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 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 : can define the

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 BiotSavart law):

A solenoidal vector field: can define the

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×107 H·m−1·1A / (2·0.02 m) = 105 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 10 Magnetic dipole in a uniform field Rectangular (a×b) current loop : 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:

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