Application of Ampere's Law - Magnetic Field of an Infinite Sheet of Current

Magnetic Field of an Infinite Sheet of Current:

We can consider an instance of a current distribution which does not have cylindrical symmetry, but which is susceptible to Ampere's Law for finding the magnitude of the magnetic field. Consider a sheet of current which is infinitesimally thin but infinitely long and wide. The sheet has a linear current density (i.e. current per unit length).

An infinite sheet of current with current per unit length. We wish to find the magnetic field direction and magnitude at point P a distance h away from the sheet.

Consider a set of wires laid in place of the current sheet. Each wire carries current out of the page. The magnetic field due to wires which are equidistant from the point directly underneath point P can only be in the horizontal direction.

We can replace the sheet with an infinite set of wires arranged so that each wire carries current consistent with an overall current per unit length of as in the case of the current sheet. We see in figure that wires which are equidistant from the line from the set of wires to point P have magnetic fields whose vertical components cancel and whose horizontal components add. Hence, the net magnetic field is in the horizontal direction. This field is uniform since the distribution of wires is infinite, i.e. any position for point P can be considered as the "middle" of an infinitely long current sheet. Note also that the magnetic field direction for the infinite sheet is independent of the distance of P from the sheet and that the same arguments for extending consideration of a finite set of wires to an infinite current sheet state that the magnetic field direction on the other side of the sheet has the field pointing in the opposite direction.

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Application of Ampere's Law - Magnetic Field of a Toroid

Magnetic Field of a Toroid:

We can use Ampere’s Law to examine the case of another device that produces a uniform magnetic field. In this case, we take advantage of our analysis of the solenoid to ask what happens if we bend a solenoid into a circle so that the ends join. The new configuration still has approximately zero field in the regions outside the volume contained by the coils, but the field inside that volume is again approximately uniform if the distance between the coils is small compared to the size of the coils. This device is a toroid.

Ampere's Law applied to a toroid. Note that Amperian loops which lie wholly outside the volume contained by the toroidal coils experience no magnetic field.

We expect the magnetic field to have circular symmetry about the center of the toroid using the same reasoning as for the solenoid. Hence, we expect it to be most useful to use circular paths for evaluating the Ampere integral of magnetic field and path. For any circular path whose area is not intersected by the coils, the magnetic field is zero and the current penetrating the area is, by definition, zero.

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