A proton and an `alpha` particle having equal kinetic energy are projected in a uniform transverse electric field as shown in figure

Options

A. Proton trajectory is more curved
B. `alpha` particle trajectory is more curved
C. Both trajectories are equally curved but in opposite direction
D. Both trajectories are equally curved and in same direction

Answer

To solve the problem of a proton and an alpha particle having equal kinetic energy projected in a uniform transverse electric field, we can follow these steps:

### Step 1: Understand the given information
We have a proton and an alpha particle (which consists of 2 protons and 2 neutrons) both having equal kinetic energy and projected into a uniform electric field.

### Step 2: Write the expression for kinetic energy
The kinetic energy (KE) of both particles can be expressed as:
\[ KE = \frac{1}{2} mv^2 \]
Since both particles have equal kinetic energy, we can denote their kinetic energies as:
\[ KE_p = KE_{\alpha} \]
where \( KE_p \) is the kinetic energy of the proton and \( KE_{\alpha} \) is the kinetic energy of the alpha particle.

### Step 3: Relate kinetic energy to velocity
For the proton:
\[ KE_p = \frac{1}{2} m_p v_p^2 \]
For the alpha particle:
\[ KE_{\alpha} = \frac{1}{2} m_{\alpha} v_{\alpha}^2 \]
Setting these equal gives:
\[ \frac{1}{2} m_p v_p^2 = \frac{1}{2} m_{\alpha} v_{\alpha}^2 \]
This simplifies to:
\[ m_p v_p^2 = m_{\alpha} v_{\alpha}^2 \]

### Step 4: Calculate the force acting on each particle
The force acting on a charged particle in an electric field is given by:
\[ F = Q \cdot E \]
For the proton (charge \( Q = e \)):
\[ F_p = eE \]
For the alpha particle (charge \( Q = 2e \)):
\[ F_{\alpha} = 2eE \]

### Step 5: Calculate acceleration for each particle
Using Newton's second law, the acceleration \( a \) can be expressed as:
\[ a = \frac{F}{m} \]
For the proton:
\[ a_p = \frac{F_p}{m_p} = \frac{eE}{m_p} \]
For the alpha particle:
\[ a_{\alpha} = \frac{F_{\alpha}}{m_{\alpha}} = \frac{2eE}{m_{\alpha}} \]
Since the mass of the alpha particle is approximately four times that of a proton (\( m_{\alpha} \approx 4m_p \)):
\[ a_{\alpha} = \frac{2eE}{4m_p} = \frac{eE}{2m_p} \]

### Step 6: Compare the accelerations
Now we can compare the accelerations:
\[ a_p = \frac{eE}{m_p} \]
\[ a_{\alpha} = \frac{eE}{2m_p} \]
This shows that the acceleration of the proton is greater than that of the alpha particle:
\[ a_p > a_{\alpha} \]

### Step 7: Determine the trajectory
Since the proton has a greater acceleration, it will experience a more significant deflection in the electric field compared to the alpha particle. Therefore, the trajectory of the proton will be more curved than that of the alpha particle.

### Conclusion
The proton will have a more curved trajectory than the alpha particle when both are projected into the electric field with equal kinetic energy.

### Final Answer
The correct option is A: The proton's trajectory is more curved.

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