Suppose that we have $2.10 \times 10^\pi$ doubly charged positive ions per cubic centimeter, all moving north with a speed of $1.40 \times 10^5 \mathrm{~m} / \mathrm{s}$. (a) Calculate the current density, in magnitude and direction. (b) Can you calculate the total current in this ion beam? If not, what additional information is needed?

A current of $4.82 \mathrm{~A}$ exists in a $12.4 \mathrm{\Omega}$ resistor for $4.60 \mathrm{~min}$. (a) How much charge and (b) how many electrons pass through any cross section of the resistor in this time?

A $1-\mu \mathrm{C}$ point charge is embedded in the center of a solid Pyrex sphere of radius $R=10 \mathrm{~cm}$. (a) Calculate the electric field strength $E$ just beneath the surface of the sphere. (b) Assuming that there are no other free charges, calculate the strength of the electric field just outside the surface of the sphere. (c) What is the induced surface charge density $\sigma_{\text {ind }}$ on the surface of the Pyrex sphere?

In a mass spectrometer used for commercial purposes, uranium ions of mass $238 \mathrm{u}$ and charge $+2 e$ are separated from related species. The ions are first accelerated through a potential difference of $105 \mathrm{kV}$ and then pass into a magnetic field, where they travel a $180^{\circ}$ arc of radius $97.3 \mathrm{~cm}$. They are then collected in a cup after passing through a slit of width $1.20 \mathrm{~mm}$ and a height of $1.14 \mathrm{~cm}$. (a) What is the magnitude of the (perpendicular) magnetic field in the separator? If the machine is designed to separate out $90.0 \mathrm{mg}$ of material per hour, calculate $(b)$ the current of the desired ions in the machine and $(c)$ the internal energy dissipated in the cup in $1.00 \mathrm{~h}$.

The given Figure shows an arrangement used to measure the masses of ions. An ion of mass $m$ and charge $+q$ is produced essentially at rest in source S, a chamber in which a gas dis. charge is taking place. The ion is accelerated by potential dif. ference $\Delta V$ and allowed to enter a magnetic field $\overrightarrow{\mathrm{B}}$. In the field it moves in a semicircle, striking a photographic plate at distance $x$ from the entry slit. Show that the ion mass $m$ is given by

$$m=\frac{B^2 q}{8 \Delta V} x^2$$

Bainbridge's mass spectrometer, separates ions having the same velocity. The ions, after entering through slits $S_1$ and $S_2$, pass through a velocity selector composed of an electric field produced by the charged plates P and $\mathbf{P}^{\prime}$, and a magnetic field $\overrightarrow{\mathbf{B}}$ perpendicular to the electric field and the ion path. Those ions that pass undeviated through the crossed $\overrightarrow{\mathbf{E}}$ and $\overrightarrow{\mathbf{B}}$ fields enter into a region where a second magnetic field $\vec{B}^{\prime}$ exists, and are bent into circular paths. A photographic plate registers their arrival. Show thal $q / m=E / r B B^{\prime}$, where $r$ is the radius of the circular orbit.

The electrons in the beam of a television tube have a kinetic energy of $12.0 \mathrm{keV}$. The tube is oriented so that the electrons move horizontally from magnetic south to magnetic north. The vertical component of the Earth's magnetic field points down and has a magnitude of $55.0 \mu \mathrm{T}$. (a) In what direction will the beam deflect? (b) What is the acceleration of a given electron due to the magnetic field? (c) How far will the beam deflect in moving $20.0 \mathrm{~cm}$ through the television tube?

An electron has an initial velocity $(12.0 \mathrm{~km} / \mathrm{s}) \hat{\mathbf{j}}+(15.0 \mathrm{~km} / \mathrm{s}) \hat{\mathbf{k}}$ and a constant acceleration of $\left(2.00 \times 10^{12} \mathrm{~m} / \mathrm{s}^2\right) \hat{\mathbf{i}}$ in a region in which uniform electric and magnetic fields are present. If $\overrightarrow{\mathrm{B}}=(400 \mu \mathrm{T}) \hat{\mathbf{i}}$. find the electric field $\vec{E}$.

An electron that is moving through a uniform magnetic field has velocity $\vec{v}=(40 \mathrm{~km} / \mathrm{s}) \hat{i}+(35 \mathrm{~km} / \mathrm{s}) \hat{\mathrm{j}}$ when it experiences a force $\vec{F}=-(4.2 \mathrm{f~N}) \hat{\mathrm{i}}+(4.8 \mathrm{f~N}) \hat{\mathrm{j}}$ due to the magnetic field. If $B_x=0$, calculate the magnetic field $\vec{B}$.

A wire 4.00 m long and 6.00 mm in diameter has a resistance of 15.0 m$\Omega$. A potential difference of 23.0 V is applied between the ends. Calculate the resistivity of the wire material.

Show that the dielectric constant of a conductor can be taken to be infinitely great.

When the lights of an automobile are switched on, an ammeter in series with them reads $10.0 \mathrm{~A}$ and a voltmeter connected across them reads $12.0 \mathrm{~V}$ as shown in given figure. When the electric starting motor is turned on, the ammeter reading drops to $8.00 \mathrm{~A}$ and the lights dim somewhat. If the intemal resistance of the battery is $50.0 \mathrm{~m} \Omega$ and that of the ammeter is negligible, what are $(a)$ the emf of the battery and $(b)$ the current through the starting motor when the lights are on?

A solenoid $95.6 \mathrm{~cm}$ long has a radius of $1.90 \mathrm{~cm}$, a winding of 1230 turns, and carries a current of $3.58 \mathrm{~A}$. Calculate the strength of the magnetic field inside the solenoid.

At a location in the Philippines, the Earth's magnetic field has a value of $39.0 \mu \mathrm{T}$ and is horizontal and due north. The net field is zero $8.13 \mathrm{~cm}$ above a long, straight, horizontal wire that carries a steady current. (a) Calculate the current and (b) find its direction.