Benedict Cumberbatch is at a point in his career where he needn't say yes to any project if he doesn't fancy it. The star of the biggest drama on British TV, Sherlock, he's proved himself on stage in Danny Boyle's innovative, award-sweeping reinvention of Frankenstein, and will become a proper Hollywood star this summer thanks to a major role in feverishly anticipated nerdgasm Star Trek into Darkness – a casting that director JJ Abrams says was "a formality" after one viewing of a Sherlock DVD.

At this rate, by 2014 Cumberbatch will simultaneously be playing Batman, Superman and James Bond – but even if that happens, on present form you can bet he'll still be a regular presence on good old BBC radio.

Christmas Day on Radio 4, just before the Queen's speech: there he was as the young Rumpole of the Bailey. Then on Wednesday, the Radio 4 sitcom Cabin Pressure came back for series 4 with Roger Allam, Stephanie Cole and, yes, Benedict Cumberbatch all returning as the staff of a tiny airline.

Now he's on the wireless yet again, playing German nuclear scientist Werner Heisenberg in a new version of Michael Frayn's modern classic play Copenhagen (Sunday 8.30pm Radio 3). The three-hander has a cast from the velvet-lined box inside the top drawer: Simon Russell Beale is Heisenberg's Danish former mentor Niels Bohr, while Greta Scacchi takes the ultimately crucial role of Bohr's wife Margrethe. Exalted company, but Cumberbatch is the big name.

The play premiered in 1998 and is a famously knotty beast, concerned with the details of atomic physics and the insoluble question of whether, when Heisenberg visited Bohr in 1941, he was trying to glean info that might help the Nazis get the bomb, or warning Bohr that Hitler wanted it. The three protagonists discuss this meeting after their deaths.

"I never saw a production of it," says Cumberbatch when RT visits during a break in recording at Broadcasting House in London. "So I'm probably going to piss a lot of people off who want to hear it the way they last heard it. There's no way I can impersonate that.

"These are such extraordinary people with so much on their shoulders. So much of what they did affected so many people. It's a ripe topic for drama and he's just a master, Crazy Phrasey Frayn. He's brilliant."

"Greta and I both saw it at the National," recalls Simon Russell Beale. "It was a brain-stretching evening at the theatre. It's the most extraordinary piece of multi-layered writing. It's like a piece of origami."

"In the theatre," says Scacchi, "it was a very stark set. No furniture or dressing that I can remember. It was an abstract space. You tried to follow the words. To lie on your sofa, shut your eyes and listen to it, just the words, is probably one of the best ways to receive this story."

So Copenhagen isn't the sort of radio play you can half-listen to while pottering about. "Put those tax returns down and listen!" laughs Cumberbatch. It's the latest in a series of roles where he's played someone very, very clever: Stephen Hawking, Christopher Tietjens and of course Sherlock Holmes all had brains to spare.

"You can't betray the intelligence of the characters for the sake of simplfying the story; at the same time, you can't completely leave them in the dust," explains Cumberbatch – who, rather thrillingly, speaks in real life with Sherlockian pace and precision. "I've struggled with science in things I've played before, and it's important to understand what's in front of you, given the speed at which [Bohr and Heisenberg] deliver it, because they are that smart." So how can the audience keep up? "Frayn relies on rhythms and repetitions and patterns within the structure of the language."

The hard science is there, which means plenty of tricky lines and a lot of names of German and Danish scientists, and their principles, to learn. "I'm really bad at pronouncing names, I have to have them drummed in. I've got better at remembering people's names. It's great to see you again, Bill. Terrible joke!"

Chewy as the play might be, when Radio Times drops in, the mood on the fifth floor of the old building at the top of Regent Street is light: Cumberbatch says the trio have "laughed a lot" as they've struggled to make sense of Frayn's layers of meaning.

"You hear stories of people doing the gravest of subjects, laughing," he says. "Gallows humour. Not that this is quite gallows humour. But these are two very wonderful people, it's fun to have a giggle with them." Later on, when Beale mentions that "for a large portion of the play, we're actually dead. And then talking," Cumberbatch says: "So we can dooooo thooooose type of voices..."

Much cameraderie seems to revolve around the simple problem of not rustling one's script when the green light is on. "I'm hopeless at the page-turning," says Beale. "I had a sleepless night, the night before we started, I'm not joking, about the f***ing pages. The speeches here are 20 pages long. So I knew that I'd have to... turn over."

Cumberbatch grabs a page of script from the table and leans forward. "This is one of the secrets," he murmurs, conspiratorially. "If you do that" – he scrunches the paper slightly, into a shell-like shape - "the molecules tighten so it becomes something hard rather than moveable. It's more solid. It doesn't make a noise."

It's an atomic approach to crunch-free radio drama? "It certainly is. It's science!"

Back to the serious business at hand: Copenhagen is really about mankind's constant, hopeless quest for understanding, of the universe and of each other. Another person's motivations and feelings can't ever be nailed down.

"The fundamental bass note of it is human puzzlement at the world," Beale agrees. "The blackness of not knowing. Inside the human brain, inside the human soul, is the dark."

Cumberbatch takes up the theme. "The way atomic physics works is a metaphor for these three people trying to understand their motivations looking through the mists of time," he says. "That's what Michael's done so beautifully, because it's not a patronising way to represent the arguments: it is what science is. Science has come from the human, not the other way round. Our idea and understanding of it is through our sensory filter. Whether that be a process of mental theory or observable, experimental, tenable science, it's a beautiful metaphor for these three people trying to understand each other - and then you've got war splitting everything apart."

Perhaps it's not too much of a stretch to suggest that, for someone at the level of fame Cumberbatch enjoys/suffers, there's special resonance in the idea of piecing together an image of a person that can never be correct. Cumberbatch's every move is now monitored, but do any of his fans really know him?

"They know you from the trail you leave with your work," he replies, the slightest edge of frustration in his voice. "They assume things about you because of who you play and how you play them, and the other scraps floating around in the ether. People try to sew together a narrative out of scant fact."

Says Beale: "Some of it's harmless. There are funny things. I recently broke my finger on stage, and apparently I struggled on through great pain. I know that's going to reappear: my bravery. Actually I wasn't in any pain at all because that's not what you do when you break your finger: some adrenaline kicks in and it just feels odd. But I love the idea that I struggled on. That's fine."

Not that Cumberbatch would, in any case, ever be so blithe or inelegant as to carp about the attention he gets. "I have been around for ten years. I don't want to complain or explain. It's a thing that will pass. It's part of a predictable pattern."

Does he worry about portraying real people and events? "Not when you're in Michael Frayn's hands, no. Yes, you have a responsibility. It's an examination of various interpretations of what happened and why, rather than definite statements or a political axe to grind. He takes care of that."

A radio studio is sanctuary from the crazy world outside, but what makes Cumberbatch a particularly good fit for the medium is his deep, commanding voice, which incidentally has given him a busy sideline in narrating adverts. Listen the next time you have to sit through a commercial break. Insurance, dog food, digital cameras: Cumberbatch's brushed-suede vowels will be there somewhere.

Radio's focus on the pure power of the human voice is something that genuinely excites him. "I love Radio 3. I'm Radio 3 in the car as well. And Radio 4: I change between the two, I'm not religious about it. Radio's something I go back to if I've been out of the country for any length of time. I still find the magical art [when acting on radio] of what effect you are having, how the space is constructed, what the mic's doing, is a mystery. It's nice to really intensely concentrate on and listen to the word and the sense of the word. Radio's a joy. It's just a joy."

Copenhagen is available on iPlayer until 20 January