We will begin our study of Amerpe's Law, , by developing an understanding of the term "magnetic element."

A magnetic element is essentially the product of magnetic field and distance. Since magnetic field strength is measured in Teslas, the magnetic element is measured in Tesla meters. For example, if we take a line that is one meter long and parallel to a uniform magnetic field that is one Tesla in strength, then its magnetic element would equal 1 Tm. Now if we were to take a square loop in this same uniform magnetic field, the net magnetic element would be 0, because one side would have 1 Tm clockwise, and one side would have 1 Tm counterclockwise, resulting in a net of 0 Tm. In fact, if there is no current running through a loop, there will always be a net of 0 Tm along that loop (hence Amperes Law).

There are many differences between electric fields and magnetic fields.

• While electric fields have a ‘positive’ end and a ‘negative’ end, magnetic fields will loop back onto themselves.
• Since magnetic fields have no beginning or end, they therefore cannot apply a force in the same direction as their flux; otherwise, they could apply infinite energy.
• Third, and finally, the net magnetic flux into and out of a volume is zero ().

Now, it is quite possible to have a line that is not parallel to the magnetic field. In such an event, you will have to take the component of the magnetic field that is parallel to the path. For example, let's calculate the magnetic element of a line that is 1 m long, and it forms a 60° angle with a 1 T-magnetic field. Well, the parallel component of the magnetic field is equal to 1 T * cos(60°), or 0.5 T. Multiplied by 1 m, and you get a total magnetic element of 0.5 Tm. This is, of course, the same path experienced by a line that is 0.5 m and parallel to the magnetic field.

Now, let us try a real life example of Ampere's Law. What would be the strength of the magnetic field a distance R away from a wire with a current I going through it? Due to symmetry, the magnetic field has the same magnitude at all points that are R away from the wire. Thus, if you were to form a circular loop around this wire with a radius R, then you will have a uniform magnetic field throughout a loop that is 2pR long, so . From this, we can deduce that . This is an equation that we already learned from our earlier study of magnetic fields around current carrying wires. But it  proves that Ampere's law works.

Ampere's Law can be used for more than proving equations that you already know. What Ampere's Law is most useful for is calculating the magnetic field strength of a solenoid. A solenoid is a wire that has been looped many times in a helix which creates a magnetic field within it. If the solenoid is longer than it is wide, then the magnetic field within it will be parallel to the axis of the solenoid. Furthermore, the magnetic field outside of the solenoid will be negligible.

So let's form a rectangle by taking a path with length x inside the solenoid, and a parallel path outside of the solenoid, and connected them in the form of a rectangle. One side will be parallel to the field, two sides will be perpendicular to the field, and one side will be parallel, but located within a negligible magnetic field. So the magnetic element is effectively measured along one line, with length x. Now, since this solenoid is helical (looped), its wire will go through our rectangular area many times. So we will say that our rectangle encompasses N loops of the wire. There is a current of I going through this wire, and since the same current flows through every point of this wire, we will say that the total current flowing through our rectangle is NI.

Applying Amperes Law, , with the length of our rectangle now encompassing the entire length of the solenoid and the current being NI, . Since any solenoid you are likely to encounter is likely to be regular, the value of N/ will be constant, and we will refer to it as the value n. We can now conclude that the magnetic field at any point within a regular solenoid is . This is demonstrated in the following figure.

A more generalized use for Ampere's Law can be used to find the inductance of our solenoid. While the strength of the magnetic field may be important for physical uses of a solenoid, the inductance is important for the solenoid as an electrical component. When a solenoid is used in an electrical circuit, the solenoid becomes known as an inductor.

Remember that Faraday's Law, E = - N (dF / dt ), tells us that an opposing emf will be induced in a coil (solenoid in our case) whenever the magnitude of the magnetic flux passing through it changes. We can restate this law as E = - L (dI / dt ) where L is called the inductance of the coil, L = NF / I. Inductance is a measure of the solenoid's opposition to a change in the magnitude of the current passing through its coils. This is analogous to resistance being a measure of a wire's opposition to the passage of current.

The inductance of an inductor is measured by the unit Henry. A Henry is equal to one weber of magnetic flux per ampere of current. This unit of measure is equivalent to the SI definition of 1 H = 1 volt • sec / amp.

Magnetic flux is the total magnetic flux through every loop, which is the product of the magnetic flux through one loop and the total number of loops, N. We will now use Ampere's Law to calculate the inductance of a solenoid.

Taking the solenoid that we were working with earlier, we already know that the strength of the magnetic field parallel to its central axis can be calculated with . Now we just need to find the magnetic flux. Magnetic flux is equal to the area of a plane multiplied by the strength of an incident magnetic field, F = BA. Since the magnetic field within a solenoid is always incident to the cross section, we can use that as our area. Assume for the purpose of this demonstration that the cross sectional area of our solenoid is A, and that the shape doesn’t matter. We can now calculate the inductance of the solenoid as

depending on whether you use the total number of loops, N, or the density of the looping, n. Note that in this formula, L represents inductance and represents the length of the solenoid. Be careful with your notation.

Ampere's Law can be used to find the inductance of many more physical arrangements of inductors, or even to create a circuit that has no self-inductance. For more practice with Ampere's Law, try pages 826-828 in your black book.

Contributed by Lawrence Camarota, August 2004