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Provided it is above the threshold frequency of the metal, when electromagnetic radiation is shone onto a metals surface photo-electrons are emitted. This occurs because 1 photon is absorbed by 1 electron giving it enough energy to be ejected.

We know that the energy of the incident photons are all equal from the equation E = hf. If this is so why does the kinetic energy of the emitted photons vary? Why is there a maximum kinetic energy, is it not the same amount every time?

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There are a couple of reasons for this. First and foremost, the electrons are ejected from the surface of the metal in random directions. When you measure things like the "stopping potential" you're only sensitive to motion in directions that would carry the photo-electron to the anode. Because you're only sensitive to motion in particular directions, you only see the part of the kinetic energy along that particular direction. It is this kinematic messiness that makes measuring the reverse potential needed to stop all current the preferred method of measuring the photoelectric effect - the last electrons stopped will be the ones where the largest possible fraction of the photon's energy went in to propelling the electron toward the anode.

Secondarily, the valence electrons in metals have energies in what are known as the "conduction band", which means the electron's energy can exist in a continuum. That fact, combined with the random messiness inherent in thermodynamics, means that no two electrons will have the same kinetic energy before the photons hit them. Thus, each electron will have a slightly different kinetic energy after it gets ejected from the metal.

Of approximately equal weight is the spread in frequencies for the incident light. See, even if your light is produced by a nice sharp atomic line, like in a low pressure mercury lamp, the mercury atoms in the gas will be undergoing thermal motion, leading to Doppler broadening of the line.

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"We know that the energy of the incident photons are all equal"

True only if the light shone is monochromatic. Look at the plot used to determine the photoelectric effect:

it uses the maximum electron energy for a given frequency, so as to not depend on the kinematic dispersion discussed in the chosen answer.

photoelectric

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Look my friend you can consider incident photon as a bigger ball having more energy(in motion) and the electron of the metal surface be a smaller ball kept at rest in a muddy area. when the both balls collide whole energy from bigger ball is tranferred to the smaller one. The energy of the bigger ball gets divided into two parts; one part of energy will help the smaller ball to overcome resistance of muddy area and the other part of energy will provide him kinetic energy. The energy required to overcome resistance is always constant no matter how much the energy the big ball is providing to it, while the rest of energy will help the smaller ball to attain varying speed. The smaller ball will absorb all the energy of the bigger ball and ultimately its kinetic energy will vary. Similarly the muddy area is nothing but work function of the metal which is holding the electron at its place. When the bigger ball i.e. photon strike electron, it tranfer whole energy into it, so the energy required to beat work function will remain constant and the rest of energy will be utilised in increasing kinetic energy of the photoelectron. ENERGY OF PHOTON VARIES WITH VARYING FREQUENCY OF LIGHT ... Hence kinetic energy varies..

Hope you got it!!!

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