Physics equations/Magnetic field calculations

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The Biot-Savart law[edit]

Geometry for the Biot-Savart law.

The Biot–Savart law is used for computing the resultant magnetic field B at position r generated by a steady current I (for example due to a wire): a continual flow of charges which is constant in time and the charge neither accumulates nor depletes at any point. The law is a physical example of a line integral: evaluated over the path C the electric currents flow. The equation in SI units is:

**Problem: Generalize this line integral to a volume integral involving current density.

Magnetic field lines for typical geometries[edit]

Magnetic field due to a current loop[edit]

Magnetic element current loop

*Problem: Show that if circular loop of radius carries a current , then the the magnetic field at the center of the loop points in the direction and as a magnitude of:

.

(This represents the magnetic field at in the figure to the right; the magnetic field points in the direction.)


***Problem:Show that if circular loop of radius carries a current , then the the magnetic field at the at a distance away from the center is in the z direction, has magnitude:

Ampère's circuital law[edit]

Closed surfaces to the left; open surfaces with boundaries to right.
Understanding Maxwell's displacement current. A current flowing through a wire produces a magnetic field, in accordance with Ampère's law. But a capacitor in the circuit represents a break in the current, so that a surface can avoid penetration by the current by passing between the plates of the capacitor. Maxwell corrected this flaw by postulating that a time-varying electric also generates a magnetic field.

The "integral form" of the original Ampère's circuital law[1] is a line integral of the magnetic field around any closed curve C (This closed curve is arbitrary but it must be closed, meaning that it has no endpoints). The curve C bounds both a surface S, and any current which pierces that surface is said to be enclosed by the surface. The line integral of the magnetic B-field (in tesla, T) around closed curve C is proportional to the total current Ienc passing through a surface S (enclosed by C):

Maxwell's correction term (displacement current)[edit]

This equation might is not generally valid if a time-dependent electric field is present, as was discovered by James Clerk Maxwell, who added the displacement current term to Ampere's law around 1861.[2][3] The need for this extra term can be seen in the figure to the right. The diagram shows a capacitor being charged by current flowing through a wire, which creates a magnetic field around it. The magnetic field is found from Ampere's law:

*****Problem: Show that with Maxwell's correction (with ), Ampere's law becomes:

where,

Magnetic field due to a long straight wire[edit]

Magnetic field near a long straight wire.

*Problem: Show that in the vicinity of a long, straight wire carrying current , Ampere's law yields:

where is a unit vector that points in the azimuthal direction, and is the magnetic constant.

Magnetic field inside a long thin solenoid[edit]

Magnetic field due to a solenoid

*Problem: Show that in the vicinity of a long, thin solenoid of length

where is the current, is the number of turns, and is the number of turns per unit length.

references[edit]

  1. https://en.wikipedia.org/w/index.php?title=Amp%C3%A8re%27s_circuital_law&oldid=578507291
  2. Example taken from Feynman, Richard; Robert Leighton; Matthew Sands (1964) The Feynman Lectures on Physics, Vol.2, Addison-Wesley, USA, p.18-4, using slightly different terminology.
  3. https://commons.wikimedia.org/w/index.php?title=File:Displacement_current_in_capacitor.svg&oldid=38260258

Basic Magnetic Terms definition with Formulas