An electron of mass m with an initial velocity \(\overrightarrow v= v_0\hat i\)$\stackrel{}{}$\( ( v_o > 0 ) \) enters in an electric field \(\overrightarrow E = -E_0 \hat i\)\((E_0 = \text{constant}>0)\) at \(t=0\). If \(\lambda_0\)${\mathrm{}}_{}$, is its de-Broglie wavelength initially, then what will be its de-Broglie wavelength at time \(t\)?
1. \(\frac{\lambda_0}{\left(1+ \frac{eE_0}{mv_0}t\right)}\)
2. \(\lambda_0\left(1+ \frac{eE_0}{mv_0}t\right)\)
3. \(\lambda_0 t\)
4. \(\lambda_0\)

When the light of frequency \(2\nu_0\) (where \(\nu_0\) is threshold frequency), is incident on a metal plate, the maximum velocity of electrons emitted is \(v_1.\) When the frequency of the incident radiation is increased to \(5\nu_0,\) the maximum velocity of electrons emitted from the same plate is \(v_2.\) What will be the ratio of \(v_1\) to \(v_2?\)

The photoelectric threshold wavelength of silver is \(3250\times 10^{-10}~\text{m}.\) What will be the velocity of the electron ejected from a silver surface by the ultraviolet light of wavelength \(2536\times 10^{-10}~\text{m}?\) 
(Given \(h= 4.14\times 10^{-15}~\text{eVs}\)$$ and $\mathrm{}$\(c= 3\times 10^{8}~\text{m/s}\)${\mathrm{}}^{}$)
1. \(\approx 0.6\times 10^{6}~\text{m/s}\)
2. \(\approx 61\times 10^{3}~\text{m/s}\)
3. \(\approx 0.3\times 10^{6}~\text{m/s}\)
4. \(\approx 0.3\times 10^{5}~\text{m/s}\)

An electron of mass m and a photon have the same energy E. Find the ratio of de-Broglie wavelength associated with the electron to that associated with the photon. (c is the velocity of light)

^{$1.$ ${\left(\frac{E}{2m}\right)}^{1/2}$
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$2.$ $c{\left(2mE\right)}^{1/2}$
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$3.$ $\frac{1}{c}{\left(\frac{2m}{E}\right)}^{1/2}$
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$4.$ $\frac{1}{c}{\left(\frac{E}{2m}\right)}^{1/2}$}