4). Ampere’s Law and Applications

4). Ampere’s Law and Applications • As far as possible, by analogy with Electrostatics • B is “magnetic flux density” or “magnetic induction” • Units: weber per square metre (Wbm-2) or tesla (T) • Magnetostatics in vacuum, then magnetic media • based on “magnetic dipole moment”

I dB(r) r-r’ r O r’ dℓ’ Biot-Savart Law • The analogue of Coulomb’s Law is • the Biot-Savart Law • Consider a current loop (I) • For element dℓ there is an • associated element field dB • dB perpendicular to both dℓ’ and r-r’ • same 1/(4pr2) dependence • o is “permeability of free space” • defined as 4p x 10-7 Wb A-1 m-1 • Integrate to get B-S Law

B B-S Law examples I q (1) Infinitely long straight conductor dℓ and r, r’ in the page dB is out of the page B forms circles centred on the conductor Apply B-S Law to get: dℓ r’ z r - r’ dB a O r q = p/2 + a sin q = cos a =

dℓ r - r’ dB r’ q r I z a dBz B-S Law examples (2) “on-axis” field of circular loop Loop perpendicular to page, radius a dℓ out of page and r, r’ in the page On-axiselementdB is in the page, perpendicular to r - r’, at q to axis. Magnitude of element dB Integrating around loop, only z-components of dB survive The on-axis field is “axial”

dℓ r - r’ dB r’ q r I z a dBz On-axis field of circular loop Introduce axial distance z, where |r-r’|2 = a2 + z2 2 limiting cases:

Magnetic dipole moment The off-axis field of circular loop is much more complex. For z >> a it is identical to that of the electric dipole m “current times area” vs p “charge times distance” m q r

B field of large current loop • Electrostatics – began with sheet of electric monopoles • Magnetostatics – begin sheet of magnetic dipoles • Sheet of magnetic dipoles equivalent to current loop • Magnetic moment for one dipole m = Ia area a for loop M = I A area A • Magnetic dipoles one current loop • Evaluate B field along axis passing through loop

I (enclosed by C) a z→-∞ z→+∞ C moI moI/2  B field of large current loop • Consider line integral B.dℓ from loop • Contour C is closed by large semi-circle which contributes zero to line integral

Np/2eo  Electric magnetic -Np/2eo Field reverses no reversal Electrostatic potential of dipole sheet • Now consider line integral E.dℓ from sheet of electric dipoles • m = IaI = m/a (density of magnetic moments) • Replace I by Np (dipole moment density) and mo by 1/eo • Contour C is again closed by large semi-circle which contributes zero to line integral

B j dℓ S Differential form of Ampere’s Law Obtain enclosed current as integral of current density Apply Stokes’ theorem Integration surface is arbitrary Must be true point wise

B B r R Ampere’s Law examples Infinitely long, thin conductor B is azimuthal, constant on circle of radius r Exercise: find radial profile of Binside and outside conductor of radius R

B I I L Solenoid Distributed-coiled conductor Key parameter: n loops/metre If finite length, sum individual loops via B-S Law If infinite length, apply Ampere’s Law B constant and axial inside, zero outside Rectangular path, axial length L (use label Bvac to distinguish from core-filled solenoids) solenoid is to magnetostatics what capacitor is to electrostatics

Relative permeability Recall how field in vacuum capacitor is reduced when dielectric medium is inserted; always reduction, whether medium is polar or non-polar: is the analogous expression when magnetic medium is inserted in the vacuum solenoid. Complication: the B field can be reduced or increased, depending on the type of magnetic medium

Magnetic vector potential For an electrostatic field We cannot therefore represent B by e.g. the gradient of a scalar since Magnetostatic field, try B is unchanged by

4). Ampere’s Law and Applications