An electron of mass ' m ' with an initial velocity v⃗=v₀ î(v₀>0) enters an electric field E⃗=-Eₒ k̂. If the initial de Broglie wavelength is λ₀, the value after time t would be

Solution

Physics sir · 4 steps

The field pushes sideways and adds a new velocity component, so speed grows and the wavelength shrinks.

1. Force on electron. The electron's charge is -e, so F⃗ = -eE⃗ = -e(-E₀k̂) = eE₀k̂. This is a constant force along +k̂, perpendicular to the initial velocity.
2. Velocity after time t. The acceleration is eE₀/m along k̂, so after time t the velocity is v₀î + (eE₀ t)/mk̂. The î part stays v₀ because no force acts along î.

   v = √(v₀² + e²E₀²t²/m²) = v₀√(1+e²E₀²t²/m²v₀²)

3. Use de Broglie. λ = h/mv, so λ is inversely proportional to speed. With λ₀ = h/mv₀, the new wavelength is λ = λ₀ v₀/v.
4. So. λ = λ₀/√(1+e²E₀²t²/m²v₀²), which is option (c).

The idea to keep. For de Broglie questions, find the speed (or momentum) at the new moment first. A constant sideways force adds a perpendicular component, so add the components in quadrature, then use λ = h/p.

Answer from the NTA final answer key. Solution by PrepWiser.