## Series Notation 2

• This problem is used in the following sequences
• assignment Series Notation 1

assignment Homework

##### Series Notation 1

Power Series Sequence (E&M)

Static Fields 2023 (6 years)

Write out the first four nonzero terms in the series:

1. $\sum\limits_{n=0}^\infty \frac{1}{n!}$

2. $\sum\limits_{n=1}^\infty \frac{(-1)^n}{n!}$
3. $$\sum\limits_{n=0}^\infty {(-2)^{n}\,\theta^{2n}}$$

• assignment Mass of a Slab

assignment Homework

##### Mass of a Slab
Static Fields 2023 (6 years)

Determine the total mass of each of the slabs below.

1. A square slab of side length $L$ with thickness $h$, resting on a table top at $z=0$, whose mass density is given by $$\rho=A\pi\sin(\pi z/h).$$
2. A square slab of side length $L$ with thickness $h$, resting on a table top at $z=0$, whose mass density is given by $$\rho = 2A \Big( \Theta(z)-\Theta(z-h) \Big)$$
3. An infinitesimally thin square sheet of side length $L$, resting on a table top at $z=0$, whose surface density is given by $\sigma=2Ah$.
4. An infinitesimally thin square sheet of side length $L$, resting on a table top at $z=0$, whose mass density is given by $\rho=2Ah\,\delta(z)$.
5. What are the dimensions of $A$?
6. Write several sentences comparing your answers to the different cases above.

• assignment Total Charge

assignment Homework

##### Total Charge
charge density curvilinear coordinates

Integration Sequence

Static Fields 2023 (6 years)

For each case below, find the total charge.

1. A positively charged (dielectric) spherical shell of inner radius $a$ and outer radius $b$ with a spherically symmetric internal charge density $$\rho(\vec{r})=3\alpha\, e^{(kr)^3}$$
2. A positively charged (dielectric) cylindrical shell of inner radius $a$ and outer radius $b$ with a cylindrically symmetric internal charge density $$\rho(\vec{r})=\alpha\, \frac{1}{s}\, e^{ks}$$

• assignment Power Series Coefficients A

assignment Homework

##### Power Series Coefficients A
Static Fields 2023 (6 years) Use the formula for a Taylor series: $f(z)=\sum_{n=0}^{\infty} \frac{1}{n!} \frac{d^n f(a)}{dz^n} (z-a)^n$ to find the series expansion for $f(z)=e^{-kz}$ to second order around $z=3$.
• assignment Power Series Coefficients B

assignment Homework

##### Power Series Coefficients B
Static Fields 2023 (6 years) Use the formula for a Taylor series: $f(z)=\sum_{n=0}^{\infty} \frac{1}{n!} \frac{d^n f(a)}{dz^n} (z-a)^n$ to find the series expansion for $f(z)=\cos(kz)$ to second order around $z=2$.
• assignment Volume Charge Density

assignment Homework

##### Volume Charge Density
Static Fields 2023 (6 years)

Sketch the volume charge density: $$\rho (x,y,z)=c\,\delta (x-3)$$

• assignment Spherical Shell Step Functions

assignment Homework

##### Spherical Shell Step Functions
step function charge density Static Fields 2023 (6 years)

One way to write volume charge densities without using piecewise functions is to use step $(\Theta)$ or $\delta$ functions. If you need to review this, see the following link in the math-physics book: https://paradigms.oregonstate.eduhttps://books.physics.oregonstate.edu/GMM/step.html

Consider a spherical shell with charge density $\rho (\vec{r})=\alpha3e^{(k r)^3}$ between the inner radius $a$ and the outer radius $b$. The charge density is zero everywhere else. Use step functions to write this charge density as a single function valid everywhere in space.

• assignment Divergence

assignment Homework

##### Divergence
Static Fields 2023 (6 years)

Shown above is a two-dimensional vector field.

Determine whether the divergence at point A and at point C is positive, negative, or zero.

• group Magnetic Field Due to a Spinning Ring of Charge

group Small Group Activity

30 min.

##### Magnetic Field Due to a Spinning Ring of Charge
Static Fields 2023 (7 years)

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in small groups to use the Biot-Savart law $\vec{B}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})\times \left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}$ to find an integral expression for the magnetic field, $\vec{B}(\vec{r})$, due to a spinning ring of charge.

In an optional extension, students find a series expansion for $\vec{B}(\vec{r})$ either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

• assignment Curl

assignment Homework

##### Curl
Static Fields 2023 (6 years)

Shown above is a two-dimensional cross-section of a vector field. All the parallel cross-sections of this field look exactly the same. Determine the direction of the curl at points A, B, and C.

• Static Fields 2023 (6 years)

Write (a good guess for) the following series using sigma $\left(\sum\right)$ notation. (If you only know a few terms of a series, you don't know for sure how the series continues.)

1. $1 - 2\,\theta^2 + 4\,\theta^4 - 8\,\theta^6 +\,\dots$

2. $\frac14 - \frac19 + \frac{1}{16} - \frac{1}{25}+\,\dots$