Physics 7B Week 12-2 Notes
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Notes by Kohtaro Yamakawa
Ampere’s Law
I will discuss and review the way to approach solving for the magnetic field for different configurations using Ampere’s Law:
I
~ · d~l = µ0 Ienc
B
Let’s break this down.
H
• d~l is the integral over a loop (a curve starting and ending at the same point) with d~l the
tangent vector along the direction of the curve. In the following we will call this loop, the
Amperian loop.
~ is the magnetic field
• B
• µ0 is the permeability of free space (a constant) which may change if we work in different
materials. For now, we will assume we are working in a vacuum/free space and use µ0 .
• Ienc is the current enclosed by the current loop. More on what this means below.
Given a configuration that is “nice” where Ampere’s law is useful, we will go through following
steps:
1. Find a Coordinate system
2. Choose an Amperian Loop.
H
~ · d~l
3. Evaluate B
4. Evaluate µ0 Ienc
~
5. Use Ampere’s law to find B(r).
The first step is something that should be obvious. A thick wire has cylindrical coordinates, a
sphere has spherical coordinates, a plane has cartesian coordinates etc. The difficulty of Ampere’s
Law questions comes from step 2: choosing an Amperian loop.
2
Choosing an Amperian loop
I will stipulate that we want an Amperian Loop which satisfies,
~ k ~d~l or B
~ ⊥ d~l for all points along the loop.
(1) B
(2) The magnitude of the magnetic field |B| is constant on the loop1 .
Find such a loop is usually not too hard. Or rather, if it is too difficult to find an Amperian loop
that satisfies the above, you should just resort to the Biot-Savart law. Doing so, however, requires
~ before doing the question. More specifically, we need
an intimate knowledge of the behavior of B
~ is at all points of interest and how it varies/along which directions it is
to know what direction B
constant. This can be found through symmetry2 or through Biot Savart’s law. I will expand on the
latter method.
1
It is okay if it is constant but different values at different points of the Amperian loop as in the case of finding
the magnetic field for an infinite slab of thickness a with current conducting through it. As long as the magnetic
field does not continuously change over the loop
2
See here for more on making purely symmetric arguments
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Physics 7B Week 12-2 Notes
Notes by Kohtaro Yamakawa
Biot-Savart’s law, while complicated, implies that for some infinitesimal element d~l with some
current I
Z
µ0
I d~l × r̂
~
B=
∝ I d~l × r̂
4π
r2
where I is the current of this small element, d~l is the tangent vector pointing in the direction the
current is going3 , and r̂ is the unit vector pointing from the differential element you are considering.
Using this, you can gain an intuition on what should occur.
Example 2.1. Consider a thick wire with current in the x̂ direction. What does the magnetic field
look like around this wire? If we consider some point P outside the wire, then consider I d~l × r̂ for
some differential element d~l = dlx̂ in the wire. Letting r̂ point from some differential element dl to
the point P , we can figure out what magnetic field this small differential element will contribute to
our point P :
~
~ = µ0 I dl × r̂
dB
4π r2
If we add the contributions from all differential elements up (taking advantage of the law of superposition), you should see that the only direction that the magnetic field can point is in the φ̂
direction. In addition, this tells us that the magnetic field is constant if we are at a fixed r distance
from the axis.
Let’s go one step further and ask what Amperian loop satisfies conditions (2) and (1). What
is a shape of a loop that has a fixed r distance from the axis and has a tangent vector in the φ̂
direction at all times? Why a circle tangent and centered on the axis!
~ looks like, why don’t we just use
You may ask, if we use Biot Savart’s law to figure out what B
Biot-Savart’s law always? While the above description is very explicit in deriving the direction and
behavior of the magnetic field in all space, we can use the above result from a thick wire and apply
it to all other configurations. Therefore, using the law of superposition, you can use the fact that:
Given a current in a direction, a magnetic field is generated in a +φ̂ direction around
this current.
Said in terms of the RH rule, if our right hand thumb points along the direction of the current, that
current will induce a magnetic field in the direction that our fingers wrap around our hand from
the knuck to the fingers.
Using this, you can derive that the magnetic field at the center of a loop of current is in the
direction of the axis of the the loop. Or, the magnetic field above (z > 0) an infinite slab in the
xy plane with current in the x̂ direction is in the −ŷ direction. Below (z < 0) the infinite slab, the
magnetic field is in the ŷ direction.
Therefore, we use our intuition built from how a single straight wire works to figure out the
behavior of the magnetic field. From that, we can pick out what Amperian loops work best.
Don’t confuse this with the d~l tangent vector of the Amperian loop! I only use d~l here because it agrees with
the convention used in textbooks.
3
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Physics 7B Week 12-2 Notes
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Notes by Kohtaro Yamakawa
Evaluate the LHS of Ampere’s Law
This step becomes quite simple due to the conditions we put on our Amperian loop. s
I
I
~
~
~ k d~l or B
~ ⊥ d~l)
B · dl =
|B| dl
(Since B
Bkdl
I
dl
(Since |B| is constant on the loop.)
= |B|
Bkdl
= |B|LBkdl
~ d~l: B
~ · d~l = |B| dl. Be aware that this is
In the above, we assumed that in all the places that B||
~ · d~dl = −|B| dl. This is the case if we consider a
not always the case. It is entirely possible for B
square Amperian loop inside a uniform magnetic field.
Note also, that we assumed that the magnetic field is constant for all parts of the loop. This
is also not always the case and it be that the magnetic field is constant on parts of the loop but
different along different parts of the loop. This is the case when we consider a square Amperian
loop half inside and half outside of a solenoid. Understanding the above two caveats, we can rewrite
the evaluation as,
I
X
~ · d~l =
~ i · d~l)|Bi |LB kdl
B
sgn(B
i
Bi kdl
~ k d~l and over which |B| is
where the summation is over all parts of the Amperian loop where B
constant. LBi kdl is the length of the Amperian loop where these constraints are satisfied. As you can
see, this looks like a mess even though it is fairly straightforward when you explicitly do Amperian
loop questions. That’s the curse of generality!
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Evaluate the RHS of Ampere’s Law
The right hand side of Ampere’s law is µ0 Ienc which is usually evaluated using the current density
~
J~ = I/A
where A is the cross section over which the current travels. Notice that if we consider a
~ = I/l
~ (where I
surface current, the cross section is one dimensional and the current density is K
~ to follow convention in textbooks). The current density is a vector field that assigns a vector
use K
at every point in space. In other words, it is 0 in places where there is no current and it is nonzero
~
where there is current. In the special case of uniform (volume) current, J~ = AI .
This current density helps evaluate Ienc :
Z
~
=⇒ Ienc = J~ · dA
where we take the integral over a surface enclosed by the Amperian loop we selected above. The
~ is the differential area element perpendicular to the surface, with a direction determined by
dA
the orientation of the Amperian loop that you select. Using the right hand rule, if your fingers
~
wrap around in the direction of the Amperian loop, then your thumb will point in the correct dA
direction. See here for a quick discussion on subtleties on finding the current enclosed. However, I
don’t think you will have to worry about these subtleties at this level. In most cases, we will find
~ and,
that J~ · dA
Z
Z
Uniform Current I
~
~
Ienc = J · dA = |J| dA −−−−−−−−−→ Aenc
A
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Physics 7B Week 12-2 Notes
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Notes by Kohtaro Yamakawa
Conclusion
~
We have spelled out how to use Ampere’s law. The left hand side consisted of guessing what the B
looks like and choosing the right Amperian loop. The right hand side consisted of understanding
what current is enclosed by that same loop. All in all, Ampere’s law states something similar to
Gauss’s law: From the outside, a uniform current (charge) looks like a wire (point charge). While
it is powerful, it is also easy to make mistakes along the way. I hope that this document clarified
your understanding of Ampere’s law. Let me know if you have any questions at all.
-Koh
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