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Electromagnetism Notes

A summary of all necessary exam knowledge, with focus on equations, de...
Academic year: 2018/2019
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CQ
Ciaran Quigley
University of Bath
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ELECTROMAGNETISM Electromagnetism is primarily involving

EB (( rr ,t) The Electric Field,t) The Magnetic Field Both are continuous functions of position and time t defined over a given field r Other vector fields such as J ( r ,t) Current Density Scalar Fieldsρ( r ) Electric Charge Density φ( r ) Electrostatic Potential

Vector Field Scalar Field Maxwell’s Equations

Where t is time and ε 0 and μ 0 are constants. All Electromagnetic Fields are caused by Charges Assume a charge carrying particle is a point charge

Coulomb’s Law

The force between two stationary charges is given as The Electric Field by a unit positive charge placed at E ( r ) is the force experienced r. ∴ Principle of superposition states that the E-field is the sum of all individual charge fields Gauss’ Law PROOF

∴,

So

Total charge within S

Integral form of Gauss’ Law Differential form of Gauss’ Law This states that the direct source of electric field is electric charge True for all r, not just within V. Away from charge, Does not mean E=0∇.E = 0. The work done in moving a charge Q around any closed path C in an electrostatic field is zero done = And since Work done = 0. Apply Stoke’s Theorem S is any surface enclosed within the path C electrostatic fields E and s(r) are solutions to Since ∇X∇ψ = 0 for any scalar field ψ EΦ(r) is the electrostatic potential can be written as Then Poisson EquationThis governs electrostatics. If we want a solution in a region away from charges where ρ(r) = 0, then

The Laplace Equation

Energy in an Electrostatic Field Solutions to the Poisson Equation Finding φ(r) by inspection It follows the electrostatic potential due to a single charge at the origin is For a general charge distribution, ρ(volume V’, r’ ) in

Let each charge move with constant velocity v1 and v We would find an additional F m

B charge Q( r ) is the Magnetic Field due to a single 1 at the origin, moving at constant velocity v 1. Units are Ns/(cm) or Tesla B = 0 if v 1 = 0 or if v 1 is parallel to

Where ΦB = magnetic flux through circuit C. ε is an electric potential difference not derivedfrom Coulomb’s Law, it must be due to an extra force Assume that the induced current consists of a F’ single charge Q moving around C.

Either 1. Circuit is static; B is static; circuit moves B changes with time In 1: u is extra velocity of Q due to motion of circuit F’ =Q( u X B ) In 2: electric field u =0 so force must be due to an extra F’ =Q E’ In 2: Stokes’ Theorem Since S is arbitrary, we have If electrostatic field Esuperposition gives the total electric field ass is also present,

So Thus 3 rd of Maxwell’s Equations A time-varying B is a rotational source of E. The Maxwell-Ampere Law To see that is incomplete, curl both sides.

Only correct when Conservation of charge means:Rate of flow of charge across S = Rate of change of charge in V Divergence theorem used Maxwell Ampere Law displacement current density ^vacuum

Electromagnetic waves in a vacuum The wave Equation Maxwell’s equations 1-4 are coupled, but may be decoupled to give separate equations for E and B. Taking the curl of Maxwell’s second Equation: Gives us Vector Identity Subbing in Maxwell’s Equations gives us This is the wave equation for E in a source-freevacuum, so ρ=0, J=0. Taking the curl of Equation 4, Identical in form to the wave equation for E.

Implies both have the same form of solution. Wave solutions for E Plane wave solutions Substitute into the wave equation for E.

∴: ∴: RHS: Wave equation is ∴ The plane waves are solutions to the wave equation if the wavenumber k and angular frequency ω are related: V.W stated for any wave,, the phase speed

∴ light is a form of EM wave Monochromatic EM waves Consider an electric plane wave travelling in the +z

∴

In a source-free vacuum, ρ( r ) = 0 so ∇·E=

KThe electric field in a plane wave in source-z can’t equal zero, so Ez=0 for this wave. free vacuum is transverse B -wave is also transverse, and perpendicular to phase. E. In a source-free vacuum, B and E are in Using Where

E X B points in the direction of energy flow. This is true for all EM fields. Maxwell’s Equations require But nothing prevents rotation of E and B perpendicular to wavevector k. H-fields and wave impedance H-fields are often used to describe magnetic fields in materials. In a vacuum: In terms of H, the plane polarized wave moving in the +z is ∴

The plane of polarization is (Eβ=π(+2nπ) also gives plane polarization of 0 ,k) plane

Circular Polarisation

So for we have Therefore |rotates in the transverse plane. The endpoint E |=E 0 but the direction of E 0 of E 0 traces out a circle.

E-vector rotates clockwise with angular frequency ω “right-circularly” polarised wave Where is a “left circularly” polarised wave polarisation also has Elliptical Polarisation, where the tip of the E-vector traces an ellipse

Natural Light Can be considered unpolarised, better to say randomly polarised with a rapidly and randomly changing polarisation state Polarisers

Only the component of Ethrough 0 parallel to TA gets

Transmitted intensityIntensity = average power per unit area So Max intensity is at θ= Malus’ Law IIinc= intensity of incident light Circularly Polarised lighttr= intensity of transmitted light rapidly 0 has constant magnitude but is rotating ∴

MARCINS PART

What’s wrong with Maxwell’s Equations in Vacuum? The derived equations are correct, but: EM waves propagate through media and induce oscillations of atom cores and electrons

Temporary dipoles and/or flow of currents in materials in response to E and B fields EM waves hit boundaries, we need to describetheir reflection/transmission Considering Gauss’ Law: The charge density in Gauss’ Law includes all charges (intentional or unintentional) Electric field is the total electric field rather than the field caused/controlled by us Atomic Polarizibility For non-ionizing fields, the induced dipole moment is: What is the polarizibility of a atom with a point nucleus (+q) and uniformly charged cloud (-q) of radius r

External field shifts inner charge q by d =dz^ E negative sphere:ext is balanced by the field charge q from the

∴ ∴ since ∴ Where V is the volume of the atom.

Far away, a polarized atom/molecule can be described as a tiny dipole induced by the applied E-field.

Polarization charge

Considering a large number of dipoles within adilectric medium, in a differential volume. All wholly enclosed dipoles contribute no net charge within V. What is the induced polarization charge? (cut by surface and in volume?) Where P is the polarization density

Relative permittivity of a vacuum is 1, relative permittivity of air is close to 1 (and used as 1) Dilectric (breakdown) strength is the max Voltage a material can sustain before breaking.

Considering a parallel plate capacitor with a dilectric permittivity.

Dilectrics lower the electric field caused by charge Q. Effectively, it can accommodate charge before reaching the same field as in vacuum and hence the capacitance is increased times Energy in a dilectric Parallel plate capacitor in vacuum: With a Dilectric:

Some electric field is cancelled by bound charges, so more work needs to be spent on moving (free) charges to achieve a set V.

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Electromagnetism Notes

Module: Electromagnetism 1 (PH20014)

University: University of Bath

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ELECTROMAGNETISM
Electromagnetism is primarily involving
E(r,t) The Electric Field
B(r,t) The Magnetic Field
Both are continuous functions of position r
and time t defined over a given field
Other vector fields such as
J(r,t) Current Density
Scalar Fields
ρ(r) Electric Charge Density
φ(r) Electrostatic Potential
Vector Field Scalar Field
Maxwell’s Equations
Where t is time and ε0 and μ0 are constants.
All Electromagnetic Fields are caused by
Charges
Assume a charge carrying particle is a point
charge
Coulomb’s Law
The force between two stationary charges is
given as
The Electric Field E(r) is the force experienced
by a unit positive charge placed at r.
∴
Principle of superposition states that the E-
field is the sum of all individual charge fields
Gauss’ Law
PROOF
∴,
So
Total charge within S

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