IA: 1P3: Electromagnetics Flashcards
How are electric flux density and electric field strength related?
D = electric flux density
E = electric field strength
What is the equation for the magnitude of the electric flux density, D, at a distance r from a point charge?
What is the equation for the magnitude of the electric field strength, E, at a distance r from a point charge?
What are the equations for the electric flux density and electric field strength at a distance r from an infinitely long wire?
ρ = static charge density (per unit length)
What are the equations for the electric flux density and electric field strength at a distance r from an infinitely large plane?
α = static charge density (per unit area)
What is Gauss’ law?
The total electric flux D passing through any closed surface S equals the charge Q enclosed by the surface. The surface is called a Gaussian surface.
What is the equation for Gauss’ law when the gaussian surface is always perpendicular to the flux?
Q = DA
What is the equation for Gauss’ law when the gaussian surface is NOT always perpendicular to the flux?
What is an equipotential?
A line or a surface of constant potential. A point charge can be moved along it without having to do any work against electric fields. They are always perpendicular to the electric field lines (because if a charge travels perpendicular to an electric field then it isn’t doing any work with or against the field).
What can an equipotential be replaced by?
Any equipotential can be replaced by a conducting surface. Conductors are equipotentials, with the electric field lines ending at right angles to their surface.
What is the method of images?
Use the diagram shown as an example
The method of images is a technique used to solve problems involving conducting boundaries - especially when dealing with point charges near conductors like planes or spheres. It uses the idea that an equipotential can be replaced by a conductor, in reverse. You can construct a system of charges which has, as one of its equipotentials, the same plane as that occupied by a conductor. The second imaginary charge is known as the “image charge”. Once you have added the “image charges” you can calculate the electric field produced by using superposition from the fields arising from the charges.
How can you use the method of images to determine the electric field between the point charge and the conductor?
A metal sphere of radius 5mm is centred at a height of 10mm from an earthed conducting sheet. The electrical breakdown strength of air, |Eₘₐₓ|, is 3x10⁶ Vm⁻¹. Determine the maximum voltage and charge which may be applied to the sphere without breakdown occuring.
Qₘₐₓ = 7.5nC
Vₘₐₓ = 9000 V
Take note of the step where they substitute in the values for r (superposition step)
What is the equation for the energy stored in a capacitor?
½CV²
Derive the expression for the energy stored in a capacitor (½CV²)
What is the equation linking the electric field strength and voltage in a parallel plate capacitor?
V = Ex
x = separation of the 2 plates
What is the electrostatic force (in a direction x) given by?
Determine the force acting between the 2 plates of a capacitor
½qE
Determine the force F acting on the dielectric
Derive the “virtual work” equation
What is the equation for the magnetic flux passing through a surface?
What is the Biot-Savart law?
a short element of wire of length dl carrying a current I produces a magnetic flux density δB at a position described by the vector r. The direction given by the cross product between the vector dl of the current in the wire at 1 and the position given by the vector r.
What is the equation for the magnetic flux density around an infinitely long current carrying wire?
What is the equation for the Biot-Savart law?
r³ in the denomindator rather than r² as the r in the numerator is NOT a unit vector
How do you find the direction of the magnetic flux density around a straight current carryign wire?
The right hand rule
What is the “circulation of B”?
The circulation of B is:
Note: The “x” is just a multiplication sign not a cross product.
What is Ampere’s law?
The circulation of B around any closed path C is μ₀I
Where I is the total current enclosed by the path
What is the equation for Ampere’s law?
Where I is the total current enclosed by the path
What is an amperian path?
The closed path chosen to evaluate the line integral in ampere’s law
What is Faraday’s law of electromagnetic induction?
Include the equation
The emf induced is proportional to the rate of change of the magnetic flux linking the circuit. It is expressed mathematically by:
The minus sign comes from Lenz’s law
What is the self inductancve of a circuit defined as?
φ’ = the total magnetic flux linking the circuit through N turns
φ = the magnetic flux linking each turn
I = the current through the circuit generating the flux
What is the unit of inductance?
Henry (H)
What is the equation for induced voltage (Faraday’s law) in terms of inductance?
What are inductance and capacitance useful quantities for describing the characteristics of electrical circuits?
They depend only on the geometry of the circuit
What are the 4 useful steps (most of the time) for solving inductor problems?
- Determine the current along the conductor
- Use Ampere’s law to calculate the magnetic flux density B produced by the current I along the conductor
- Calculate the total flux linking the circuit
- Calculate the inductance
Determine the self-inductance per unit length of a coaxial cable
Lₗ = μ₀ln(b/a) / 2π
Determine the capacitance per unit length of an air filled coaxial cable
Cₗ = 2πε₀ / ln(b/a)
How are the capacitance per unit length and the inductance per unit length of a coaxial cable related?
What is Z₀
It is the “characteristic impedance of free space”. It is a fundamental constant that represents the impedance that an electromagnetic wave encounters when it propagates through free space. It is ~377Ω
What is mutual inductance?
When two electrical circuits are close to one another, a current I₁ in one may cause magnetic flux φ₂ to link the other
What is the equation for mutual inductance?
M = mutual inductance
Determine the mutual inductance, M
Why is magnetic energy stored in current-carrying circuits?
Whenever a current is established in an electrical circuit, work must be done by the voltage sources against the induced emfs which are created by the changing magnetic fields as the currents are built up
What is the equation for the magnetic energy stored in a system with 2 inductors?
What does each part of this equation mean?
How is this expression derived?
What is the force acting between 2 inductors?
What is the skin effect?
Inside a current-carrying wire, there are circular lines of magnetic flux. If the current is a high-frequency AC current, these magnetic flux lines will oscillate at that same high frequency. According to Faraday’s law, this changing magnetic flux induces an emf within the wire. By Lenz’s law, the direction of this emf opposes the original change in current, which leads to the formation of eddy currents that partially cancel the original current in the wire’s interior.
This opposing emf is strongest at the centre of the wire, which reduces the current flow there. As a result, the original current tends to flow near the surface of the wire, within a thin layer of thickness δ, known as the skin depth. Because the current is confined to a smaller cross-sectional area, the resistance of the wire increases. The higher the frequency, the thinner the skin depth becomes.
What is the formula for skin depth?
σ = conductivity
f = frequency of current
μ₀ = permeability of free space
ρ = resistivity
When the skin effect occurs, what is the equation for the resistance of the wire?
When is the skin effect significant and when can it be ignored?
- If δ > r, skin effect can be ignored. In this case the skin depth is greater than the radius of the wire and so the area through which the current flows is unaffected. Therefore the resistance of the wire does not change
- If δ < r, skin effect is significant. In this case the skin depth is less than the radius of the wire, therefore the area through which the current flows reduces and so the resistance of the wire increases
δ = skin depth
r = radius of wire
Why is it difficult to generate high magnetic flux densities in air-filled coils?
To generate a high magnetic flux density you must use a very high current, but this can case some issues:
* Resistive heating in the wires
* The magnetic flux density will exert forces (F = BIL) on the current-carrying wires, placing them under tension
Therefore if the current is high enough, these effects will break the wires in the coil
Alternatively you could increase the number of turns per unit length, however this would require using a thinner diameter wire which would:
* Increase the resistance
* Be mechancically weaker
* Only be able to sustain a smaller current
Without increasing the current or the number of turns in the coil, how can you incrase the magnetic flux density of a coil?
Fill the coil with a magnetic material
How does filling a coil with a magnetic material increase the magnetic flux density?
A magnetic material contains many randomly oriented magnetic domains. When subject to an external magnetic flux density these magnetic domains are aligned. This can be thought of as intially random atomic current loops, Iₘ, being brought into line by the field produced in the coil. The overall effect is that the atomic currents Iₘ will produce a magnetic flux density Bₘ which reinforces the field from the coil, and results in a much higher total magnetic flux density. B = Bᵢ + Bₘ
What is the circulation of B around an air filled coil of wire?
μ₀NI
The “total” current enclosed is NI and so the circulation is μ₀NI
What is the equation for the circulation of B around a coil of wire filled with a magnetic material?
What is the equation for the atomic current in a linear magnetic material?
χₘ = magnetic susceptibility
What is a linear magnetic material?
A magnetic material is linear when the atomic currents are proportional to the total current in the coil aligning them
What is the equation for the circulation of B around a coil of wire filled with a linear magnetic material?
Bl = μ₀μᵣNI
What is μᵣ?
Include equation
It is the relative permeability of the material
What is the magnetic field intensity, H?
It is defined such that the circulation of H around a closed loop equals the total current in the wires linking the loop
What is the quantity NI often called?
The magnetomotive force (mmf)
What are the units of the magnetic field intensity, H?
A m⁻¹ (sometimes called Oe, oersted)
How can the magnetic flux density be defined in terms of magnetic field intensity?
What is ampere’s law in terms of magnetic field intensity?
What is the concept of magnetic flux conservation?
Lines of magnetic flux density always form closed loops. If there is a confined region of space where the B lines are converging/contained/confined, then the flux of B across areas A₁ and A₂ will be the same. Hence, if the flux travels from a magnetic material into another material without significant divergence, then the flux will be conserved
How can a magnetic field be created in a thin air gap?
Due to the conservation of magnetic flux, when magnetic flux density travels from one material to another or across a thin gap, B will be conserved as long as it passes through the same cross sectional area. This means a magnetic field can be created inside an air gap.
What is fringing?
Fringing is when the magnetic flux lines bend outwards at edges/corners/bends. In most scenarios the assumption is made that there are no fringing magnetic fields
When considering the magnetic flux density, how can we treat magnetically susceptible materials?
We can treat them like wires in a magnetic circuit
- It is assumed that we can ignore fringing effects at sharp corners
- It is assumed that high μᵣ material effectively contains the mnagnetic flux density
If A₁ = A₂, what happens to the magnetic flux density passing through this system?
If this is a linear magnetic material, determine the magnetic flux density in the gap (Bg)
Amperes law can be expressed in terms of H or in terms of B, when can you use the equations below?
IMPORTANT
- You can always use the equation in terms of H
- The equation in terms of B is only valid in free space!!
