Magnetic field of a circular loop of wire

  1. 1. The problem statement, all variables and given/known data
    I know how to find the magnetic field at the center of a circular loop of wire carrying current. If the radius of the loop is R, how do you find the magnetic field at a distance a from the center of the loop where a<R?


    2. Relevant equations
    [tex]
    d{\mathbf{B}} = \frac{{\mu _0 }}{{4\pi }}\frac{{Id{\mathbf{s}} \times {\mathbf{\hat r}}}}{{r^2 }}
    [/tex]


    3. The attempt at a solution
    The small current element ds is always tangent to the loop. r varies from R-a to R+a. The angle θ between ds and [tex]\mathbf{\hat r}[/tex] seems to vary from 90° to some maximum angle that depends on a.
    Also, if you define ϕ to be the angle around P from the place you first start to integrate to ds, then [tex]ds\neq rd\phi[/tex].

    This is as far as I can analyze. I have no idea what to do with the angles. I am guessing there is some kind of relationship between r and θ or between r and ϕ or between θ and ϕ that I can't see.
     
  2. jcsd
  3. Cyosis

    Cyosis 1,495
    Homework Helper

    First thing you want to do is draw a picture and realise this is a highly symmetric problem. From symmetry you can determine the direction of [itex]\boldsymbol{B}[/itex], draw this vector. Then draw the x and y components. Due to symmetry can you tell what the magnitude of [itex]B_y[/itex] will be? Can you express [itex]B_x[/itex] in terms of B?

    Why do you think the angle between [itex]d\boldsymbol{s}[/itex] and [itex]\mathbf{\hat r}[/itex] changes? It does not. So [itex]d\boldsymbol{s} \times \mathbf{\hat r}=ds[/itex].

    How does the distance r from the loop to a depend on known variables and does the magnitude of B vary when you rotate over the angle [itex]\phi[/itex]?

    Try to enter all information into the Bio Savart law now.
     
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