ECE 430: Review 1

Instructions

The name for this ~~torture~~ technique is active recall. Your brain does not like being required to work like this.

It is best to avoid notes or book or Internet or friend or enemy or even mere acquaintance for the first legitimate attempt at these activites. Wait until the next day before then seeking out additional review; your brain works better this way since it can sleep on it.

1. Coulomb’s Law

Please state what you know about Coulomb’s Law. Include the equation for electric field that is defined/derived from this law.

2. Gauss’s Law

State what you know about Gauss’s Law. Include the point and integral forms. Also, state the simplification that has been most commonly been used thus far in calculating the electric field

3. Electric vector field

Consider the given three equal-valued charges. Select a grid of points around these charges and draw the E-field vector at each point, i.e. draw a vector field (not the field lines). Make the length of each \(\vec{E}\) proportional to its magnitude as accurately as you can.

4. Coulomb v. Gauss

The Coulomb’s Law version of the electric field includes a 1 / R² term, while Gauss’s simplified version does not include a radius term. Describe how these two equations are not conflicting with each other.

5. You nits

a Willy Wonka reference

Symbol

Units

Name

\(Q\)

\(\vec{F}\)

\(\vec{E}\)

\(\rho_V\)

\(\rho_S\)

\(\rho_L\)

\(\epsilon_0\)

\(W_e\)

\(V\)

\(\vec{P}\)

\(\rho_p\)

\(\vec{D}\)

\(\chi_e\)

\(\epsilon_r\)

\(\epsilon\)

\(C\)

\(E_{DS}\)

6. First of Maxwell’s equations

What is the first full equation from the set of Maxwell’s equations that we’ve covered thus far? What is the first constituitive equation?

Give the point and integral forms.

7. Coordinate systems

Set

System

Right or Left-handed

\((x, y, z)\)

\((z, x, y)\)

\((z, y, x)\)

\((\rho, \phi, z)\)

\((r, \theta, \phi)\)

8. Name it and claim it

	*ρ_v*	\(\mathbf{\vec{E}}\)	\(\mathbf{V}\)
*ρ_v*		① \(\vec{\nabla}\bullet \vec{E} = \dfrac{\rho_v}{\epsilon_0}\)	② \(\nabla^2 V = - \dfrac{\rho_v}{\epsilon_0}\)
\(\mathbf{\vec{E}}\)	③ \({\displaystyle \vec{E} = \frac{1}{4\pi\epsilon_0} \int_{\Delta v} \frac{\rho_v(r')}{R^2}\hat{a}_R\,\mathrm{d}v'}\)		④ \(\vec{E} = - \nabla V\)
\(\mathbf{V}\)	⑤ \({\displaystyle V(x, y, z) = \frac{1}{4\pi\epsilon_0} \int_{\Delta v} \frac{\rho_v(x', y', z')}{R} \,\mathrm{d}x' \mathrm{d}y' \mathrm{d}z'} \\ \text{ with } \;R\!=\!\sqrt{(x-x')^2 + (y-y')^2 + (z-z')^2}\)	⑥ \({\displaystyle \Delta V_{ab} = \int_a^b \vec{E}\bullet \mathrm{d}\vec{l}}\)
\(\mathbf{W_e}\)	⑦ \({\displaystyle W_e = \frac{1}{2} \int_{\Delta v} \rho_v V \,\mathrm{d}v}\)	⑧ \({\displaystyle W_e = \frac{\epsilon_0}{2} \int_{\Delta v} E^2 \,\mathrm{d}v}\)	⑨ \({\displaystyle W_e = \frac{1}{2} \int_{\Delta v} \rho_v V\,\mathrm{d}v}\)

ρ_v

\(\mathbf{\vec{E}}\)

\(\mathbf{V}\)

ρ_v

① \(\vec{\nabla}\bullet \vec{E} = \dfrac{\rho_v}{\epsilon_0}\)

② \(\nabla^2 V = - \dfrac{\rho_v}{\epsilon_0}\)

\(\mathbf{\vec{E}}\)

③ \({\displaystyle \vec{E} = \frac{1}{4\pi\epsilon_0} \int_{\Delta v} \frac{\rho_v(r')}{R^2}\hat{a}_R\,\mathrm{d}v'}\)

④ \(\vec{E} = - \nabla V\)

\(\mathbf{V}\)

⑤
\({\displaystyle V(x, y, z) = \frac{1}{4\pi\epsilon_0} \int_{\Delta v} \frac{\rho_v(x', y', z')}{R} \,\mathrm{d}x' \mathrm{d}y' \mathrm{d}z'} \\ \text{ with } \;R\!=\!\sqrt{(x-x')^2 + (y-y')^2 + (z-z')^2}\)

⑥
\({\displaystyle \Delta V_{ab} = \int_a^b \vec{E}\bullet \mathrm{d}\vec{l}}\)

\(\mathbf{W_e}\)

⑦ \({\displaystyle W_e = \frac{1}{2} \int_{\Delta v} \rho_v V \,\mathrm{d}v}\)

⑧ \({\displaystyle W_e = \frac{\epsilon_0}{2} \int_{\Delta v} E^2 \,\mathrm{d}v}\)

⑨ \({\displaystyle W_e = \frac{1}{2} \int_{\Delta v} \rho_v V\,\mathrm{d}v}\)

Describe the meaning of each of the 9 numbered equations, include its name if it has an official one. For the integral equations, show its equivalent version using a summation. ^[1]

①
②
③
④
⑤
⑥
⑦
⑧
⑨

9. Problem maker

Consider the list of § 9.1.

Use the random number generator tool to select a random learning objective.
Create a problem whose solution demonstrates meeting the objective. (Pay attention to the objective’s verb.)
- Alternatively, show a situation where the objective’s concept shows up in a circuit / system / structure in real life.
Solve the problem. At minimum describe the method for finding a solution and set up the solving thereof. (e.g. some geometries are difficult to compute the integrals for, but are real situations in current technologies)
Do this for 12 unique objectives, beginning a fresh page each time.

9.1. Segment 1 Learning Objectives

Chapters 01 — 09

Vector calculus and coordinate systems

Calculate the magnitude of a vector and determine a unit vector between two points.
Calculate the dot product between two vectors and use the dot product to determine the angle between vectors.
Calculate the cross product between two vectors.
Convert a vector among rectangular, cylindrical, and spherical coordinates.
Convert a function among rectangular, cylindrical, and spherical coordinates.
Calculate the divergence and gradient of a vector field.

Charges and electric fields

Use Coulomb’s Law to determine the force exerted by one electric charge on another.
Determine the electric field surrounding a point charge.
Determine the force on a charge due to an electric field.
Use superposition to determine the electric field due to multiple discrete charges.
Use integrals to calculate the electric field near approximately one-dimensional and two-dimensional regions of uniform charge density.

Electric fields near objects

Use Gauss’s Law to calculate the electric field in the vicinity of highly symmetric hollow three-dimensional objects.
Use Gauss’s Law to calculate the electric field in the vicinity of highly symmetric solid or thin-walled three-dimensional objects.

Potentials and voltage

Calculate the total electric potential energy contained within a system of discrete or continuous charges.
Calculate the voltage between two points due to an electric field.
Calculate the absolute potential in a region near discrete or continuous charges.
Calculate the voltage at all points near a charged object.
Calculate the electric field and charge density given the voltage.

Flux density and capacitance

Explain how electric flux density, polarization, and electric fields are related to each other.
Calculate surface charge density, polarization charge density, electric flux density, polarization, and electric field and sketch it for a parallel plate capacitor containing a combination of vacuum and dielectric materials.
Sketch and calculate the electric field near a conductive surface.
Analyze and design parallel plate capacitors and the charge they contain for a given voltage.
Calculate the parasitic capacitance of other shapes.
Calculate the breakdown voltage of a capacitor given its dimensions and dielectric material.

10. Cohere into a connected whole

Summarize the course’s content so far: Draw a concept map of the terminology, concepts, and connections.

Begin first on paper, brain dumping any words and ideas before you begin to arrange and group them.

Examples electronics-world, voltage-current, semiconductor-physics.
Use the free tool Cmap Cloud, or draw your own in a similar style.

1. The summation version is also how you commonly compute (approximate) the solution in a computer program.