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Question
An electron (mass m) with an initial velocity `v = v_0hati (v_0 > 0)` is in an electric field `E = - E_0hati `(E0 = constant > 0). It’s de Broglie wavelength at time t is given by ______.
Options
`λ/((1 + (eE_0)/m t/v_0))`
`λ_0 (1 + (eE_0t)/(mv_0))`
`λ_0`
`λ_0t`
Solution
An electron (mass m) with an initial velocity `v = v_0hati (v_0 > 0)` is in an electric field `E = - E_0hati `(E0 = constant > 0). It’s de Broglie wavelength at time t is given by `underline(λ/((1 + (eE_0)/m t/v_0)))`.
Explanation:
Initial de-Broglie wavelength `λ_0 = h/(mv_0)`
Force on electron = F = qE v F = (– e)(– E0i)
`ma = eE_0i`
`a = (eE_0)/m i`
Velocity of electron after time t is v = v0 + at
`v = v_0i + (eE_0)/m i * t`
`v = [v_0 + (eE_0t)/m]hati`
∴ New de-Broglie wavelength `λ = h/(mv)`
`λ = h/(m[v_0 + (eE_0t)/m]hati) = h/(mv_0 [1 + (eE_0t)/(mv_0)])`
`λ = λ_0/([1 + (eE_0t)/(mv_0)])` ......`[∵ h/(mv_0) = λ_0 "from equation" I]`
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