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 medals

Explanation

To determine which metals will emit photoelectrons when exposed to radiation of wavelength 4100 Å, we will use Einstein's photoelectric equation. The steps are as follows:

### Step 1: Convert the wavelength to energy
We need to calculate the energy of the incident radiation using the formula:
\[ E = \frac{hc}{\lambda} \]
Where:
- \( E \) is the energy of the incident radiation in electron volts (eV),
- \( h \) is Planck's constant (\( 4.135667696 \times 10^{-15} \) eV·s),
- \( c \) is the speed of light (\( 3 \times 10^8 \) m/s),
- \( \lambda \) is the wavelength in meters.

First, convert the wavelength from angstroms to meters:
\[ \lambda = 4100 \, \text{Å} = 4100 \times 10^{-10} \, \text{m} = 4.1 \times 10^{-7} \, \text{m} \]

Now, substituting the values into the energy formula:
\[ E = \frac{(4.135667696 \times 10^{-15} \, \text{eV·s})(3 \times 10^8 \, \text{m/s})}{4.1 \times 10^{-7} \, \text{m}} \]

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

### Step 2: Compare the energy with the work functions
Now we compare the calculated energy with the work functions of metals A, B, and C:
- Work function of metal A = 1.92 eV
- Work function of metal B = 2.0 eV
- Work function of metal C = 5.0 eV

### Step 3: Determine which metals will emit photoelectrons
According to the photoelectric effect, photoelectrons will be emitted if the incident energy is greater than or equal to the work function of the metal.

- For metal A: \( 3.03 \, \text{eV} > 1.92 \, \text{eV} \) (Emission occurs)
- For metal B: \( 3.03 \, \text{eV} > 2.0 \, \text{eV} \) (Emission occurs)
- For metal C: \( 3.03 \, \text{eV} < 5.0 \, \text{eV} \) (No emission)

### Conclusion
Metals A and B will emit photoelectrons when exposed to radiation of wavelength 4100 Å, while metal C will not.

### Final Answer
The metals which will emit photoelectrons are A and B.

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