Ampere's law, field of straight wire

Ampere's law. The line integral of the magnetic field B around any closed loop equals μ0 times the net current enclosed by the loop:

∮ B·dl = μ0 I_enc

Long straight conductor of radius R carrying current I (uniform current density)

Current density J = I/(πR²). By symmetry the field lines are circles about the axis, with B tangential and constant on a circular loop of radius r. Use circular Amperian loops.

      Amperian loop (radius r)
          .--------.
        /   .----.   \
       |   | wire |   |  -> current I out of page
        \   '----'   /
          '--------'

1. Outside (r > R)

Enclosed current = I.

B (2πr) = μ0 I

B = μ0 I / (2πr)

2. On the surface (r = R)

B = μ0 I / (2πR) (maximum value)

3. Inside (r < R)

Enclosed current = J (πr²) = I r²/R².

B (2πr) = μ0 I r²/R²

B = μ0 I r / (2πR²)

The field inside grows linearly with r; outside it falls as 1/r.

Graph of B versus r

   B
   |        . B max = μ0I/(2πR)
   |      /  `.
   |    /      ` .   B ~ 1/r
   |  /             ` . .
   | /  B ~ r
   |/_____________________ r
   0            R

B is zero at the axis, rises linearly to the maximum at r = R, then decays as 1/r.