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    Say a steady current flows down a long cylindrical wire of radius a. I need to work out the magnetic field within it. But that's not just it. The current density within the wire is proportional to the distance from the centre of its axis. That is  j= ka . Where a is the distance from the centre of the wire and k is a constant.

    To find the magnetic field I use Ampere's law, but I also need to know the current enclosed by a loop placed within the wire. So imagine I place a circle within the wire and I'm meant to work out the current passing though that circle. We use the rule;

     I_{enclosed} = \int j.dS = \int ka.dS

    The thing is, I'm confused because I'm thinking there's two ways to find out the current enclosed.

    1)

    Surely if I integrate  ka with respect to 'a' between limits zero and a this finds me the current per unit when I go from the centre of the circle to its outer edge (the radius). THEN times that by  \pi a^2 to find the total current enclosed by the circle, I get the answer  I_{enclosed} = \frac {1}{2} k\pi a^4?

    2) But if I don't do that and just integrate the above equation with respect to dS (dS is the surface, i.e. the circle [this has limits 0 and \pi a^2 ) I get;


     I_{enclosed} = \int^{\pi a^2}_{0} ka.dS = [kaS]^{\pi a^2}_{0} = k\pi a^3


    Giving me two different answers, which is the correct one tough? Please met me know if this makes no sense. I'm terrible at explaining.
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    Your first method sound more correct then the second, but thats just me...
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    (Original post by Anti Elephant Mine)
    Say a steady current flows down a long cylindrical wire of radius a. I need to work out the magnetic field within it. But that's not just it. The current density within the wire is proportional to the distance from the centre of its axis. That is  j= ka . Where a is the distance from the centre of the wire and k is a constant.

    To find the magnetic field I use Ampere's law, but I also need to know the current enclosed by a loop placed within the wire. So imagine I place a circle within the wire and I'm meant to work out the current passing though that circle. We use the rule;

     I_{enclosed} = \int j.dS = \int ka.dS

    The thing is, I'm confused because I'm thinking there's two ways to find out the current enclosed.

    1)

    Surely if I integrate  ka with respect to 'a' between limits zero and a this finds me the current per unit when I go from the centre of the circle to its outer edge (the radius). THEN times that by  \pi a^2 to find the total current enclosed by the circle, I get the answer  I_{enclosed} = \frac {1}{2} k\pi a^4?

    2) But if I don't do that and just integrate the above equation with respect to dS (dS is the surface, i.e. the circle [this has limits 0 and \pi a^2 ) I get;


     I_{enclosed} = \int^{\pi a^2}_{0} ka.dS = [kaS]^{\pi a^2}_{0} = k\pi a^3


    Giving me two different answers, which is the correct one tough? Please met me know if this makes no sense. I'm terrible at explaining.
    I'm no expert on integration, but I'd say that you should stick to one variable (a in this case). The point would be to find the differential of S in terms of a and replace it replace with \mathrm{d}S.

    S=\pi a^2 \quad \Longrightarrow \quad \dfrac{\mathrm{d}S}{\mathrm{d}a}  =2\pi a \quad \Longrightarrow \quad \mathrm{d}S=2\pi a \, \mathrm{d}a.

    Now you can substitute this back into your equation.
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    jaroc is correct, you simply need to find an expression for the Surface element in the wire and integrate it from 0 to the radius of the wire. Essentially what you are doing is summing the contributions of current density flux of a bunch of differently sized disks.
 
 
 
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