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.

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

Magnetic Field of a Solenoid:

We can readily find examples where Ampere's Law is more useful, by virtue of being easier to apply, than the Biot-Savart Law. For example, as was the case for the electric field, it was deemed to be very practical to have a device which could "store" magnetic field as a capacitor "stores" an electric field. We know that what the capacitor really stores is electric charge separation, but one of the properties for which a parallel-plate capacitor is useful is its ability to produce a uniform electric field between its plates. The equivalent device for magnetic fields is the solenoid. It's simply a conducting wire wrapped into a cylindrical shape. Even though the actual shape of the wire is helical, for densely packed wrapping we can actually consider the solenoid to be a bunch of closely spaced coils. Near the center of each coil we know that the magnetic field is nearly perpendicular to the plane of the coil.

The magnetic field near the center of a single coil carrying a current I.

The magnetic field near the center of a set of three coils all carrying a current I.

If we place coils on either side of the first and let them all carry the same current I in the same direction, then the magnetic field lines will link together near the center to make a field that is approximately uniform in direction and strength near the center of the set of coils. Placing many coils in close proximity to each other yields the solenoid field.

A solenoid approximates many current-carrying coils spaced much more closely than the coil diameter.

A perfect solenoid has coils so close together that the magnetic field is zero outside the solenoid and perfectly uniform inside.

Approximating the field as constant in direction and magnitude near the center of the solenoid allows us to use Ampere's Law to calculate its magnitude.

Applying Ampere's Law to a solenoid.

Choosing a flat rectangle as the Amperian loop, we see that the contributions to the loop integral can be broken into four parts. The area bounded by the flat rectangle is penetrated by N loops over a length L.

where n = N/L is the linear density of current loops.

Ampere's Law

Ampere's Law:

The magnetic field in space around an electric current is proportional to the electric current, which serves as its source, just as the electric field in space is proportional to the charge that serves as its source. Ampere's Law states that for any closed loop path, the sum of the length elements times the magnetic field in the direction of the length element is equal to the permeability times the electric current enclosed in the loop.

In the electric case, the relation of field to source is quantified in Gauss's Law, which is a very powerful tool for calculating electric fields.

Modulus of Rigidity

Modulus of Rigidity:
In materials science, modulus of rigidity or shear modulus is defined as the ratio of shear stress to the shear strain. The rigidity modulus is one of several quantities for measuring the strength of materials. All of them arise in the generalized Hooke's law:
Young's modulus describes the material's response to linear strain (like pulling on the ends of a wire),
The bulk modulus describes the material's response to uniform pressure.

This modulus is concerned with the deformation of a solid when it experiences a force parallel to one of its surfaces while its opposite face experiences an opposing force (such as friction).

Description:

The apparatus consists of a rod clamped at one end and attached to a wheel at the other. The rod passes through a bearing at the wheel end and known torques may be applied by a string wrapped around the wheel. The twist in the rod is measured with an angular scale.