Two conducting strips, having infinite length in the z direction, lie in the xz plane. One occupies the region d/2 < x < b + d/2 and carries surface current density $\mathbf{K}=K_{0} \mathbf{a}_{z};$ the other is situated at −(b + d/2) < x < −d/2 and carries surface current density $-K_{0} \mathbf{a}_{z}.$ (a) Find the force per unit length in z that tends to separate the two strips. (b) Let b approach zero while maintaining constant current, $I=K_{0} b,$ and show that the force per unit length approaches $\mu_{0} I^{2} /(2 \pi d) \mathrm{N} / \mathrm{m}.$

The measure of $\angle P$ is $3 \frac { 1 } { 2 }$ times the measure of $\angle Q$. If $\angle P$ and $\angle Q$ are complementary, what is $m \angle P$ in degrees?

Let P be a real matrix that is normal and has real eigenvalues. Prove that P is symmetric.

Express the thought of each sentence in no more than four words.

The paper money that they have been using as a medium of exchange is not worth anything.

The electrostatic force between the top and the bottom of this thundercloud is ________.

Expand $\sum_{k=1}^{n} k$.

Estimate the total cost of the following items for your dorm room:

Loft bed               $159.95
Beanbag chair$49.95
Storage cubes          $29.95
Lava lamp$19.95

If possible, maximize and minimize z subject to the given constraints.

$$z=8 x+3 y$$

$$\begin{aligned} 4 x+y & \geq 12 \\ x+2 y & \geq 6 \\ x \geq 0, y & \geq 0 \end{aligned}$$

Determine the solution set. Explain each step and graph your solution.

21 x + 28 < 10 - 3x

State whether the inequality describes the empty set, all real numbers, two intervals of real numbers, or one interval of real numbers.

$x<-3$ or $x>2$

An insect's legs and wings are attached to its (a) head (b) thorax $(c)$ abdomen (d) labrum.

Minimize $C=5 x+4 y$ with the constraints

$$\left\{\begin{aligned} 3 x+4 y & \geq 32 \\ x+4 y & \geq 24 \\ x \leq 12, y & \leq 15 \end{aligned}\right.$$

Use mathematical induction to show that the given statement is true.

$$100 n \leq n ^ { 2 } \text { for all } n \geq 100$$

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Quadrilateral $A B C D$ is inscribed in circle $P$. If $\angle A$ is an acute angle, what type of angle is $\angle C$?