The work function for metals `A , B` and `C` are respectively `1.92 eV , 2.0 eV` and `5 eV`. According to Einstein's equation , the metals which will emit photoelectrons for a radiation of wavelength `4100 Å` are

Options

A. None
B. A only
C. A and B only
D. All the three metals

Explanation

To determine which metals will emit photoelectrons when exposed to radiation of wavelength 4100 Å, we can use Einstein's photoelectric equation. Here's a step-by-step solution:

### Step 1: Convert Wavelength to Nanometers
The given wavelength is in angstroms. To convert angstroms to nanometers, we use the conversion factor:
1 Å = 0.1 nm.

So,
\[ 4100 \, \text{Å} = 4100 \times 0.1 \, \text{nm} = 410 \, \text{nm}. \]

### Step 2: Calculate the Energy of the Incident Radiation
Using the formula for energy of a photon:
\[ E = \frac{hc}{\lambda}, \]
where:
- \(h\) (Planck's constant) = \(4.1357 \times 10^{-15} \, \text{eV s}\),
- \(c\) (speed of light) = \(3 \times 10^8 \, \text{m/s}\),
- \(\lambda\) = \(410 \, \text{nm} = 410 \times 10^{-9} \, \text{m}\).

Substituting the values:
\[ E = \frac{(4.1357 \times 10^{-15} \, \text{eV s})(3 \times 10^8 \, \text{m/s})}{410 \times 10^{-9} \, \text{m}}. \]

Calculating this gives:
\[ E \approx 3.03 \, \text{eV}. \]

### Step 3: Compare the Energy with the Work Functions
Now we compare the calculated energy \(E\) with the work functions of metals A, B, and C.

- Work function of Metal A: \(1.92 \, \text{eV}\)
- Work function of Metal B: \(2.0 \, \text{eV}\)
- Work function of Metal C: \(5.0 \, \text{eV}\)

### Step 4: Determine Which Metals Will Emit Photoelectrons
For a metal to emit photoelectrons, the energy of the incident radiation must be greater than the work function of the metal.

- For Metal A: \(E = 3.03 \, \text{eV} > 1.92 \, \text{eV}\) (will emit)
- For Metal B: \(E = 3.03 \, \text{eV} > 2.0 \, \text{eV}\) (will emit)
- For Metal C: \(E = 3.03 \, \text{eV} < 5.0 \, \text{eV}\) (will not emit)

### Conclusion
The metals that will emit photoelectrons for the radiation of wavelength 4100 Å are Metal A and Metal B.

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