backpack for commuting by bike

Carradice Nelson saddlebag - wrap your laptop in a towel!

A bit leftfield, but how about a trailer?

Something like a Bob Yak, although obviously not a Bob Yak as they are bloody expensive, however they turn up on ebay on a regular basis at far more sensible prices.

Personally I use an Osprey Hike Elite 18ltr, no laptop sleeve but roomy enough to carry one if needed, but then again computers are for doing work stuff, so why would I be taking one home with me anyway?

If a bag was the only go to option I'd look at a messenger bag, somethong like a Timbuk2, bulletproof design and materials and ample room for carrying even the biggest laptops going. Doubt you'll get one new for less than £70 but there's always ebay.

Probably not in your price range, unless you can find it on special, but I have an Osprey Radial 34 and find it quite good.

The laptop sleeve is fine, the carrying capacity is big enough, there are enough pockets and the light aluminium frame which makes it stand up on its own is very handy.

I do still get a sweaty back, but the breathability is much better with the Radial than anything else I've tried in the last couple of years - apart from other Osprey packs like a Talon 22, but that didn't have the laptop sleeve.

Expensive, not super light due to the aluminium frame, but works fine for my commute and as breathable for my back as I can find.

I too am obviously going to recommend panniers...however...

I have an Evoc Trail 20 (20litre) mountain bike back pack for big mtb rides. 

I also use it for anything else back pack related, including a lot of flights as hand luggage. It comes with a back protector which is easily removable and then a Mac Book Air fits perfectly in the protector/liquid bladder compartment (also removed).

It has big, wide waist straps with velcro and a clip so it fits real snug to the body. Imagine a back pack that you barely notice when hooning down a trail. 

Oh and it comes with a rain cover.

If you HAD to get a back pack, i'd recommend it. 

I'd recommend my Osprey Syncro 20 backpack. It's got a design that gives decent ventilation for your back, a water reservoir pocket that's big enough for a decent sized (not huge) laptop, a separate front pocket and a top keys/phone/wallet etc pocket. There are stretchy mesh outside pockets for stuffing in gloves, armwarmers etc in mid-ride and a built-in waterproof cover. It's reasonably slim and not too high, so it doesn't get in the way. I've used it daily for years on my 13 mile each way commute, and am very happy with it.

You could try a courier bag. Not so stable, but you get much less sweaty and can see over your shoulder.

I've gone through a few backpacks in my time and none of them are breathable enough to stop a sweaty back. I've tried "airflow" ones that have an open mesh against your back and space for air to go between that and the bag, but if you're cycling hard, just having your clothes pushed against your back will cause sweating.

I can recommend the ProViz ones though they're not especially waterproof.

My current choice is a huge Earth-Pak one: https://www.amazon.co.uk/Earth-Pak-Waterproof-Backpack-Front-Zippered/dp/B07BDQG6MG

I went for the 55L as I needed something that could fit a large boardgame into it (it can fit two easily) and as the top rolls down, it's not too big for ordinary use. I daresay that the 55L might be too big in terms of blocking your line of sight though I don't find it a problem, personally. The big advantage of them is the cheapness and complete waterproofness. No laptop sleeve though.

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It needs to be breathable so I don't look like I rode to work underwater.

You're going to be in the backpack while cycling?

To be fair, when the weather gets warm it's nigh on impossible to not have sweaty parts if you cycle with a rucksack, no matter how good it's circulation. Have you considered panniers?