Stopping Potential And Frequency

Combining Einstein's equation with the stopping potential relation, NCERT shows that $e{V}_0=h\nu -\varphi$, which rearranges to $V_0=\frac{h}{e}\nu -\frac{\varphi }{e}$. This is the equation of a straight line when $V_0$ is plotted against frequency $\nu$, with slope $\frac{h}{e}$ and a negative intercept $\frac{\varphi }{e}$ on the vertical axis. Remarkably, the slope $\frac{h}{e}$ is the same for all metals, a striking confirmation of Einstein's theory that Millikan verified experimentally. The option naming the work function is wrong because the work function appears in the intercept, not the slope. The option naming the threshold frequency is wrong because the threshold frequency is the horizontal-axis intercept where $V_0=0$, not the slope. The option giving the product $h\times e$ is wrong because dimensional analysis requires the slope to be $\frac{h}{e}$, a ratio, so that volts result when multiplied by frequency. As stated in NCERT Class 12, Chapter 11, the universal slope of these lines allowed an accurate determination of Planck's constant. This linear graph is one of the most elegant confirmations of quantum theory, because both the universal slope and the metal-specific intercept can be read off a single straight line. The intercept on the frequency axis, where the stopping potential is zero, directly gives the threshold frequency of that particular metal, tying the whole picture together. A dimensional check confirms that $\frac{h}{e}$ carries units of volt-seconds, which multiplied by frequency in per-second yields volts, exactly matching the stopping potential.