A 0.200-A current is charging a capacitor that has circular plates 10.0 cm in radius. If the plate separation is 4.00 mm (a) what is the time rate of increase of electric field between the plates? (b) What is the magnetic field between the plates 5.00 cm from the center?

Answer the following questions about force on a moving charge. A particle with a positive unit charge (q=1) enters a constant magnetic field B=i+j with a velocity v=20k. Find the magnitude and direction of the force on the particle. Make a sketch of the magnetic field, the velocity, and the force.

$$\begin{array} { l } { \text { Suppose that } F \subset K \subset E \text { are fields and } E \text { is the splitting field of } } \\ { \text { some polynomial in } F [ x ] . \text { Show, by means of an example, that } K } \\ { \text { need not be the splitting field of some polynomial in } F [ x ] . } \end{array}$$

When a charged particle moves at an angle of

$$25^\circ$$

with respect to a magnetic field, it experiences a magnetic force of magnitude F. At what angle (less than

$$90^\circ$$

) with respect to this field will this particle, moving at the same speed, experience a magnetic force of magnitude 2F?

A current, I= 15 A, is directed along the positive x-axis and perpendicular to a uniform magnetic field. The conductor experiences a magnetic force per unit length of 0.12 N/m in the negative y-direction. Calculate the magnitude and direction of the magnetic field in the region through which the current passes.

A power line consists of two wires, each carrying a current of 400 A in the same direction. The lines are perpendicular to the earth's magnetic field and are separated by a distance of 5.0 m. Which is larger: the force of the earth's magnetic field on each wire or the magnetic force between the wires?

A long cylinder of copper of radius 3 cm is charged so that it has a uniform charge per unit length on its surface of 3 C/m. (a) Find the electric field inside and outside the cylinder. (b) Draw electric field lines in a plane perpendicular to the rod.

A flow field is characterized by the stream function $\psi$ = 3x$^{2}$y − y$^{3}$. Demonstrate that the flow field represents a two dimensional incompressible flow. Show that the magnitude of the velocity depends only on the distance from the origin of the coordinates. Plot the stream line $\psi$ = 2.

A cosmic-ray electron moves at $7.5 \times 10 ^ { 6 } \mathrm { m } / \mathrm { s }$ perpendicular to Earth’s magnetic field at an altitude where the field strength is $1.0 \times 10 ^ { - 5 } \mathrm { T }$. What is the radius of the circular path the electron follows?

A pulse of radiation propagates with velocity $\vec{v}=\langle 0, c, 0\rangle.$ The electric field in the pulse is $\vec{E}=\left\langle 0,0,1 \times 10^{6}\right\rangle\ \mathrm{N} / \mathrm{C}.$ What is the magnetic field (vector) in the pulse?

An incompressible flow field is characterized by the stream function $\psi=3 A x^{2} y-A y^{3}$, where A = 1m$^{−1} \cdot s^{−1}$. Show that this flow field is irrotational. Derive the velocity potential for the flow. Plot the streamlines and potential lines, and visually verify that they are orthogonal.

The three components of velocity in a flow field are given by

$$\begin{aligned} u &=x^{2}+y^{2}+z^{2} \\ v &=x y+y z+z^{2} \\ w &=-3 x z-z^{2} / 2+4 \end{aligned}$$

(a) Determine the volumetric dilatation rate and interpret the results. (b) Determine an expression for the rotation vector. Is this an irrotational flow field?

A football kicker scores 1 point for each extra point and 3 points for each field goal. One season, a kicker made 34 extra points and scored a total of 112 points. How many field goals did the kicker make?

A steady, incompressible, two-dimensional velocity field is given by the following components in the xy-plane: u = 0.205 + 0.97x + 0.851y, v = -0.509 + 0.953x - 0.97y. Calculate the acceleration field (find expressions for acceleration components ax and ay) and calculate the acceleration at the point (x, y) = (2, 1.5).