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I used the formula for the magnetic field of a circular loop of current on its axis and integrated it along the solenoid:

[itex]

\vec{dH}=\frac{I R^2 \frac{N}{L} \hat{z} dz}{[(d-z)^2+R^2]^{\frac 3 2}} \Rightarrow \vec{H}=\frac{N I R^2 \hat{z}}{L} \int_\alpha^\beta \frac{R\csc^2\varphi d\varphi}{(R^2\cot^2\varphi+R^2)^{\frac 3 2}} \Rightarrow \vec B =\frac{\mu_0 N I R^2 \hat{z}}{L}(\cos\beta-\cos\alpha)

[/itex]

I want to know how the magnetic material filling the solenoid affects the field outside it and how can I calculate its effect?Is it right to just calculate H,as I did, and then simply multiply it by [itex] \mu_0 [/itex] to obtain the field outside the solenoid?

Thanks