1. Use the threshold condition for photoelectric emission

For the longest wavelength that can just cause emission, the photon energy must be equal to the work function:

$E = \phi = 4.0\,\text{eV}$

Also,

$E = \frac{hc}{\lambda}$

Using the standard relation in electron-volts and nanometers:

$E(\text{eV}) = \frac{1240}{\lambda(\text{nm})}$

2. Substitute the work function

At threshold,

$4.0 = \frac{1240}{\lambda}$

So,

$\lambda = \frac{1240}{4.0} = 310\,\text{nm}$

3. Match with the options

The closest and exact option is:

• A: $310\,\text{nm}$

4. Conclusion

The longest wavelength that can cause photoelectron emission is:

$\boxed{310\,\text{nm}}$