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Magnetic Field Sources. Recall that:. Magnetic fields are caused by currents. Currents in magnetic fields can experience magnetic forces. (Also recall that a “current” can be a single charged particle in motion with some velocity v .).
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Magnetic Field Sources Recall that: • Magnetic fields are caused by currents. • Currents in magnetic fields can experience magnetic forces. (Also recall that a “current” can be a single charged particle in motion with some velocity v.) In the previous chapter we focused on item (2), the forces experienced by currents (or by individual moving charges) in a magnetic field. In this chapter we focus on (1), the calculation of several magnetic field configurations caused by currents (or by point charges in motion).
The magnetic field of a single point charge Here is the picture of the magnetic field created as a charge q moves with velocity v. As you can see, it’s a bit complicated. The B field circulates around the particle’s direction of travel. Try out the right hand rule! B decreases in strength with increasing distance from the charge. v, r, and B appear to involve a cross product relationship. P is the point where the B field is being calculated.
The magnetic field of a single point charge This B field is described by the Biot-Savart Law. with This is the magnetic equivalent of Coulomb’s Law. Notice the similarities and differences: It has a force constant, km , in place of the Coulomb constant, k, and, like the electric force, it is proportional to q. Also, like the electric force, it decreases as the square of the distance! As “expected”, the B vector depends on the cross product of v with r. Jean-Baptiste Biot1774-1862 “Permeability” of free space. Félix Savart 1791-1841
Magnetic fields superpose. Example: Find the direction and magnitude of the magnetic field at the origin in terms of the given quantities.
Biot-Savart Law for an element of current This looks like the field of a point charge in motion, but with qv replaced by Idl. The Biot-Savart Law for current can be derived from the point charge equation, by first putting that equation into infinitesimal form: Work on the infinitesimals: The Biot-Savart Law for an element of current:
Magnetic field of a line of current. (We looked at this last chapter.) This B field has cylindrical symmetry. Compare and contrast to E field of line charge.
Magnetic field of a line of current, from the Biot-Savart Law for an element of current Notice that at point P the magnetic field from all current elements along the line is pointed in the same direction—into the page. The sine factor below comes only from the cross product. and Integrating along line from –a to +a: For infinitely long wire, a >> x. Here is the B field with cylindrical symmetry, falling off as “1/r” from the wire:
Magnetic fields superpose. Another example: Can be used at each location Pi to find the total field at that point by superposing the fields from both wires.
Magnetic field on the axis of a current loop (single turn) Notice that at point P the magnetic field element components dB due to dl “add” in the x direction and cancel in the transverse directions as we integrate around the loop. Integrating around the loop from 0 to 2p: At the center of the loop, x = 0:
Magnetic field on the axis of a current loop (for N turns we multiply B by N)
Total force between two point charges (as they pass) This case is exceptionally simple since, at the time these two charges pass each other, the cross products are evaluated at 90o. So we will be finding magnitudes. The particle at the origin creates electric and magnetic fields felt by the other particle: The particle at y = r is moving to the left at speed –v. It experiences a total force due to both fields: Putting the E and B equations into F gives: Looks like a magnified electric field! Discuss.
Magnetic force between parallel currents How does the force depend on whether the currents are going in the same direction or in opposite directions???
Magnetic force between parallel currents Magnetic field due to current I: Force felt by I’ in this magnetic field: If the two currents are equal:
Ampere’s Law André-Marie Ampère – 1775-1836 On September 11, 1820 Ampere heard of H. C. Ørsted's discovery that a magnetic needle is acted on by a voltaic current. Only a week later, on September 18, he presented a paper containing a far more complete exposition of that and other phenomena. The SI unit of measurement of electric current, the ampere, is named after him. Ampère's fame mainly rests on the service that he rendered to science in establishing the relations between electricity and magnetism, and in developing the science of electromagnetism, or, as he called it, electrodynamics.
Ampere’s Law Recall, for a moment, the electric field calculations we did earlier. Coulomb’s Law was the basis for these calculations, but for problems involving symmetric systems it was easier to use Gauss’s Law whenever possible. The situation is similar for magnetic field calculations. The Biot-Savart Law describes sources of magnetic fields at the most fundamental level. But for systems with symmetric magnetic fields, Ampere’s Law, which we shall now introduce, can greatly simplify the calculation. We’ll write down the equation first, then interpret it, and do a number of examples: B A The integral of B around a closed loop is proportional to the total current enclosed by the loop. J Recall that Gaussian surfaces are chosen for purposes of calculation. So are “Amperian loops”
Magnetic field of a line of current. This time, we use Ampere’s Law. Ampere’s Law: Integrate around any circle centered on wire, in direction of B. Cylindrical symmetry means we can bring B outside integral: Solve for the magnetic field: Simple!
Magnetic field of a solenoid Field approximately uniform inside, and close to zero outside. S N
Magnetic field of a solenoid, from Ampere’s Law Ampere’s Law: Using the integration path shown, the only contribution to the integral comes from the segment a b: Solve for the magnetic field: N = number of turns in loop n = turns per unit length in the solenoid Simple!
Magnetic field on the axis of a solenoid: realistic field • In a real solenoid: • The central field is slightly lower than the value given by Ampere’s Law. • The end field is exactly half the Amperean value. Note: This curve cannot be found by using Ampere’s Law. It must be found by integrating over a “stack of loops”.
Magnetic field of a toroid A donut is an example of a “torus” (so is a coffee cup). A “toroid” is a solenoid that has been bent into a circle, with its ends connected to form a torus. The magnetic field is contained inside the toroid, and the field lines are circles centered at the center of the torus. But, unlike a solenoid, the field is not uniform: it varies with r. Ampere’s Law: The magnetic field will be constant around any circle inside the toroid. The total current passing through any such circle is NI. So: Solve for the magnetic field:
Magnetic field for cylinder with uniform current density We already know the magnetic field outside the cylinder. The enclosed current is I, so the field is the same as that for a wire (by cylindrical symmetry). For the field inside the cylinder, we first must find the current density: So at r, the enclosed current is: And Ampere’s Law gives:
Magnetic fields of sheets of current • Sketch magnetic fields for finite sheets of current. • Extend to infinite sheets of current. • Find magnetic field near an infinite sheet of current.
