Light of different frequencies is incident on a metal plate. The sketch graph below shows the relationship between the maximum kinetic energy, Ekm, of the photoelectrons and the energy of the incident photons.

10.1 The work function of a metal is the minimum energy required to eject an electron from the surface of the metal.

10.2 Step 1: The photoelectric effect equation is given by $E_{photon} = W_0 + E_{k,max}$, where $E_{photon}$ is the energy of the incident photon, $W_0$ is the work function, and $E_{k,max}$ is the maximum kinetic energy of the emitted photoelectron. Step 2: Rearranging the equation to match the graph's axes ($E_{k,max}$ on y-axis, $E_{photon}$ on x-axis): $E_{k,max} = E_{photon} - W_0$ This equation is in the form $y = mx + c$, where $m$ is the gradient. Step 3: Comparing the equation to the graph, the gradient $m$ is the coefficient of $E_{photon}$, which is 1. The numerical value of the gradient is $\boxed{\text{1}}$.

10.3 10.3.1 Step 1: From the graph, when the energy of photons is Y, the maximum kinetic energy of the photoelectrons ($E_{k,max}$) is $2.99 \times 10^{-19}$ J. Step 2: Use the formula for maximum kinetic energy: $E_{k,max} = \frac{1}{2}mv^2$. The mass of an electron ($m_e$) is $9.11 \times 10^{-31}$ kg. $2.99 \times 10^{-19} \text{ J} = \frac{1}{2} (9.11 \times 10^{-31} \text{ kg}) v^2$ Step 3: Solve for $v^2$: $v^2 = \frac{2 \times 2.99 \times 10^{-19} \text{ J}}{9.11 \times 10^{-31} \text{ kg}}$ $v^2 = \frac{5.98 \times 10^{-19}}{9.11 \times 10^{-31}} \text{ m}^2/\text{s}^2$ $v^2 \approx 6.564 \times 10^{11} \text{ m}^2/\text{s}^2$ Step 4: Calculate $v$: $v = \sqrt{6.564 \times 10^{11} \text{ m}^2/\text{s}^2}$ $v \approx \boxed{8.10 \times 10^5 \text{ m/s}}$

10.3.2 Step 1: From the graph, the x-intercept represents the work function ($W_0$), which is the minimum energy required to eject an electron (where $E_{k,max} = 0$). $W_0 = 3.66 \times 10^{-19} \text{ J}$ Step 2: At point Y, the maximum kinetic energy ($E_{k,max}$) is $2.99 \times 10^{-19}$ J. Use the photoelectric equation: $E_{photon} = W_0 + E_{k,max}$. $Y = 3.66 \times 10^{-19} \text{ J} + 2.99 \times 10^{-19} \text{ J}$ Step 3: Calculate the value of Y: $Y = (3.66 + 2.99) \times 10^{-19} \text{ J}$ $Y = \boxed{6.65 \times 10^{-19} \text{ J}}$

10.4 The maximum kinetic energy of the photoelectrons will REMAIN THE SAME.

Reason: The maximum kinetic energy of photoelectrons depends only on the energy (or frequency) of the individual incident photons and the work function of the metal. It does not depend on the intensity (number of photons per second) of the incident light. Increasing the number of photons per second only increases the number of photoelectrons emitted, not their individual maximum kinetic energy.