BII-1 Currents make B-fields We have seen that charges make E-fields: 201dQˆdE r4rπε=K. Currents make B-fields according to the Biot-Savart Law: 02ˆIdrdB4rµπ×=KKA where µ0 = constant = 4π×10–7 (SI units). dB is the element of B-field due to the element of current I dl . dl is an infinitesimal length of the wire, with direction given by the current. The total B-field due to the entire current is 02ˆIdBdB4rµπ×==∫∫KrKKA . This can be a very messy integral! I d l r ^ r dB I This law was discovered experimentally by two French scientists (Biot and Savart) in 1820, but it can be derived from Maxwell's equations. Example of Biot-Savart: B-field at the center of a circular loop of current I, with radius R. Here the integral turns out to be easy. 0ˆˆdrdrsin90×= =KAA 1 ⇒ 02IddB4Rµπ=A We can replace the vector integral B=∫KKdB with the scalar integral B dB=∫ because all of the dB's point in the same direction. N002IIBdB d4R 2R2Rµµππ== =∫∫A. The full field at all positions near a current loop requires very messy integrations, which are usually done numerically, on a computer. The full field looks like this: I R B r ^ dl I B Last update: 10/21/2009 Dubson Phys1120 Notes, ©University of ColoradoBII-2 Another, more difficult example of the Biot-Savart Law: B-field due to a long straight wire with current I. The result of a messy integration is 0IB(r)2rµπ= dB r This formula can be derived from a fundamental law called Ampere's Law, which we describe below. The B-field lines form circular loops around the wire. To get directions right for both these examples (B due to wire loop, B due to straight wire), use "Right-hand-rule II": With rhand, curl fingers along the curly thing, your thumb points in direction of the straight thing. ight BK Force between two current-carrying wires: Current-carrying wires exert magnetic forces on each other. Wire2 creates a B-field at position of wire 1. Wire1 feels a force due to the B-field from wire 2: on1 1 2from2FIL=×KK⇒ 02on 1 1 2 1from 2IFILBIL2rµπ== . force per length between wires = 120IIFL2µrπ= • Parallel currents attract • Anti-parallel currents repel. "Going my way? Let's go together. Going the other way? Forget you!" I dlr ^ I B I2I1Bfrom I2LF F r I1I2I1I2FF F FrLast update: 10/21/2009 Dubson Phys1120 Notes, ©University of ColoradoBII-3 Gauss's Law for B-fields B-field lines are fundamentally different from E-field lines in this way: E-field lines begin and end on charges (or go to ∞ ). But B-field lines always form closed loops with no beginning or end. A hypothetical particle which creates B-field lines in the way a electric charge creates E-field lines is called a magnetic monopole. As far as we can tell, magnetic monopoles, magnetic charges, do not exist. There is a fundamental law of physics which states that magnetic monopoles do not exist. Recall the electric flux through a surface S is defined as ESEdaΦ = ⋅∫KK. In the same way, we define the magnetic flux through a surface as BSBdaΦ = ⋅∫KK. Gauss's law stated that for any closed surface, the electric flux is proportional to the enclosed electic charge: BBE M OK ! Yes! Impossible! enc0qEdaε⋅ =∫KKv. But there is no such thing as "magnetic charge", so the corresponding equation for magnetic fields is Bda 0⋅ =∫KKv This equation, which has no standard name, is one of the four Maxwell Equations. It is sometimes called "Gauss's Law for B-fields". Ampere's Law Ampere's Law gives the relation between current and B-fields: For any closed loop L, 0enclosedBd Iµ⋅ =∫KKAvL , where Ienclosed is the current through the loop L . Last update: 10/21/2009 Dubson Phys1120 Notes, ©University of ColoradoBII-4 (It will turn out the Ampere's Law is only true for constant current I. If the current I is changing in time, Ampere's Law requires modification.) Ampere's Law for steady currents, like Gauss's Law, is a fundamental law of physics. It can be shown to be equivalent to Biot-Savart Law. We can use Ampere's Law to derive the B-field of a long straight wire with current I. B-field of a long straight wire: L = imaginary circular loop of radius r We know that B must be tangential to this loop; B is purely azimuthal; B can have no radial component toward or away from the wire. How do we know this? A radial component of B is forbidden by Gauss's Law for B-fields. Alternatively, we know from Biot-Savart that B is azimuthal. So, in this case, Also, by symmetry, the magnitude of B can only depend on r (distance from the wire): B = B(r). Bd Bd⋅ =KKAr loop L I A0(Bconst )(B||d )Bd Bd B d B2 r Iπµ⋅ ===∫∫ ∫KKAKKAA Avv vL= ⇒ 0IB2rµπ= Like Gauss's Law, Ampere's Law is always true, but it is only useful for computing B if the situation has very high symmetry. B-field due to a solenoid. solenoid = cylindrical coil of wire It is possible to make a uniform, constant B-field with a solenoid. In the limit that the solenoid is very long, the B-field inside is uniform and the B-field outside is virtually zero. Consider a solenoid with N turns, length L, and n = N/L = # turns/meter Last update: 10/21/2009 Dubson Phys1120 Notes, ©University of ColoradoBII-5 N0thru 0 0#turnsthruBd B I n I B nIµµ µ⋅ == = ⇒ =∫KKAA AvLLL ⇒ B-field inside solenoid is 00NBnILµµ==I Permanent Magnets Currents make B-fields. So where's the current in a permanent magnet (like a compass needle)? An atom consists of an electron orbiting the nucleus. The electron is a moving charge, forming a tiny current loop –– an "atomic current". In most metals, the atomic currents of different atoms have random orientations, so there is no net current, no B-field. In ferromagnetic materials (Fe, Ni, Cr, some alloys containing these), the atomic currents can all line up to produce a large net current. I I B End View: I B uniform inside L Side View I(out) B I(in) B = 0 outside loop L l Last update: 10/21/2009 Dubson Phys1120 Notes, ©University of ColoradoBII-6 In interior, atomic currents cancel: In a magnetized iron bar, all the atomic currents are aligned, resulting in a large net current around the rim of the bar. The current in the iron bar then acts like a solenoid, producing a uniform B-field inside: Why do permanent magnets sometimes attract and sometimes repel? Because parallel currents attract and anti-parallel current repel. atoms cross-section of magnetized
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