W09D1: Sources of Magnetic Fields: Ampere’s Law Today’s Reading Assignment Course Notes: Sections 9.3-9.4, 9.7, 9.10.2 Class 18

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 Faraday’s Law Course Notes: Sections 10.1-10.4

Today: Biot-Savart vs. Ampere Biot-Savart Law general current source ex: finite wire wire loop Ampere’s law symmetric ex: infinite wire infinite current sheet Class 20

Last Time: Creating Magnetic Fields: Biot-Savart Class 20

The Biot-Savart Law Current element of length carrying current I produces a magnetic field at the point P: Class 20

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/ProblemSolving/solution05.pdf More generally: Class 20

3rd Maxwell Equation: Ampere’s Law (we will eventually add one more term to this equation) Open surface is bounded by closed path Class 20

Ampere’s Law: The Idea In order to have a B field around a loop, there must be current punching through the loop Class 20

Concept Question: Line Integral The integral expression is equal to the magnetic work done around a closed path. 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. is always zero. is equal to the magnetic potential energy between two points. None of the above. Class 20

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. Class 20

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. Class 12

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 : Class 20

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 : Class 20

Concept Questions: Ampere’s Law Class 20

Concept Question: Ampere’s Law Integrating B around the loop shown gives us: a positive number a negative number zero Class 20

C.Q. Answer: Ampere’s Law Week 07, Day 2 C.Q. Answer: Ampere’s Law Answer: 3. Total enclosed current is zero, so Class 16

Concept Question: Ampere’s Law Integrating B around the loop in the clockwise direction shown gives us: a positive number a negative number zero Class 20

C.Q. Answer: Ampere’s Law Week 07, Day 2 C.Q. Answer: Ampere’s Law Answer: 2. Net enclosed current is out of the page, so field is counter-clockwise (opposite to circulation direction) Class 16

Applying Ampere’s Law Identify regions in which to calculate B field. Choose Amperian closed path such that by symmetry B is zero or constant magnitude on the closed path! Calculate Calculate current enclosed: Apply Ampere’s Law to solve for B: check signs Class 20

Week 06, Day 1 Infinite Wire I 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 < R) Class 14

Worked Example: Ampere’s Law Infinite Wire Week 06, Day 1 Worked Example: Ampere’s Law Infinite Wire I B I Amperian Closed Path: B is Constant & Parallel Current penetrates surface Class 14

Example: Infinite Wire Week 06, Day 1 Example: Infinite Wire Region 1: Outside wire (r ≥ R) Cylindrical symmetry  Amperian Circle B-field counterclockwise Class 14

Group Problem: Magnetic Field Inside Wire Week 06, Day 1 Group Problem: Magnetic Field Inside Wire I We just found B(r>R) Now you find B(r<R) Class 14 23

Infinite Wire: Plot of B vs. r Week 06, Day 1 Infinite Wire: Plot of B vs. r Class 14

Group Problem: Non-Uniform Cylindrical Wire A cylindrical conductor has radius R and a non-uniform current density with total current: Find B everywhere Class 20

Other Geometries Class 20

Two Loops http://web.mit.edu/viz/EM/visualizations/magnetostatics/MagneticFieldConfigurations/tworings/tworings.htm Class 20

Two Loops Moved Closer Together Class 20

Multiple Wire Loops Class 20

Multiple Wire Loops – Solenoid http://youtu.be/GI2Prj4CGZI Class 20

Demonstration: Long Solenoid http://tsgphysics.mit.edu/front/?page=demo.php&letnum=G%2018&show=0 Class 20

Magnetic Field of Solenoid Horiz. comp. cancel loosely wound tightly wound For ideal solenoid, B is uniform inside & zero outside Class 20

Magnetic Field of Ideal Solenoid Using Ampere’s law: Think! Class 20

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. Class 20

Ampere’s Law: Infinite Current Sheet B I B Amperian Loops: B is Constant & Parallel OR Perpendicular OR Zero I Penetrates Class 20

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 Class 20

Solenoid is Two Current Sheets Consider two sheets each of thickness d with current density J. Then surface current per unit length Use either Ampere’s Law or superposition principle Class 20

infinite current sheet Biot-Savart vs. Ampere Biot-Savart Law general current source ex: finite wire wire loop Ampere’s law symmetric ex: infinite wire infinite current sheet Class 20

(Infinite) Current Sheet Ampere’s Law: I B Long Circular Symmetry (Infinite) Current Sheet X B X = 2 Current Sheets Solenoid Torus Class 20