Stationary Magnetic field Comparison of Electric and Magnetic field Definition of Magnetic field Biot-Savart Law Ampere’s Circuital Law Application of Ampere’s Circuital Law Differential form of Ampere’s Circuital Law

Electric Field & Magnetic Field Electric forces acting at a distance through electric field. Vector field, E. Source: electric charge. Positive charge (+) and negative charge (-). Opposite charges attract, like charges repel. Electric field lines visualizing the direction and magnitude of E. Magnetic forces acting at a distance through Magnetic field. Vector field, B Source: moving electric charge (current or magnetic substance, such as permanent magnet). North pole (N) and south pole (S) Opposite poles attract, like poles repel. Magnetic field lines visualizing the direction and magnitude of B.

Test charge and electric field Definition of Test charge and electric field Magnetic poles are always found in pairs. A single magnetic pole has never been isolated. N S N S N S Define B at some point in space in terms of the magnetic force FB that the field exerts on a charged particle moving with a velocity v : The magnitude FB is proportional to the charge q and to the speed v of the particle. FB = 0 when the charged particle moves parallel to the magnetic field vector. When velocity vector makes any angle θ≠0 with the magnetic field, FB is perpendicular to both B and v. FB on a positive charge is opposite on a negative charge. The magnitude FB is proportional to sinθ.

Magnetic Fields Magnetic force Right-hand rule determine the direction of magnetic force. So the magnetic force is always perpendicular to v and B. The magnitude of the magnetic force is The electric force is along the direction of the electric field, the magnetic force is perpendicular to the magnetic field. The electric force acts on a charged particle regardless of whether the particle is moving, the magnetic force acts on a charged particle only when the particle is in motion. The electric force does work in displacing a charged particle, the magnetic force does no work when a particle is displaced.

Biot-Savart Law The analogue of Coulomb’s Law is the Biot-Savart Law r-r’ dB(r) I dℓ’ 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

Direction of Magnetic field Ampere’s Law: The magnetic field in space around an electric current is proportional to the electric current which serves as its source, just as the electric field in space is proportional to the chargewhich serves as its source. Ampere's Law states that for any closed loop path, the sum of the length elements times the magnetic field in the direction of the length element is equal to the permeability times the electric current enclosed in the loop. Direction of Magnetic field Curl your right hand around the Amperian loop, with the fingers pointing in the direction of integration. A current through the loop in the general direction of your outstretched thumb is assigned a plus sign, and a current generally in the opposite direction is assigned a minus sign.

Application of Ampere’s Law: Solenoid B I 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 solenoid is to magnetostatics what capacitor is to electrostatics I L

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 S j B dℓ