Does the magnetic field produced by a current carrying wire, exert a magnetic force on the wire itself? I have to calculate the pressure on a current carrying wire. Since there is a pressure on the wire, there must be a force on it, which is a magnetic force. Does the magnetic field produced by the wire, exert a magnetic force on the wire itself? If this is true, why?
 A: The answer is: no.
The 'magnetic force' on the wire is due (indirectly) to magnetic Lorentz forces acting on the moving electrons in it. It is true that there will be attractive magnetic forces between electrons moving in parallel paths at different points in the wire's cross-section (for example between electrons at opposite ends of a diameter). But these forces are equal and opposite, so there will be no resultant force on these electrons taken together, and no resultant force on the wire.
It's a different story when we apply an external magnetic field with a component at right angles to the wire. The electrons will then experience forces in a direction given by $\textbf F = -e \mathbf v \times \mathbf B$, (or by Fleming's left hand rule). [The moving electrons would be forced out of the wire, were it not for 'bonding forces'  that stop them from leaving. Strictly, it is these 'bonding' forces (or their Newton's Third Law partners) that the rest of the wire experiences, rather than the magnetic Lorentz forces directly. However the magnitude of the force on the wire can be correctly calculated as the vector sum of the Lorentz forces, which is easily shown to be equal in magnitude to $F=BIL\ \sin \theta$ with the usual notation.]
A: If the wire is straight, then no, due to axial symmetry magnetic field is just compressing the wire a little but no net force is present.
However, if the wire isn't straight, then net magnetic force due to wire on itself may be non-zero. For example, consider wire in shape of upside-down letter J, in which current flows from bottom to top.
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The shorter oblique segment will experience magnetic force in north-west direction from the longer vertical segment. At the same time, the longer segment will experience force in the west direction. There forces add up to net non-zero magnetic force on the wire due to itself.
A: 
I have to calculate the pressure on a current carrying wire. Since there is a pressure on the wire

This is not relevant to the main question, but what do you mean by “pressure”? Pressure normally means the ratio of force to area. What area would you be using here?

Does the magnetic field produced by the wire, exert a magnetic force on the wire itself?

No.
Perhaps the concept will be easier to understand in the context of gravity. For example, consider the Earth. Does the gravitational field exerted by the Earth exert a force on the Earth itself? Of course not; it is not possible for an object to exert a force on itself. This applies to all forces, including magnetic forces.
