Photons of energy 7eV fall on the surface of a metal X resulting in emission of photoelectrons having maximum kinetic energy `E_(1) = 1eV`. Y is another metal on the surface of which photons of energy 6eV are incident and result in emission of photoelectrons of maximum kinetic energy `E_(2)= 2eV`. The ratio of work functions of metals X and Y, `(phi_(x))/(phi_(y))` is ____________

Explanation

To solve the problem, we will use the photoelectric effect equation, which relates the energy of the incident photons, the work function of the metal, and the maximum kinetic energy of the emitted photoelectrons. The equation is given by:

\[ E_i = \phi + KE_{max} \]

Where:
- \( E_i \) is the energy of the incident photon,
- \( \phi \) is the work function of the metal,
- \( KE_{max} \) is the maximum kinetic energy of the emitted photoelectrons.

### Step 1: Analyze the first metal (Metal X)

For metal X:
- The energy of the incident photon \( E_{iX} = 7 \, eV \)
- The maximum kinetic energy of the emitted electrons \( KE_{maxX} = 1 \, eV \)

Using the photoelectric equation:

\[ E_{iX} = \phi_X + KE_{maxX} \]

Substituting the known values:

\[ 7 \, eV = \phi_X + 1 \, eV \]

Now, we can solve for \( \phi_X \):

\[ \phi_X = 7 \, eV - 1 \, eV = 6 \, eV \]

### Step 2: Analyze the second metal (Metal Y)

For metal Y:
- The energy of the incident photon \( E_{iY} = 6 \, eV \)
- The maximum kinetic energy of the emitted electrons \( KE_{maxY} = 2 \, eV \)

Using the photoelectric equation again:

\[ E_{iY} = \phi_Y + KE_{maxY} \]

Substituting the known values:

\[ 6 \, eV = \phi_Y + 2 \, eV \]

Now, we can solve for \( \phi_Y \):

\[ \phi_Y = 6 \, eV - 2 \, eV = 4 \, eV \]

### Step 3: Calculate the ratio of work functions

Now, we need to find the ratio of the work functions of metals X and Y:

\[ \frac{\phi_X}{\phi_Y} = \frac{6 \, eV}{4 \, eV} \]

Simplifying this gives:

\[ \frac{\phi_X}{\phi_Y} = \frac{6}{4} = \frac{3}{2} \]

### Final Answer

The ratio of the work functions of metals X and Y is:

\[ \frac{\phi_X}{\phi_Y} = \frac{3}{2} \]

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