Explanation:

To understand the threshold wavelength for the photoelectric effect from cesium, we need to consider the relationship between the energy of a photon and the work function of the material.

1. Energy of a Photon:
The energy of a photon can be calculated using the equation:

E = hc/λ

where E is the energy of the photon, h is the Planck's constant (6.63 x 10^-34 J.s), c is the speed of light (3 x 10^8 m/s), and λ is the wavelength of the light.

2. Work Function:
The work function is the minimum amount of energy required to remove an electron from the surface of a material. It can be represented as:

φ = hc/λ0

where φ is the work function and λ0 is the threshold wavelength.

3. Threshold Wavelength:
The threshold wavelength is the wavelength of the light below which the photoelectric effect does not occur. It can be determined by rearranging the equation for the work function:

λ0 = hc/φ

4. Calculation:
In this case, we are given that the yellow light incident on the cesium surface has a wavelength of 557 nm. We are also given that the cathode-anode voltage drops below 0.25 V when no photo-electron flow occurs.

To find the threshold wavelength, we can rearrange the equation for the work function:

λ0 = hc/φ

Since the cathode-anode voltage drops below 0.25 V, the maximum kinetic energy of the emitted electrons is given by:

KEmax = eV

where e is the elementary charge (1.6 x 10^-19 C) and V is the voltage.

The energy of a photon can be related to the maximum kinetic energy of an electron using the equation:

E = KEmax + φ

Substituting the values and rearranging the equation, we get:

hc/λ = eV + φ

Since no photo-electron flow occurs when the cathode-anode voltage drops below 0.25 V, the maximum kinetic energy of the emitted electrons is zero. Therefore:

0 = e(0.25) + φ

Simplifying the equation, we find:

φ = -0.25e

Now we can substitute this value into the equation for the threshold wavelength:

λ0 = hc/φ

λ0 = hc/(-0.25e)

Calculating the value, we find:

λ0 ≈ 625 nm

Therefore, the threshold wavelength for the photoelectric effect from cesium is approximately 625 nm.