Mdthbs

The 6th-level Evocation spell Sunbeam reads as follows:

A beam of brilliant light flashes out from your hand in a
5-foot-wide, 60-foot-long line. [...] For the duration, a mote of
brilliant radiance shines in your hand.

Bearing in mind that you need to have at least one hand busy with the Material/Somatic component of the spell (as remarked by NautArch in the comments), the text doesn't directly say "you need an extra free hand to cast this spell", but the wording makes it seem so. I need some clarifications.

• Can I cast Sunbeam if both my hands are busy, even if one is holding the Material component?


• Otherwise, do I need to hold the Material component (or focus) in one hand while keeping the other one free?


• How does the casting of Sunbeam interact with the War Caster feat?

One hand alone can do all three: perform the somatic component, perform the material component, and emit the beam of light.

• 
  

  Can I cast Sunbeam if both my hands are busy, even if one is holding the Material component?

  
  
  

  Yes.

  
  


• 
  

  Otherwise, do I need to hold the Material component (or focus) in one hand while keeping the other one free?

  
  
  

  No.

  
  


• 
  

  How does the casting of Sunbeam interact with the War Caster feat?

  
  
  

  Not in a way that frees up a hand.

  
  

The rules for spellcasting state that you can use one free hand to fulfil both the material and somatic components of a spell. You also do not need to hold the material component in one hand and emit the beam from the other; the same hand can do both.

PHB p. 203

A spellcaster must have a hand free to access these components but it can be the same hand that he or she uses to perform somatic components

For sunbeam, the material component is a magnifying glass. There seems to be some confusion about how one can emit the spell from his or her hand while that hand is 'busy' performing the components.

The rules make no mention of requiring your other hand to emit the beam; it can emit from the one casting the spell. To aid with this explanation, simply imagine that the hand holding the magnifying glass has the beam of light travelling from the palm, through the glass as the fingers hold the glass in front of the palm, and expanding to fill the cone AoE. (Pretty cool looking, and satisfies the mechanics in the book.)

The second bullet of War Caster is pertinent here, and sadly, does not help with sunbeam's material component.

PHB p. 170

You can perform the somatic components of spells even when you have weapons or a shield in one or both hands.

Since sunbeam requires the use of a material component, and since war caster makes no mention of material components, this part of the feat will not aid with the spell; a hand will still be required to manipulate the magnifying glass regardless of the other one now being able to perform the somatic component while holding weapons or shields.

The term "a free hand" is not defined in the rules, so it is up to a GM; regardless, the second bullet of the War Caster feat does not help when casting the sunbeam spell

What we do know know from the rules on Spellcasting Components is that:

[...] If a spell requires a somatic component, the caster must have free use of at least one hand to perform these gestures.

A spellcaster must have a hand free to access a spell's material components -- or to hold a spellcasting focus -- but it can be the same hand that he or she uses to perform somatic components.

The sunbeam spell has both material and somatic components, so we can use one hand to perform both of them; the only requirement is that the hand be "free".

The rules never define the word "free" so it is left up to standard English; if your hand is free to do something, then it is "able or permitted to take a specified action." (definition from lexico).
Unless a specific rule says something, any rule regarding the "freeness" of a hand would be up to the GM

However, the sunbeam spell never says it requires a free hand

Note, the sunbeam spell has Verbal, Somatic, and Material components and its description states:

A beam of brilliant light flashes out from your hand in a 5-foot-wide, 60-foot-long line...

You can create a new line of radiance as your action on any turn until the spell ends...

For the duration, a mote of brilliant radiance shines in your hand...

No part of the spell requires that the hand emitting light must be free (the hand performing the material components, however, must be free). When things do require a free hand they typically say so (weapons are "handheld", components require a free hand, as does grappling). Thus it is likely that the spell does not require a free hand.

How does the War Caster feat interact?

Note, the War Caster feat states:

You can perform the somatic components of spells even when you have weapons or a shield in one or both hands.

This doesn't help us much at all, the spell still has material components and in order to use material components (or a focus) we would need a free hand. The feat only helps in two scenarios:

1. You have a spell with Somatic components but not Material components. It allows you to use a hand that is holding something else to perform the components.


2. You have a spell with all three but you have a focus engraved into a shield or some similar armor or weapon which you are already wielding. It allows you to use a hand that is holding something else to perform the Somatic components only.