Light of photon energy hv is incident on a metallic plate having work function φ to eject electrons. The most energetic electrons are then allowed to enter in a region of uniform magnetic field as shown in figure. The electrons are projected in x-z plane making an angle θ with the x-axis. Pitch of the helix described by an electron is found to be P. Take mass of electron as m and charge as q. Then

Engineering

Physics

Photoelectric Effect Einsteins Equation

Motion of Charged Particle in Uniform Magnetic Field

Question

Light of photon energy hv is incident on a metallic plate having work function φ to eject electrons. The most energetic electrons are then allowed to enter in a region of uniform magnetic field $\vec{B} = B_{0} \hat{i}$ as shown in figure. The electrons are projected in x-z plane making an angle θ with the x-axis. Pitch of the helix described by an electron is found to be P. Take mass of electron as m and charge as q. Then

$P = \frac{2 πcosθ \sqrt{2 (hν - ϕ) m}}{{qB}_{0}}$

If the electron is crossing origin at t = 0, then the z coordinate of the electron as a function of time t is $\frac{- \sqrt{2 m (h v - ϕ)}}{{qB}_{0}} \times sinθ \times \sin [\frac{{qB}_{0} t}{m}]$

If the electron is crossing origin at t = 0, then the z coordinate of the electron as a function of time t is $\frac{\sqrt{2 m (h v - ϕ)}}{{qB}_{0}} \times sinθ \times \sin [\frac{{qB}_{0} t}{m}]$

$P = \frac{2 πm}{{qB}_{0}} \sqrt{(h v - ϕ)}$

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At any instant of time 't' the location of electron is shown as P. In two dimensional view of electron in y-z plane

$v = \sqrt{\frac{2 KE}{m}} = \sqrt{\frac{2 (h v - φ)}{m}}$

$p = v cos θ \times \frac{2 πm}{{qB}_{0}}$

⇒ pqB₀ = 2πm v cos θ

$\Rightarrow {pqB}_{0} = 2 π cos θ . m . \sqrt{\frac{2 (h v - φ)}{m}}$

$\Rightarrow {pqB}_{0} = 2 π cos θ \sqrt{2 m (h v - φ)}$

x - coordinate, x = v cos θ × t

y = – [R – R cos ωt

z = R sin ωt

$z = \frac{mvsinθ}{{qB}_{0}} \times \sin [\frac{{qB}_{0}}{m} t]$

$z = \frac{\sqrt{2 m (h v - φ)}}{{qB}_{0}} \times sinθ \times \sin [\frac{{qB}_{0} t}{m}]$