A beam of light of wavelength \(\lambda\) falls on a metal having work function \(\phi\) placed in a magnetic field \(B\). The most energetic electrons, perpendicular to the field, are bent in circular arcs of radius \(R\). If the experiment is performed for different values of \(\lambda\), then the \(B^2 \, \text{vs} \, \frac{1}{\lambda}\) graph will look like (keeping all other quantities constant).
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In the photoelectric effect experiment with magnetic fields, the radius of the electron’s path is proportional to thesquare of the magnetic field strength. Use the energy conservation principle to relate the magnetic field to the wavelength of light.
Step 1: Energy of the Emitted Electrons The energy of the emitted electrons is related to the wavelength of the incident light by the photoelectric equation: \[E = \frac{hc}{\lambda} - \phi,\] where \(E\) is the kinetic energy of the emitted electrons, \(h\) is Planck’s constant, \(c\) is the speed of light, and \(\phi\) is the work function. Step 2: Relation Between Kinetic Energy and Magnetic Field The kinetic energy \(E\) of the electrons is also related to the magnetic field \(B\) by the radius \(R\) of the circular path they follow: \[E = \frac{eB^2R^2}{2m},\] where \(e\) is the charge of the electron and \(m\) is the mass of the electron. Step 3: Combining the Relations Combining the above relations and solving for \(B^2\), we obtain: \[B^2 \propto \frac{1}{\lambda}.\]
