W09D1: Sources of Magnetic Fields: Ampere

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W09D1Sources of Magnetic Fields Amperes Law
Todays Reading Assignment Course Notes
Sections 9.3-9.4, 9.7, 9.10.2
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Announcements
Math Review Week Nine Tuesday from 9-11 pm in
26-152 PS 7 due Tuesday April 9 at 9 pm in
boxes outside 32-082 or 26-152 Next reading
Assignment W09D2 Faradays Law Course Notes
Sections 10.1-10.4
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Today Biot-Savart vs. Ampere
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Last TimeCreating Magnetic FieldsBiot-Savart
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The Biot-Savart Law
Current element of length carrying current
I produces a magnetic field at the point P
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The Biot-Savart Law Infinite Wire
Magnetic Field of an Infinite Wire Carrying
Current I from Biot-Savart See W06D3 Problem
Solving http//web.mit.edu/8.02t/www/materials/Pr
oblemSolving/solution05.pdf
More generally
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3rd Maxwell Equation Amperes Law(we will
eventually add one more term to this equation)
Open surface is bounded by closed path
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Amperes Law The Idea
In order to have a B field around a loop, there
must be current punching through the loop
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Concept Question Line Integral
The integral expression

1. is equal to the magnetic work done around a
   closed path.
2. is an infinite sum of the product of the tangent
   component of the magnetic field along a small
   element of the closed path with a small element
   of the path up to a choice of plus or minus sign.
   
3. is always zero.
4. is equal to the magnetic potential energy between
   two points.
5. None of the above.

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C.Q. Answer Line Integral
2. A line integral by definition is the sum

• We need to make a choice of integration
  direction (circulation) for the line integral.
  The small line element is tangent to the
  line and points in the direction of circulation.
  The dot product therefore is the product of the
  tangent component of the magnetic field in the
  direction of the line element. So the answer
  depends on which way we circulate around the
  path.

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Current Enclosed
Current density
Current enclosed is the flux of the current
density through an open surface S bounded by the
closed path. Because the unit normal to an open
surface is not uniquely defined this expression
is unique up to a plus or minus sign.
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Sign Conventions Right Hand Rule
Integration direction clockwise for line integral
requires that unit normal points into page for
open surface integral Current positive into page,
negative out of page
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Sign Conventions Right Hand Rule
Integration direction counterclockwise for line
integral requires that unit normal points out of
page for open surface integral Current positive
out of page, negative into page
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Concept QuestionsAmperes Law
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Concept Question Amperes Law
Integrating B around the loop shown gives us

1. a positive number
2. a negative number
3. zero

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C.Q. Answer Amperes Law

• Answer 3. Total enclosed current is zero, so

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Concept Question Amperes Law
Integrating B around the loop in the clockwise
direction shown gives us

1. a positive number
2. a negative number
3. zero

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C.Q. Answer Amperes Law

• Answer 2.
• Net enclosed current is out of the page, so field
  is counter-clockwise (opposite to circulation
  direction)

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Applying Amperes Law

1. Identify regions in which to calculate B field.
2. Choose Amperian closed path such that by symmetry
   B is zero or constant magnitude on the closed
   path!
3. Calculate
4. Calculate current enclosed
5. Apply Amperes Law to solve for B check signs

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Infinite Wire
A cylindrical conductor has radius R and a
uniform current density with total current I. we
shall find the direction and magnitude of the
magnetic field for the two regions (1) outside
wire (r R) (2) inside wire (r lt R)
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Worked Example Amperes Law Infinite Wire
I
B
I
Amperian Closed Path B is Constant
Parallel Current penetrates surface
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Example Infinite Wire
Region 1 Outside wire (r R)
Cylindrical symmetry ? Amperian Circle B-field
counterclockwise
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Group Problem Magnetic Field Inside Wire
We just found B(rgtR) Now you find B(rltR)
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Infinite Wire Plot of B vs. r
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Group Problem Non-Uniform Cylindrical Wire
A cylindrical conductor has radius R and a
non-uniform current density with total
current Find B everywhere
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Other Geometries
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Two Loops
http//web.mit.edu/viz/EM/visualizations/magnetost
atics/MagneticFieldConfigurations/tworings/tworing
s.htm
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Two Loops Moved Closer Together
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Multiple Wire Loops
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Multiple Wire Loops Solenoid
http//youtu.be/GI2Prj4CGZI
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DemonstrationLong Solenoid
http//tsgphysics.mit.edu/front/?pagedemo.phplet
numG2018show0
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Magnetic Field of Solenoid
Horiz. comp. cancel
loosely wound
tightly wound
For ideal solenoid, B is uniform inside zero
outside
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Magnetic Field of Ideal Solenoid
Using Amperes law Think!
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Group Problem Current Sheet
A sheet of current (infinite in the y z
directions, of thickness d in the x direction)
carries a uniform current density Find the
direction and magnitude of B as a function of x.
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Amperes LawInfinite Current Sheet
B
I
B
Amperian Loops B is Constant Parallel OR
Perpendicular OR Zero I Penetrates
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Surface Current Density
A very thin sheet of current of width w carrying
a current I in the positive z-direction has a
surface current density For sheet of thickness d
, width w, and current I
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Solenoid is Two Current Sheets
Consider two sheets each of thickness d with
current density J. Then surface current per unit
length Use either Amperes Law or
superposition principle
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Biot-Savart vs. Ampere
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Amperes Law
Long Circular Symmetry
(Infinite) Current Sheet
2 Current Sheets
Solenoid
Torus