Graphs are plotted showing the variation of de Broglie wavelength ( ) with energy ( E ) for a photon and an electron. The plots are drawn on a logarithmic scale, with ( ) on the y-axis and (E) on the x-axis. What is the ratio of the slope of the photon's graph to the slope of the electron's graph?

Step-by-step solution

For a photon, the energy is E = hc . Rearranging for wavelength gives = hc E . Taking the natural logarithm on both sides: ( ) = (hc) - (E) . Comparing this with the equation of a straight line y = mx + c , the slope of the photon's graph is m_p = -1 . For an electron, the kinetic energy is E = p^2 2m , so its momentum is p = 2mE . The de Broglie wavelength is = h p = h 2mE . Taking the natural logarithm on both sides: ( ) = ( h 2m ) - 1 2 (E) . The slope of the electron's graph is m_e = - 1 2 . The ratio of the sl