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Department of Physics and Applied PhysicsPHYS.1440 Lecture15 A.Danylov
Lecture 15
Chapter 32
Ampere’s law
Course website:http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsII
Physics II
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Department of Physics and Applied PhysicsPHYS.1440 Lecture15 A.Danylov
Ampere’s LawElectric Field Magnetic Field
From Coulomb’s law1
4̂
Bio-Savart law
4̂
Gauss’s Law
∙There must be
something similar for B
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Department of Physics and Applied PhysicsPHYS.1440 Lecture15 A.Danylov
Amperian loop
Ampere’s Law
I1
I2
I3
I3
1) An Amperian loop is imaginary2) It is a closed loop (any path can be used)
It gives us which current is positive/negative(use a right-hand rule:
So I1 is positive; I2 is negative
The line integral of the magnetic field around the curve is given by Ampère’s law:
∙
Ampère’s law is very useful for a problem with a high degree of symmetry.
Closed loop (Amperian)
Amperian loop
B
B
∙
∙
∙
+
+
∙
These currents pass through thearea bounded by the loop, sothey are enclosed, Iin
3) Choose direction (up to you).
curl your fingers in a chosen direction and anoutstretched thumb shows a positive current direction)
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ConcepTest 1 Ampere’s LawA) 0 AB) 1 A out of the screenC) 1 A into the screenD) 5 A out of the screenE) 5 A into the screen
The line integral of B around the loop is 0 · 7.0 A.
Current I3 is
∮ ∙ =7
Minus means our original assumption was wrong, it is into the screen
1
Assume I3 is out of the page
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Department of Physics and Applied PhysicsPHYS.1440 Lecture15 A.Danylov
Magnetic field of a current-carrying wireThe wire has cylindrical symmetry so that we can easily use Ampere’s law.
One moving charge creates magnetic field lines centered on the motion line:
Now we have many moving charges (not just one). The field pattern must be the same.
So we’ll take our Amperianloop to be a concentric circles of r.
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Department of Physics and Applied PhysicsPHYS.1440 Lecture15 A.Danylov
Solenoid
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Department of Physics and Applied PhysicsPHYS.1440 Lecture15 A.Danylov
Solenoid
A uniform magnetic field can be generated with a solenoid.
A solenoid is a helical coil of wire with the same current I passing through each loop in the coil.
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Department of Physics and Applied PhysicsPHYS.1440 Lecture15 A.Danylov
Steps to make a solenoidMagnetic field lines produced with a straight wire
Let’s bend the wire into a loop
Now, let’s add more loops
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Department of Physics and Applied PhysicsPHYS.1440 Lecture15 A.Danylov
where n N/l is the number of turns per unit length.
The Magnetic Field of a Solenoid
Along the bottom (ab), the line integral is zero since B 0outside the solenoid.
Along the sides (bc, da), the line integral is zero since the field is perpendicular to the path.
dsB
B
B=0dsBds
∙abcda
0 0 0
0 ∥
There are N loops with current I enclosed by an Amperian loop, so ∙
∙ ∙ ∙ ∙
Uniform field
Amperian loopabcda
∙
/
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Department of Physics and Applied PhysicsPHYS.1440 Lecture15 A.Danylov
This patient is undergoing magnetic resonance imaging (MRI). The large cylinder surrounding the patient contains a solenoid that is wound with superconducting wire to generate a strong uniform magnetic field.
B=1.2 T, I=100 A
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Department of Physics and Applied PhysicsPHYS.1440 Lecture15 A.Danylov
The Magnetic Field Outside a Solenoid• The magnetic field outside a solenoid looks like that of a bar magnet.• Thus a solenoid is an electromagnet
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Department of Physics and Applied PhysicsPHYS.1440 Lecture15 A.Danylov
Electric Field Magnetic Field
From Coulomb’s law1
4̂
Biot-Savart law
4̂
Gauss’s Law
∙
Ampere’s Law
∙
So, now we know how to find magnetic fields using Bio-Savart and Ampere’s laws.Now, the question is “how does a magnetic field interact with material (which consists of charges and current)?”
Magnetic force on a moving charge
Magnetic force on current
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Department of Physics and Applied PhysicsPHYS.1440 Lecture15 A.Danylov
Magnetic force on a moving charge
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Department of Physics and Applied PhysicsPHYS.1440 Lecture15 A.Danylov
The Magnetic Force on a Moving ChargeAfter Oersted’s discovery, there were many other experiments with magnetic fields, currents, charges, etc. It was found that B exerts a force on a moving charge.
The magnetic force on a charge q as it moves through a magnetic field B with velocity v is:
where is the angle between v and B.
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Department of Physics and Applied PhysicsPHYS.1440 Lecture15 A.Danylov
What you should readChapter 32 (Knight)
Sections 32.6 32.5 (skip)
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Department of Physics and Applied PhysicsPHYS.1440 Lecture15 A.Danylov
Thank youSee you in a week.
Enjoy your spring break