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Transcript
Lecture 9 : Maxwell’s Equations for 𝐵 • Determining magnetic fields • Maxwell’s Equations for 𝐵 • Magnetic vector potential Recap (1) • Magnetic forces are a fundamental phenomenon of nature generated by electric currents • We say that a current (or magnet) sets up a magnetic field 𝐵 around it, which causes another current (or magnet) to feel a force Recap (2) • The Biot-Savart Law and Ampere’s Law are two equivalent methods for describing how magnetic fields 𝐵 are generated by a current 𝐼 Ampere’s Law for 𝐵 Biot-Savart Law for 𝐵 𝑑𝐵 𝑃 𝑟 𝐼 𝜇0 𝐼 𝑑 𝑙 × 𝑟 𝑑𝐵 = 4𝜋 𝑟 2 𝐼 𝑑𝑙 𝐵. 𝑑 𝑙 = 𝜇0 𝐼𝑒𝑛𝑐 Determining magnetic fields • We now introduce the current loop (or solenoid), which is equivalent to a bar magnet Similar 𝐵-field lines Determining magnetic fields • What is the 𝐵 field at the centre of a current loop? 𝐼 𝑟 𝐵 • Biot-Savart Law : 𝑑𝐵 = 𝜇0 𝐼𝑑 𝑙 × 𝑟 4𝜋 𝑟 2 • The contribution 𝑑𝐵 from all current elements reinforce 𝜇 𝐼 • Hence 𝐵 = 𝑑𝐵 = 0 2 𝑑𝑙 4𝜋 𝑟 where 𝑑𝑙 = 2𝜋𝑟 • We find: 𝐵= 𝜇𝑜 𝐼 2𝑟 Determining magnetic fields • Now, consider a solenoid of length 𝑙 with 𝑁 turns carrying current 𝐼. What is the 𝐵-field in the middle? Loop for Ampere’s Law • Apply Ampere’s Law 𝐵 . 𝑑𝑙 = 𝜇0 𝐼𝑒𝑛𝑐 to the closed loop shown • 𝐵. 𝑑 𝑙 = 𝐵 × 𝑙, neglecting 𝐵 outside coil • 𝐼𝑒𝑛𝑐 = 𝑁 × 𝐼 • Hence 𝐵= 𝜇0 𝑁 𝐼 𝑙 Maxwell’s Equations for 𝐵 • We have already met Maxwell’s two equations for 𝝆 electrostatics: 𝜵. 𝑬 = and 𝜵 × 𝑬 = 𝟎. What are 𝜺𝟎 the equivalent relations for the magnetic field 𝐵? • Take the divergence (𝛻.) of the Biot-Savart law: 𝛻. 𝑑𝐵 = 𝛻. 𝜇0 𝐼 𝑑 𝑙×𝑟 4𝜋 𝑟 2 = (where we have used the relation 𝛻 𝜇0 𝐼 𝑑𝑙. 𝛻 4𝜋 1 𝑟 = ×𝛻 1 𝑟 =0 𝑟 ) 𝑟2 • We deduce the fundamental result for any current distribution, 𝜵. 𝑩 = 𝟎 Maxwell’s Equations for 𝐵 • If 𝛻. 𝐵 = 0 everywhere, what does this mean physically? • Divergence theorem implies 𝐵. 𝑑 𝐴 = 0 : there is no inward/outward flux of magnetic field from any point Maxwell’s Equations for 𝐵 • Hence there are no magnetic charges (“monopoles”) and magnetic field lines never end and always form closed loops Maxwell’s Equations for 𝐵 • The final Maxwell’s Equation uses a fundamental result in vector calculus known as Stokes’ Theorem • If 𝑆 is any surface bounded by a closed loop 𝐿, then for any vector field 𝐵 we can say: 𝐵. 𝑑 𝑙 = Integral around closed loop (𝛻 × 𝐵). 𝑑 𝐴 Integral over surface Maxwell’s Equations for 𝐵 • But from Ampere’s Law, we have 𝐵. 𝑑𝑙 = 𝜇0 𝐼𝑒𝑛𝑐 = 𝜇0 𝐽. 𝑑𝐴 where 𝐽 is the current density of the enclosed current 𝐼𝑒𝑛𝑐 • We hence deduce: (𝛻 × 𝐵). 𝑑 𝐴 = 𝜇0 𝐽. 𝑑 𝐴 𝜵 × 𝑩 = 𝝁𝟎 𝑱 • This describes how magnetic fields 𝐵 are produced by current 𝐽 Maxwell’s Equations for 𝐵 • If 𝛻 × 𝐵 = 𝜇0 𝐽 everywhere, what does this mean physically? • The curl (𝛻 ×) of a vector field describes its tendency to circulate around a point, so this relation is saying that magnetic fields circulate around currents Maxwell’s Equations for 𝐵 • We have now deduced all four Maxwell’s Equations for the electric field 𝐸 and magnetic field 𝐵 generated by stationary charge density 𝜌 and current density 𝐽: 𝝆 𝜵. 𝑬 = 𝜺𝟎 𝜵×𝑬=𝟎 𝜵. 𝑩 = 𝟎 𝜵 × 𝑩 = 𝝁𝟎 𝑱 • In later lectures, we will discuss how these equations must be modified to account for time-varying situations Maxwell’s Equations for 𝐵 • The equations can be expressed in either differential form or integral form, and these are equivalent Differential form 𝜌 𝛻. 𝐸 = 𝜀0 𝛻×𝐸 =0 𝛻. 𝐵 = 0 𝛻 × 𝐵 = 𝜇0 𝐽 Integral form Name or effect 𝑄𝑒𝑛𝑐 𝐸. 𝑑𝐴 = 𝜀0 Gauss’s Law (Coulomb’s Law) 𝐸. 𝑑𝑙 = 0 Electrostatic potential 𝐸 = −𝛻𝑉 𝐵. 𝑑𝐴 = 0 No monopoles, magnetic potential 𝐵 = 𝛻 × 𝐴 𝐵. 𝑑𝑙 = 𝜇0 𝐼𝑒𝑛𝑐 Ampere’s Law (Biot-Savart Law) Magnetic vector potential • In electrostatics, we saw that the electric field 𝐸 could be generated from a scalar field 𝑉(𝑟) called the electrostatic potential, such that 𝑬 = −𝜵𝑽 • This relation always satisfies 𝛻 × 𝐸 = 0, and the zero-point of 𝑉 is arbitrary (i.e., an arbitrary constant can be added) • The magnetic field 𝐵 can likewise be generated from a magnetic potential, however this is a vector field 𝐴(𝑟) such that 𝑩 = 𝜵 × 𝑨 • This relation always satisfies 𝛻. 𝐵 = 0, since 𝛻. (𝛻 × 𝐴) = 0 Magnetic vector potential • The magnetic vector potential 𝐴 also has a freedom in its definition: we can add an arbitrary constant to 𝛻. 𝐴 • We choose to add a constant such that 𝜵. 𝑨 = 𝟎 • (More advanced comment:) To see why, we need to use a relation from vector calculus that comes from applying two curls! That relation is 𝛻 × 𝛻 × 𝐴 = 𝛻 𝛻. 𝐴 − 𝛻. 𝛻 𝐴 • Substituting in 𝛻 × 𝐵 = 𝜇0 𝐽 we find 𝛻 2 𝐴 − 𝛻 𝛻. 𝐴 = −𝜇0 𝐽 • Note that if a constant is added to 𝛻. 𝐴, the gradient 𝛻(𝛻. 𝐴) does not change, hence the generation of 𝐴 from currents 𝐽 is not affected Magnetic vector potential • There are hence analogous relations between the generation of electrostatic potential 𝑉 from charge density 𝜌, and magnetic vector potential 𝐴 from current density 𝐽 • These are useful for higher physics development Electrostatic potential 𝑉: Magnetic potential 𝐴: 𝜌 𝛻 𝑉=− 𝜀0 𝛻 2 𝐴 = −𝜇0 𝐽 2 Summary • The magnetic field 𝐵 satisfies the fundamental relation 𝜵. 𝑩 = 𝟎. In physical terms, 𝐵-field lines form closed loops • The generation of the 𝐵-field from currents is described by 𝜵 × 𝑩 = 𝝁𝟎 𝑱, completing Maxwell’s Equations • The magnetic field can be generated from a vector potential 𝐴, via 𝐵 = 𝛻 × 𝐴