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 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:

\[ K_{\text{max}} = E - \phi \]

where:
- \( K_{\text{max}} \) is the maximum kinetic energy of the emitted photoelectrons,
- \( E \) is the energy of the incident photons,
- \( \phi \) is the work function of the metal.

### Step 1: Analyze Metal X
For metal X:
- Energy of incident photons, \( E_X = 7 \, \text{eV} \)
- Maximum kinetic energy of emitted photoelectrons, \( K_{1} = 1 \, \text{eV} \)

Using the photoelectric equation:
\[ K_{1} = E_X - \phi_X \]
Substituting the known values:
\[ 1 \, \text{eV} = 7 \, \text{eV} - \phi_X \]

Rearranging gives:
\[ \phi_X = 7 \, \text{eV} - 1 \, \text{eV} = 6 \, \text{eV} \]

### Step 2: Analyze Metal Y
For metal Y:
- Energy of incident photons, \( E_Y = 6 \, \text{eV} \)
- Maximum kinetic energy of emitted photoelectrons, \( K_{2} = 2 \, \text{eV} \)

Using the photoelectric equation:
\[ K_{2} = E_Y - \phi_Y \]
Substituting the known values:
\[ 2 \, \text{eV} = 6 \, \text{eV} - \phi_Y \]

Rearranging gives:
\[ \phi_Y = 6 \, \text{eV} - 2 \, \text{eV} = 4 \, \text{eV} \]

### Step 3: Find the Ratio of Work Functions
Now we have the work functions:
- \( \phi_X = 6 \, \text{eV} \)
- \( \phi_Y = 4 \, \text{eV} \)

To find the ratio of the work functions:
\[ \frac{\phi_X}{\phi_Y} = \frac{6 \, \text{eV}}{4 \, \text{eV}} = \frac{6}{4} = \frac{3}{2} \]

### 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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