Two protons are having same kinetic energy. One proton enters a uniform magnetic field at right angles ot it. Second proton enters a uniform electric field in the direction of field. After some time their de Broglie wavelengths are `lambda_1 and lambda_2` then (a) `lambda_1 = lambda_2` (b) `lambda_1 lt lambda_2` (c) `lambda_1 gt lambda_2` (d) some more information is required

Explanation

To solve the problem, we need to analyze the situation of two protons with the same initial kinetic energy entering different fields: one in a magnetic field and the other in an electric field.

### Step-by-Step Solution:

1. **Understanding the Kinetic Energy**:
Both protons have the same kinetic energy, denoted as \( K \). The kinetic energy of a proton can be expressed as:
\[ K = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the proton and \( v \) is its velocity.

2. **Proton in a Magnetic Field**:
The first proton enters a uniform magnetic field at right angles to its velocity. The magnetic field does not do work on the proton, so its kinetic energy remains constant. Therefore, the velocity of the first proton remains the same throughout its motion in the magnetic field.

3. **Proton in an Electric Field**:
The second proton enters a uniform electric field in the direction of the field. The electric field does work on the proton, which results in an increase in its kinetic energy. The work done by the electric field on the proton can be expressed as:
\[ W = qEd \]
where \( q \) is the charge of the proton, \( E \) is the electric field strength, and \( d \) is the distance moved in the field. This work increases the kinetic energy of the second proton, leading to a higher velocity.

4. **Calculating de Broglie Wavelengths**:
The de Broglie wavelength \( \lambda \) of a particle is given by:
\[ \lambda = \frac{h}{p} \]
where \( h \) is Planck's constant and \( p \) is the momentum of the particle. The momentum \( p \) can be expressed in terms of kinetic energy \( K \):
\[ p = \sqrt{2mK} \]
Therefore, the de Broglie wavelength can be rewritten as:
\[ \lambda = \frac{h}{\sqrt{2mK}} \]

5. **Comparing the Wavelengths**:
- For the first proton (in the magnetic field), the kinetic energy \( K_1 \) remains constant, so:
\[ \lambda_1 = \frac{h}{\sqrt{2mK}} \]
- For the second proton (in the electric field), the kinetic energy \( K_2 \) has increased, thus:
\[ \lambda_2 = \frac{h}{\sqrt{2mK'}} \quad \text{(where \( K' > K \))} \]

6. **Conclusion**:
Since \( K' > K \), it follows that \( \sqrt{K'} > \sqrt{K} \), which implies:
\[ \lambda_2 < \lambda_1 \]
Therefore, we conclude that:
\[ \lambda_1 > \lambda_2 \]
The correct answer is (c) \( \lambda_1 > \lambda_2 \).