magnitude of the velocity) remains constant for both particles throughout their motion.

### Conclusion
Based on the analysis:
1. They do not move in circular paths of the same radii (False).
2. They experience equal forces (True).
3. The trajectory of the electron is more curved (True).
4. Their velocities do not remain equal (False), but their speeds remain equal (True).