Magnetic Field of a Solenoid Step 1 Cut

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Magnetic Field of a Solenoid Step 1: Cut up the distribution into pieces Step

Magnetic Field of a Solenoid Step 1: Cut up the distribution into pieces Step 2: Contribution of one piece origin: center of the solenoid one loop: B Number of loops per meter: N/L Number of loops in z: (N/L) z Field due to z:

Magnetic Field of a Solenoid Step 3: Add up the contribution of all the

Magnetic Field of a Solenoid Step 3: Add up the contribution of all the pieces B Magnetic field of a solenoid:

Magnetic Field of a Solenoid Special case: R<<L, center of the solenoid: in the

Magnetic Field of a Solenoid Special case: R<<L, center of the solenoid: in the middle of a long solenoid

Triangular coil There is a current going through a triangular coil. Which direction is

Triangular coil There is a current going through a triangular coil. Which direction is B at the center? How would you find the magnitude of B?

Helmholtz Coils How what is B near the origin? Assume that the positions of

Helmholtz Coils How what is B near the origin? Assume that the positions of the loops are large compared to their radii.

Patterns of Magnetic Field in Space Is there current passing through these regions? There

Patterns of Magnetic Field in Space Is there current passing through these regions? There must be a relationship between the measurements of the magnetic field along a closed path and current flowing through the enclosed area. Ampere’s law

Quantifying the Magnetic Field Pattern Curly character – introduce: Similar to Gauss’s law (Q/

Quantifying the Magnetic Field Pattern Curly character – introduce: Similar to Gauss’s law (Q/ 0) Will it work for any circular path of radius r ?

A Noncircular Path Need to compare and Where in loop doesn’t matter!

A Noncircular Path Need to compare and Where in loop doesn’t matter!

Currents Outside the Path Need to compare and for currents outside the path

Currents Outside the Path Need to compare and for currents outside the path

Three Current-Carrying Wires Ampere’s law

Three Current-Carrying Wires Ampere’s law

Ampère’s Law All the currents in the universe contribute to B but only ones

Ampère’s Law All the currents in the universe contribute to B but only ones inside the path result in nonzero path integral Ampere’s law is almost equivalent to the Biot-Savart law: but Ampere’s law is relativistically correct

Inside the Path Ampere’s law 1. 2. 3. 4. Choose the closed path Imagine

Inside the Path Ampere’s law 1. 2. 3. 4. Choose the closed path Imagine surface (‘soap film’) over the path Walk counterclockwise around the path adding up Count upward currents as positive, inward going as negative

 What is ?

What is ?

Ampere’s Law: A Long Thick Wire Can B have an out of plane component?

Ampere’s Law: A Long Thick Wire Can B have an out of plane component? Is it always parallel to the path? for thick wire: (the same as for thin wire) Would be hard to derive using Biot-Savart law

Ampere’s Law: A Solenoid Number of wires: (N/L)d What is on sides? B outside

Ampere’s Law: A Solenoid Number of wires: (N/L)d What is on sides? B outside is very small (solenoid) Uniform: same B no matter where is the path

Ampere’s Law: A Toroid Symmetry: B || path Is magnetic field constant across the

Ampere’s Law: A Toroid Symmetry: B || path Is magnetic field constant across the toroid?