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Lesson 8 Ampère’s Law and Differential Operators

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## Lesson 8 Ampère’s Law and Differential Operators

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**Amperian Loop**An Amperian loop is any closed loop Amperian loops include: a circle a square a rubber band Amperian loops do not include: a balloon a piece of string (with two ends)**Take a wire with current coming out of the screen.**The Field Contour of a Wire**The field contour is made of half-planes centered on the**wire. The Field Contour of a Wire**We draw arrows on each plane pointing in the direction of**the magnetic field. The Field Contour of a Wire**We draw an Amperian loop around the wire.**Amperian Loops**We wish to count the “net number” of field lines pierced**by the Amperian loop. Amperian Loops**First, we put an arrow on the loop in an overall**counterclockwise direction. Amperian Loops**To count the net number of surfaces pierced by Amperian**loop, we add +1 when the loop is “in the direction’ of the plane and −1 when it is “opposite the direction” of the plane. Counting Surfaces Pierced +1 +1 +1 +1 +1 −1 +1 +1 −1 +1 +1 +1 +1 −1 +1 +1 +1 +1 −1 +1 +1 +1 +1 +1**Note there are “+1” appears 20 times and “-1”**appears 4 times. Counting Surfaces Pierced +1 +1 +1 +1 +1 −1 +1 +1 −1 +1 +1 +1 +1 −1 +1 +1 +1 +1 −1 +1 +1 +1 +1 +1**The net number of surfaces pierced by the Amperian loop us**therefore +16. Counting Surfaces Pierced +1 +1 +1 +1 +1 −1 +1 +1 −1 +1 +1 +1 +1 −1 +1 +1 +1 +1 −1 +1 +1 +1 +1 +1**What is the net number of surfaces pierced by each of these**Amperian loops? Other Amperian Loops**What is the net number of surfaces pierced by each of these**Amperian loops? Other Amperian Loops**What is the net number of surfaces pierced by each of these**Amperian loops? Other Amperian Loops**The net numbers of surfaces pierced by each of these loops**is 16. Other Amperian Loops**What is the net number of surfaces pierced by this Amperian**loop? Other Amperian Loops**This time the net number of surfaces pierced by the loop is**0. Why? Other Amperian Loops**This is the same loop we saw earlier, but now only 8**surfaces are pierced, since there are only 8 surfaces extending outward from the wire. Other Wires.**There are 8 surfaces coming from the wire because the**current through the wire is half as much as it was before. Other Wires.**The net number of perpendicular surfaces pierced by an**Amperian loop is proportional to the current passing through the loop. Ampère’s Law**Always traverse the Amperian loop in a (generally)**counterclockwise direction. • If the number of surfaces pierced N>0, the current comes out of the screen. • If N<0, the current goes into the screen. Sign Convention**There are three kinds of charge density (ρ,σ,λ)**• There is one kind of current density (current/unit area) Current Density**The current passing through a small gate of area ΔA is**Current Density small gate**The total current passing through the wire is the sum of**the current passing through all small gates. Current Density small gates**The current density, j, can vary with r only.**Below, we assume that the current density is greatest near the axis of the wire. Cylindrically Symmetric Current Distribution**Cylindrically Symmetric Current Distribution**Outside the distribution, the field contour is composed of surfaces that are half planes, uniformly spaced.**Cylindrically Symmetric Current Distribution**Inside the distribution, it is difficult to draw perpendicular surfaces, as some surfaces die out as we move inward. – We need to draw many, many surfaces to keep them equally spaced as we move inward.**Cylindrically Symmetric Current Distribution**But we do know that if we draw enough surfaces, the distribution of the surfaces will be uniform, even inside the wire.**Cylindrically Symmetric Current Distribution**Let’s draw a circular Amperian loop at radius r. r**Cylindrically Symmetric Current Distribution**Now we split the wire into two parts – the part outside the Amperian loop and the part inside the Amperian loop. r r**Cylindrically Symmetric Current Distribution**The total electric field at r will be the sum of the electric fields from the two parts of the wire. r r**The total number of perpendicular surfaces pierced by the**Amperian loop is zero because there is no current passing through it. Inside a Hollow Wire r**1. We could have all the surfaces pierced twice, one in the**positive sense and one in the negative… How can we get zero net surfaces? … but this violates symmetry!**2. We could have some surfaces oriented one way and some the**other… How can we get zero net surfaces? … but this violates symmetry, too!**How can we get zero net surfaces?**3. Or we could just have no surfaces at all inside the hollow wire. This is the only way it can be done!**The Magnetic Field inside a Hollow Wire**If the current distribution has cylindrical symmetry, the magnetic field inside a hollow wire must be zero.**Since the magnetic field inside a hollow wire is zero, the**total magnetic field at a distance r from the center of a solid wire is the field of the “core,” the part of the wire within the Amperian loop. Cylindrically Symmetric Current Distribution r r**r**Cylindrically Symmetric Current Distribution Outside the core, the magnetic field is the same as that of a thin wire that has the same current as the total current passing through the Amperian loop.**r**Cylindrically Symmetric Current Distribution Inside a cylindrically symmetric current distribution, the magnetic field is:**A wire of radius R with a uniform current distribution has a**total charge i passing through it. What is the magnetic field at r < R ? Example: Uniform Current Distribution**r**A wire of radius R with a uniform current distribution has a total charge i passing through it. What is the magnetic field at r < R ? Example: Uniform Current Distribution**r**A wire of radius R with a uniform current distribution has a total charge i passing through it. What is the magnetic field at r < R ? Example: Uniform Current Distribution The current density is uniform, so:**r**A wire of radius R with a uniform current distribution has a total charge i passing through it. What is the magnetic field at r < R ? Example: Uniform Current Distribution Therefore: