The diagram above represents the graph of electron energy against the frequency of the radiation incident on a metal surface. Interpret the: (a) slope of th...

Answer Details

The straight-line graph is a plot of the maximum kinetic energy \(E_k\) of the emitted electrons (vertical axis) against the frequency \(f\) of the radiation falling on the metal surface (horizontal axis). It obeys Einstein's photoelectric equation:

\[E_k = hf - W_0 = hf - hf_0,\]

where \(h\) is Planck's constant, \(W_0\) is the work function of the metal and \(f_0\) is the threshold frequency. Comparing this with the equation of a straight line \(y = mx + c\) lets each feature of the graph be interpreted directly, as annotated below.

(a) Slope of the graph

The line has the form \(E_k = hf - hf_0\). Matching term by term with \(y = mx + c\), the gradient \(m\) corresponds to \(h\). Therefore

\[\text{slope} = \frac{\Delta E_k}{\Delta f} = h \approx 6.6 \times 10^{-34}\ \text{J s}.\]

The slope of the graph represents Planck's constant \(h\). Because \(h\) is a universal constant, the graph has the same gradient for every metal.

(b) Intercept, OC

Point C is where the line, produced backwards, cuts the energy axis at \(f = 0\). Substituting \(f = 0\) into the equation gives

\[E_k = h(0) - hf_0 = -hf_0 = -W_0.\]

The intercept OC is therefore negative, and its magnitude equals the work function \(W_0 = hf_0\) of the metal, i.e. \(OC = hf_0 = W_0\). This is the minimum energy needed to free an electron from the metal surface.

(c) Intercept, OK

Point K is where the line crosses the frequency axis at \(E_k = 0\). Setting \(E_k = 0\):

\[0 = hf - hf_0 \quad\Rightarrow\quad f = f_0.\]

The intercept OK represents the threshold (cut-off) frequency \(f_0\) of the incident radiation: the minimum frequency below which no electrons are emitted from the surface, however intense the radiation.