Lesson 17 - Ampere's Law
I.Ampere's Law (Special Case of Biot-Savart Law)
The Biot-Savart Law is very powerful. It will work for any current carrying wire regardless of the shape and it doesn't require you to know anything about the shape of the magnetic field lines. However, the math required to solve the Biot- Savart equation is difficult and often requires numerical techniques.
When we were solving electric field problems, we faced a similar problem and discovered that we could solve many electrostatic problems with far less math using Gauss' Law of Electric Fields as long as the problem exhibited symmetry (i.e. we knew the shape of the field).
In magnetic fields, we have a similar situation in which the magnetic field can be determined using far less difficult math as long as the system exhibits symmetry and we know the basic shape of the magnetic fieldbefore we do the problem.
To heuristically develop Ampere's Law, we consider our results for the infinitely long wire example. The magnetic field strength was given by
After rearranging the equation, we have
The term inside the parenthesis is the circumference for a circle surrounding the wire and passing through the point P where we wanted to calculate the magnetic field as shown below
From calculus, we know that for the circular path we have
Thus, we have
and since B is constant and parallel to the path of integration (i.e. to the displacement vector,)we find that
Where Ip is the total net current (in our case I) that penetrates the surface bounded by our integration curve (in our case a circle).
Ampere's Law is very similar to Gauss's Law in electrostatics is that it can greatly simplify our math work but can only be used on problems that exhibit a great amount of symmetry.
Ampere's Law is useful if the following conditions are met:
1)You know the basic shape of the B-filed before you do the problem
2)An integration curve can be found in which the B-field is either perpendicular to the path or is constant in magnitude and parallel to the integration path.
EXAMPLE 1: Solve for the magnetic field due to the infinitely long current wire using Ampere's Law.
II.Magnetic Dipole Revisited
In order to take advantage of Ampere's Law, we must be able to predict the shape of the magnetic field for our system (i.e. know its symmetry). The magetic dipole (current loop) is the basic building block for several important engineering structures including the solenoid (coil or inductor) and the toroid.
In the diagram below, we have a current loop along with a cross section view (what you would see if you chopped it in half using a knife). Using our right handrule, we can qualitatively discover the strength of the B-field as well asits directionboth inside and outside the loop.
As the radius of the loop is made smaller, the magnetic field outside the loop
approaches ______.
III.Very Long Solenoid (Coil With Many Loops)
We are now ready to use Ampere's Law and our work with the current loop to solve for the magnetic field due to a very long coil (i.e. when the length of the coil is much greater than the coil's diameter). In the figure below, we have a coil with N turns (loops) and a current I. The cross sectional view of the coil is also shown.
SOLN:
From our work with current loops, we know that the magnetic field strength
outside the coil is approximately ______.
To find the magnetic field inside the coil, we draw ______
curve (because of the symmetry of the problem) and use Ampere's Law.
IV.Toroid (Donut)
We now consider another important structure, the toroid. A toroid is a solenoid whose ends have been wrapped around to form a donut. Toroid'swere used to construct the original computer memory, "core memory." Toroid's with a break are still in use today to record video and audio onto magnetic tapes.
A.Case r < a "inside the hole"
B.Case r > b "outside toroid"
C.Case a < r < b "inside the toroid"
V.Comparing The Solenoid And Toroid
A.The solenoid has many advantages:
1)It is simple to build.
2)It's magnetic field is constant for a given current and number of loops.
B.The problem with the solenoid is that to be an ideal solenoid it must be
______long. Although, we can reduce the radius of the
solenoid to approximate an ideal solenoid it will reduce our working space inside
the solenoid.
C.The toroid has the advantage of limiting the field completely inside the toroid.
D.The problems with the toroid is that
1)they are more difficult to build than solenoids
2)their magnetic field is not uniform since it depends on r.