The de Broglie wavelength is a concept in quantum mechanics that describes the wave-like behavior of particles, such as electrons. It is given by the equation:

λ = h / p

where λ is the de Broglie wavelength, h is the Planck's constant, and p is the momentum of the particle.


In this problem, an electron is released from rest in a region of uniform electric field with intensity E. We need to determine the de Broglie wavelength of the electron as a function of time t.


When an electron is placed in an electric field, it experiences a force given by the equation:

F = qE

where F is the force, q is the charge of the electron (e), and E is the electric field intensity.

Since the electron is initially at rest, the force acting on it is equal to the mass of the electron (m) multiplied by its acceleration (a):

F = ma

Therefore, we can write:

qE = ma

Simplifying, we have:

a = qE / m


The velocity of the electron can be determined using the equation of motion:

v = u + at

where v is the final velocity, u is the initial velocity (which is zero in this case), a is the acceleration, and t is the time.

Substituting the value of acceleration from the previous equation, we get:

v = (qE / m) t


The momentum of the electron can be calculated using the equation:

p = mv

Substituting the value of velocity from the previous equation, we have:

p = m(qE / m) t

Simplifying, we get:

p = qEt


Now, we can substitute the value of momentum (p) in the equation for de Broglie wavelength (λ):

λ = h / p

Substituting the value of p, we get:

λ = h / (qEt)

Simplifying further, we have:

λ = h / (eEt)

Therefore, the de Broglie wavelength of the electron as a function of time (t) is:

λ = h / (eEt)

The correct answer is option 3) h / (eEt).