Experimental ObservationsMagnetic ForceMagnetic FieldUnits of Magnetic FieldDirection of Magnetic ForceE and B DualityMagnetic Charge (I)Magnetic Charge (II)Divergence of BDivergence of CurlDivergence of CurlAmp`{e}re's LawApplication of Amp`{e}re's LawMagnetic Vector PotentialEquations for PotentialWhy use Vector Potential?From Vector A to BYet Another Vector IdentifyBack to BEECS 117Lecture 14: Static Magnetic FieldsProf. NiknejadUniversity of California, BerkeleyUniversity of California, Berkeley EECS 117 Lecture 14 – p. 1/20Experimental ObservationsConsider a pair of parallel wires carrying steadycurrents I1and I2.Last lecture we found that steady currents imply zeronet charge distribution. Therefore, there should be noelectrostatic force between these current carrying wires.But experimentally we do observe a force which tendsto be attractive if the currents are in the same directionand repulsive if the currents are in opposite direction.This new force is in fact an electrostatic force if weconsider the problem from a relativistic point of view!Even though the net charge on each current carryingconductor is zero in a static reference frame, in amoving reference frame there is net charge density andhence force.University of California, Berkeley EECS 117 Lecture 14 – p. 2/20Magnetic ForceThrough careful observations, Ampère demonstratedthat this force can be computed using the followingequationdFm=µ04πI2dℓ2× I1dℓ1׈RR2The resemblance to the Coulomb force equation isnotable. Both forces fall like 1/R2.For steady currents, ∇ · J = 0 implies that the currentsmust flow in loops. Thus we can calculate the forcebetween two loops as followsFm=IC1IC2µ04πI2dℓ2× I1dℓ1׈RR2University of California, Berkeley EECS 117 Lecture 14 – p. 3/20Magnetic FieldJust as in the case of electric forces, the concept of“action at a distance” is disturbing and counterintuitive.Thus we prefer to think of the current in loop C1generating a “field” and then we say that this fieldinteracts with the current in loop C2to generate a force.Just reordering the magnetic force equation givesFm=IC2I2dℓ2×µ04πIC1I1dℓ1׈RR2| {z }BHere loop 2 is the source and loop one is the field point.The unit of B is the tesla (T), where 1T = 104G, in termsof the CGS units of gauss (G).University of California, Berkeley EECS 117 Lecture 14 – p. 4/20Units of Magnetic FieldThe tesla (T) and gauss (G) are derived units.Since F ∝ I2µ, the units of µ are simply N · A−2. This ismore commonly known as H · m−1.The units of the magnetic field is therefore[B] = [µ]A · m · m−2= H · A · m−2Not that the units of D are C · m−2, which can be writtenas F · V · m−2From circuit theory we know that voltage is proportionalto ωLI, so LI has units ofVω. So the unit of [B] isV · s · m−2For reference, the magnetic field of the earth is only.5G, so 1T is a very large fieldUniversity of California, Berkeley EECS 117 Lecture 14 – p. 5/20Direction of Magnetic ForceDue to the vector cross product, the direction of theforce of the magnetic field is perpendicular to thedirection of motion and the magnetic fieldUse the right-hand rule to figure out the direction of Fmin any given situation.University of California, Berkeley EECS 117 Lecture 14 – p. 6/20E and B DualityFor a point charge dq, the electric force is given byFe= qEThe magnetic force for a point charge in a current loop,we haveFm= Idℓ × B = qNdℓv × BThe equations for E and B are also similar when weconsider an arbitrary current density J and chargedensity ρE =14πǫZVρ(r′)RR2dV′B =14πµ−1ZVJ(r′) ׈RR2dV′University of California, Berkeley EECS 117 Lecture 14 – p. 7/20Magnetic Charge (I)We may now compare the magnetic field to the electricfield and look for similarity and differences.In this class we shall not discuss the relativisticviewpoint that explains the link between electrostaticsand the magnetic field. Instead, we shall assume thatthe magnetic field is an entity of its own.Apparently, the source of magnetic field is movingcharges (currents) whereas the source of electric fieldsis charges. But what about magnetic charges? Is thereany reason to believe that nature should be asymmetricand give us electrical charge and not magnetic charge?University of California, Berkeley EECS 117 Lecture 14 – p. 8/20Magnetic Charge (II)If magnetic charge existed, then the argument forGauss’ law would applyISB · dS = QmWhere Qmis the amount of magnetic charge inside thevolume V bounded by surface S.But no one has ever observed any magnetic charge!So for all practical purposes, we can assume thatQm≡ 0 and so Gauss’ law applied to magnetic fieldsyieldsISB · dS = 0University of California, Berkeley EECS 117 Lecture 14 – p. 9/20Divergence of BBy the divergence theorem, locally this relationtranslates intoISB · dS =ZV∇ · BdV = 0Since this is true for any surface S, the integrand mustbe identically zero∇ · B = 0A vector field with zero divergence is known as asolenoidal fieldWe already encountered such a field since ∇ · J = 0.Such a field does not have any sources and thus alwayscurls back onto itself. B fields are thus always loops.University of California, Berkeley EECS 117 Lecture 14 – p. 10/20Divergence of CurlLet’s calculate the divergence of the curl of an arbitraryvector field A, ∇ · ∇ × ALet’s compute the volume of the above quantity andapply the divergence theoremZV∇ · ∇ × AdV =IS∇ × A · dSTo compute the surface integral, consider a new surfaceS′with a hole in it. The surface integral of ∇ × A can bewritten as the line integral using Stoke’s TheoremZS′∇ × A · dS =ICA · dℓUniversity of California, Berkeley EECS 117 Lecture 14 – p. 11/20Divergence of CurlWhere C is the perimeter of the hole. As we shrink thishole to a point, the right hand side goes to zero and thesurface integral turns into the closed surface integral.ThusZV∇ · ∇ × AdV =IS∇ × A · dS = 0Since this is true for any volume V , it must be that∇ · ∇ × A = 0Thus a solenoidal vector can always be written as thecurl of another vector. Thus the magnetic field B can bewritten asB = ∇ × AUniversity of California, Berkeley EECS 117 Lecture 14 – p. 12/20Ampère’s LawOne of the fundamental relations for the magnetic fieldis Amère’s law. It is analogous to Gauss’ law.We can derive it by taking the curl of the magnetic field∇ × B = ∇ ×14πµ−1ZVJ(r′)