1 LECTURE 14 • Fundamental Laws for Calculating B-field • Biot-Savart Law (“brute force”) • Ampere’s Law (“high symmetry”) • Example: B-field of an Infinite Straight Wire • from Biot-Savart Law • from Ampere’s Law • Other examples 10/3/18 2 Our Study of Magnetism • Lorentz Force Equation • Motion in a uniform B-field • Forces on charges moving in wires • Magnetic dipole • Today: fundamentals of how currents generate magnetic fields N S Calculation of Electric Field What are the analogous equations for the Magnetic Field? • Two ways to calculate "Brute force" Coulomb’s Law "High symmetry" Gauss’ Law Calculation of Magnetic Field Two Ways to calculate "Brute force" I Biot-Savart Law "High symmetry“ also, only the ENCLOSED Current Ampere’s Law Amperian Loop Integral These are the analogous equations
Transcript
1
LECTURE 14
• Fundamental Laws for Calculating B-field• Biot-Savart Law (“brute force”)• Ampere’s Law (“high symmetry”)
• Example: B-field of an Infinite Straight Wire• from Biot-Savart Law• from Ampere’s Law
• Other examples
10/3/18 2
Our Study of Magnetism
• Lorentz Force Equation
• Motion in a uniform B-field
• Forces on charges moving in wires
• Magnetic dipole
• Today: fundamentals of how currents generate magnetic fields
NS
Calculation of Electric Field
What are the analogous equations for theMagnetic Field?
• Two ways to calculate
"Brute force"
Coulomb’s Law
"High symmetry"
Gauss’ Law
Calculation of Magnetic FieldTwo Ways to calculate
"Brute force" I
Biot-Savart Law
"High symmetry“ also, only the ENCLOSED Current
Ampere’s Law Amperian Loop Integral
These are the analogous equations
2
Biot-Savart Law: B-field due to a current in a wire
The magnetic field “curls” or “loops” around the wire
No dB field at P2 since d! is parallel to r
Vector Form:
Scalar Form:
10/3/18 6
Observations about Biot-Savart Law
Note: A moving charge produces a B field.
10/3/18 7
Observations about Biot-Savart• Right Hand Rules
*Grasp element in your right hand with thumb pointing in the direction of the current. Your fingers naturally curl around in the direction of the magnetic field.
*To find at point P,
10/3/18 8
Switch open: I = 0Compass points north.
Switch closed: I≠ 0
Oersted’s Experiment DEMO6B-01
3
Magnetic Field of an Infinite Straight Wire• Calculate field at point P using Biot-Savart Law:
y
R
rφ
P
I
dlx
dlφ
• Calculate field at distance Rfrom wire using Ampere's Law: dl
RI
Ampere's Law simplifies the calculation thanks to symmetry around the current! (axial/cylindrical)
B
Magnetic Field of an Infinite Straight Line
10/3/18 11
Magnetic Field Lines of a Straight Wire
DEMO – 6B-03
10/3/18 12
Practice both right hand rules here:
Magnetic Field at the Center of a Current Loop
4
10/3/18 13
y
x
Id!= Rdφ
Only if φ in radians
φ/2π is just the fraction of a full circle that carries current.
Note that as usual, the current comes in and leaves via un-shown wires!
Magnetic Field at the center of curvature for a partial loop
φ
10/3/18 14
I I
I
R
P
1
2
3Find B at point P.
Line Segment Combinations
10/3/18 15
Double ArcFind B at point P.Only the arcs contribute.
Inner Arc
Outer Arc
Magnitude of B at point P:
Direction of B at point P:
PR1
R2
II
10/3/18 16
Magnetic Field on Axis of Current Loop
5
10/3/18 17
Magnetic Field Lines of a Current Loop
DEMO – 6B-04 & 0510/3/18 18
Solenoid (DEMO) DEMO – 6B-04 & 05
10/3/18 19
Solenoid
10/3/18 20
Bx at the end of a long solenoid is exactly half of the value below, which is for places far from either end of a long solenoid.
This makes sense, since putting two such long equal solenoids end to end gets you back to full strength, and so each of the halves must contribute Bx/2
