Light of wavelength, \(\lambda\), falls on a metal having work function \(h c / \lambda_{0} .\) Photoelectric effect will take place only if (a) \(\lambda \geq \lambda_{0}\) (b) \(\lambda \geq 2 \lambda_{0}\) (c) \(\lambda \leq \lambda_{0}\) (d) \(\lambda \leq \lambda_{0} / 2\)

Imagine you're sitting inside a metal, and you are an electron. Sounds a bit strange, right? Well, in the world of physics, this is how we think about concepts like the work function. The work function represents the minimum energy needed to coax you, the electron, out of the metal and into the freedom of space. It's a bit like the amount of convincing a friend might need to go on a spontaneous adventure. In scientific terms, the work function is denoted by the Greek letter phi (\( \text{Φ} \) ) and is measured in electron-volts (eV).

When light, which consists of particles called photons, hits the metal's surface, each photon carries a certain amount of energy. If one of these photons has enough energy to meet or exceed the work function, it can give an electron the 'nod' it needs to leave the metal's surface. This is the starting line of the photoelectric effect. Without enough energy, the photons won't be able to convince any electrons to leave - no matter how many photons hit the surface. This all-or-nothing principle underlies the idea that the work function is a critical threshold in the photoelectric effect.

When discussing the universe's building blocks, Planck's constant is like a VIP pass to the quantum realm. It defines the size of the 'energy packets' (quanta) carried by photons. This might sound a bit abstract, so let's put it in simpler terms. If energy were currency, Planck's constant (\( h \) ) decides the denomination of bills that energy can come in when we're dealing with light and other electromagnetic waves.

Discovered by Max Planck, this fundamental constant has the value of approximately 6.626 x 10^(-34) joule-seconds. That's incredibly small, which tells us that the energy quanta of light are also minuscule. But Planck's constant is monumental in the quantum mechanics field, allowing us to calculate the energy of photons when multiplied by the speed of light (\( c \) ) and divided by the wavelength of the light (\( \text{E} = \frac{h c}{\text{λ}} \) ). It is the pivot point around which the photoelectric effect revolves, linking the energy of light to its wavelength.

The threshold wavelength is the longest wavelength of light that can still pack enough punch—enough energy—to trigger the photoelectric effect. Think of it as a bouncer at the club who decides the minimum height you need to be to get in. In this scenario, the 'club' is the photoelectric emission, and the 'height' is the energy of the photons.

This critical value, denoted by (λ_0) , hinges on the work function of the metal in question. If a photon's wavelength is longer than the threshold wavelength ({λ > λ_0}), it means it doesn't have enough energy the electron needs to overcome the metal's work function. In the case of shorter wavelengths (λ ≤ λ_0), each photon carries more energy—more than enough to help electrons overcome the work function barrier. Thus, understanding the threshold wavelength is key for scientists and engineers when designing devices that utilize the photoelectric effect, such as solar panels or light sensors.