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Electric Dipole in a Uniform Field Electric Dipole Moment of a System Electromagnetism II (spring term 2020) Lecture 1 Static electric and magnetic fields E Goudzovski [email protected] http://epweb2.ph.bham.ac.uk/user/goudzovski/Y2-EM2 Practical details The course homepage (linked to Canvas): http://epweb2.ph.bham.ac.uk/user/goudzovski/Y2-EM2 Notes for each lecture preliminary version: before each lecture; final version: within 24 hours after each lecture. References (from basic to more advanced): I.S. Grant and W.R. Phillips, Electromagnetism, 2nd ed. (Wiley) D.J. Griffiths, Introduction to Electrodynamics, 4th ed. (Cambridge) J.D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley) Lectures, problem sheets, continuous assessment: Twenty-two lectures: Mondays at 5pm and Fridays at 4pm. Assessed problem sheets: weeks 2, 4, 6, 8, 10. Non-assessed problems will be available. 1 Introduction Outline of the course Revision of the EM1 course: electric and magnetic fields. Physics of dielectric materials and magnetism. Maxwell’s equations. Electromagnetic waves: properties; polarisation; energy flux; propagation in dielectrics (dispersion, reflection, refraction) and conductors; wave guides. Motion of charged particles in static fields. System of units The SI system is used in this course. Fundamental drawbacks of SI wrt the Gaussian system: Need conversion factors: the electric and magnetic constants 12 6 2 0 8.9×10 C/(Vm) and 0 1.3×10 N/A . Different units of E, D, B, H fields (V/m, C/m2, T, A/m), though the electric vs magnetic field distinction is relative. 2 This lecture Lecture 1: Static electric and magnetic fields Electrostatic field and Gauss’s law Magnetic field and Ampere’s law Electric and magnetic dipoles in uniform fields 3 Electrostatic field Electric field of a point charge: A conservative vector field: can define the scalar potential Potential energy of a test charge: This also means that around any closed curve Principle of superposition: ; Equipotentials (perpendicular to field lines) Field lines (tangent to the field vector) 4 Gauss’s law Gauss’s law following from Coulomb’s law: dS Electric flux through Electric charge any closed surface within that surface Convenient to compute fields for systems with a sufficient degree of symmetry. Not sufficient in the general case: a single scalar relation and three unknowns E , E , E . x y z Coulomb law and Newton’s law of gravity both state F~1/r2: Gauss’s law applies also to the gravitational field. 5 Gauss’s law: examples 1) Infinite surface charge of density [C/m2]. Symmetry: field is perpendicular to the plate. Gauss’s law: ; Area S 2) For a planar capacitor, use superposition: between the plates; outside. 3) A uniformly charged ball of radius R and charge Q. Inside the ball, Outside the ball, 6 Magnetic field Field of a linear current element (the BiotSavart law): A solenoidal vector field: can define the vector potential The principle of superposition holds for and Magnetic charges (monopoles) do not exist; field lines are continuous 7 Ampere’s law Ampere’s law for the magnetic field produced by steady currents: Line integral of Current through the magnetic field any surface enclosed around a closed curve by the curve is the current density vector [A/m2] Sign convention: the direction of B relative to J given by the “right-hand rule”, or the “corkscrew rule”. Magnetic field is conservative in regions free from electrical currents 8 Ampere’s law: examples 1) Straight infinite wire: cylindrical symmetry. Outside the wire, For I = 1 A and r = 2 cm, B = 4×107 H·m−1·1A / (2·0.02 m) = 105 T (comparable to Earth’s magnetic field) Inside the wire, 2) Long solenoid with N turns per unit length. Outside, B=0 by considering it part of a very large toroid and using cylindrical symmetry. Inside the solenoid, 9 Electric dipole in a uniform field Electric dipole moment of a system: Here is the point of observation. For neural systems ( ), no dependence on . Electric dipole: Dipole moment: Uniform field (Ex<0): A +q F Torque on a dipole in a uniform field: F q Potential energy in uniform field: B x Dipole moment tends to align with the external electric field 10 Magnetic dipole in a uniform field Rectangular (a×b) current loop Lorentz force: in a uniform external field Force on the sides KL and MN: y Top view θ Density of charges Wire cross-section b Torque on the loop: Magnetic moment of the current loop Side view x (directions: the corkscrew rule) z L M I I a Then and Identical to electric dipole: magnetic K N moment tends to align with the field x In the static case, a current loop is equivalent to a pair of magnetic charges 11 Summary The principal laws in free space in the integral form: Gauss’s law Absence of magnetic poles Ampere’s law Potential energy of electric and magnetic dipoles in a uniform external field: 12 .
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