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How do I make a blurry background with a compact digital camera?

Compact digital cameras generally aren't capable of doing that very well, if at all, because of the size of the sensor. It's too small.

Depth of field normally depends on the lens, so it's more appropriate to say the size of the lens is too small, though the size of the sensor do have a significance too.
 
You want a camera with a larger sensor and preferably a long lens -- especially with low focal ratio. Laws of physics come into play here.The three things that contribute to this are: (1) larger physical sensor sizes. (2) lower focal ratio (f-stop), and (3) longer focal lengths.If you really want this, think about getting a large camera -- such as an entry level DSLR -- as your next camera.

Contrary to popular belief, longer focal lengths doesn't really add much to Bokeh at the same magnification.
 
DoF Calculator: Depth of Field Table
Put the real focal length, not the 35mm equivalent. You camera has a 4.3-68.8mm zoom (and this is the reason why it is imprecisely told that the small sensor size influence DoF: because to have fields of view similar to usual camera, focal length should be very short, and short focals are not good for shallow DoF).
 
You want a camera with a larger sensor and preferably a long lens -- especially with low focal ratio. Laws of physics come into play here.The three things that contribute to this are: (1) larger physical sensor sizes. (2) lower focal ratio (f-stop), and (3) longer focal lengths.If you really want this, think about getting a large camera -- such as an entry level DSLR -- as your next camera.
Contrary to popular belief, longer focal lengths doesn't really add much to Bokeh at the same magnification.
Now you're changing the issue. Longer lenses dont have the same magnification unless you crop. Technically the DoF does not get narrower and the bokeh doesn't get stronger because the lens is "longer". Really it's the true angle of view becoming narrower that does this. That means you can crop in on a short lense to make it seem like a long lens and the magnified view magnifies everything... including blur. If you do this, you're also simulating a smaller sensor -- so the issue is a little more complex. But the simple answer is that if you shoot with a longer lens at the same low focal ratio then a short lens, and you don't crop either, then the shot with the longer lens will have a narrower DoF and more intensified bokeh.
 
You want a camera with a larger sensor and preferably a long lens -- especially with low focal ratio. Laws of physics come into play here.The three things that contribute to this are: (1) larger physical sensor sizes. (2) lower focal ratio (f-stop), and (3) longer focal lengths.If you really want this, think about getting a large camera -- such as an entry level DSLR -- as your next camera.
Contrary to popular belief, longer focal lengths doesn't really add much to Bokeh at the same magnification.
Now you're changing the issue. Longer lenses dont have the same magnification unless you crop. Technically the DoF does not get narrower and the bokeh doesn't get stronger because the lens is "longer". Really it's the true angle of view becoming narrower that does this. That means you can crop in on a short lense to make it seem like a long lens and the magnified view magnifies everything... including blur. If you do this, you're also simulating a smaller sensor -- so the issue is a little more complex. But the simple answer is that if you shoot with a longer lens at the same low focal ratio then a short lens, and you don't crop either, then the shot with the longer lens will have a narrower DoF and more intensified bokeh.

If your subject size on the sensor is constant, DOF will be the same.
 
You want a camera with a larger sensor and preferably a long lens -- especially with low focal ratio. Laws of physics come into play here.The three things that contribute to this are: (1) larger physical sensor sizes. (2) lower focal ratio (f-stop), and (3) longer focal lengths.If you really want this, think about getting a large camera -- such as an entry level DSLR -- as your next camera.
Contrary to popular belief, longer focal lengths doesn't really add much to Bokeh at the same magnification.
Now you're changing the issue. Longer lenses dont have the same magnification unless you crop. Technically the DoF does not get narrower and the bokeh doesn't get stronger because the lens is "longer". Really it's the true angle of view becoming narrower that does this. That means you can crop in on a short lense to make it seem like a long lens and the magnified view magnifies everything... including blur. If you do this, you're also simulating a smaller sensor -- so the issue is a little more complex. But the simple answer is that if you shoot with a longer lens at the same low focal ratio then a short lens, and you don't crop either, then the shot with the longer lens will have a narrower DoF and more intensified bokeh.

Equivalence Read this.
 
It is going to be a little hard. But you can at least try.
- Subject to camera distance: Shorter the better.
- Subject to background distance: Longer the better.
- Aperture: Wider the better (smaller the f number).
- Focal length: Longer the better.
.

Also, if you cannot set aperture, you may try days with less light so the aperture would automaticly open more.
 

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