Question: The figure shows the variation of maximum kinetic energy ($K_{\text{max}}$) of emitted electrons as a function of the frequency $\nu$ of radiation incident on a photosensitive surface. The slope of the curve is:
Step 1 (Formula / Governing Law & Given Data): [0.5 Mark]
According to Einstein’s Photoelectric Equation, the maximum kinetic energy ($K_{\text{max}}$) of emitted photoelectrons is related to the incident frequency ($\nu$) by the following expression:
$$K_{\text{max}} = h\nu - \Phi_0$$Where:
- $h$ is Planck’s constant.
- $\nu$ is the frequency of incident radiation.
- $\Phi_0$ is the work function of the metal surface.
Step 2 (Step-by-Step Mathematical Working): [0.25 Mark]
To find the slope, we compare the photoelectric equation with the standard equation of a straight line, $y = mx + c$:
- Let $y = K_{\text{max}}$
- Let $x = \nu$
- The equation becomes: $K_{\text{max}} = (h)\nu + (-\Phi_0)$
By comparing this to $y = mx + c$, we identify the slope ($m$) as the coefficient of the variable $x$ ($\nu$).
$$m = \frac{d(K_{\text{max}})}{d\nu} = h$$Step 3 (Final Concluded Answer): [0.25 Mark]
The slope of the $K_{\text{max}}$ versus $\nu$ graph is equal to Planck's constant ($h$).
Correct Option: (A) $h$
💡 EL10 Concept Insight: Think of the photoelectric effect like a vending machine: you need a specific amount of energy (the work function) just to get the machine to open, and any extra energy you put in (the frequency) turns directly into the speed (kinetic energy) of the item coming out; because $h$ is the "price tag" per unit of frequency, it determines how steeply the energy increases.