http://msuperl.org/wikis/pcubed/ 2026-04-21T10:20:30+00:00 http://msuperl.org/wikis/pcubed/ http://msuperl.org/wikis/pcubed/lib/tpl/bootstrap3/images/favicon.ico text/html 2018-05-17T15:56:17+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week2_charged_thing_neutral_thing&rev=1526572577&do=diff Return to Charge and Matter Example: Interactions Between Charged and Neutral Objects Suppose we have a positively charged object near a conductor. What happens to the charge distribution of the conductor when we bring an identical positively charged object near to the other side of the conductor? The situation is pictured below.$0 \text{ C}$ text/html 2021-01-25T00:25:11+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week2_conducting_insulating_balls&rev=1611534311&do=diff Return to Charging and Discharging Example: Attempting to Charge Insulators by Induction In the notes on Charging and Discharging, we saw how to charge a pair of conductors using induction. The relevant figure is shown below as a representation. Is it possible to charge a pair of insulators using induction? Why or why not?$t=t_0$$t=t_1$$t=t_2$$t=t_0$ text/html 2021-05-19T15:11:50+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week2_electric_field_negative_point&rev=1621437110&do=diff Return to Electric Field Example: Electric Field from a Negative Point Charge Suppose we have a negative charge $-Q$. What is the magnitude of the electric field at a point $P$, which is a distance $R$ from the charge? Draw the electric field vector on a diagram to show the direction of the electric field at $P$$-Q$$P$$R$$$\vec{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{r},$$$q$$r$$\hat r$$P$$-Q$$P$$$\vec{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{r}$$$-Q$$R$$$\vec{E} = \frac{1}{4\pi\e… text/html 2018-05-17T16:49:17+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week2_electric_potential_negative_point&rev=1526575757&do=diff Return to Electric Potential Example: Electric Potential from a Negatively Charged Balloon Suppose we have a negatively charged balloon with total charge $Q=-5.0\cdot 10^{-9} \text{ C}$. What is the electric potential (also called voltage) at a point $P$, which is a distance $R=20 \text{ m}$ from the center of the balloon? Facts $Q=-5.0\cdot 10^{-9} \text{ C}$$P$$R=20 \text{ m}$$$V = \frac{1}{4\pi\epsilon_0}\frac{q}{r},$$$q$$r$$P$$P$$R=20 \text{ m}$$P$$20 \text{ m}$$0 \text{ V}$$P$\begin{alig… text/html 2018-05-17T16:48:45+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week2_electric_potential_positive_point&rev=1526575725&do=diff Return to Electric Potential Example: Electric Potential from a Positively Charged Balloon Suppose we have a positively charged balloon with total charge $Q=5.0\cdot 10^{-9} \text{ C}$. What is the electric potential (also called voltage) at a point $P$, which is a distance $R=50 \text{ cm}$ from the center of the balloon? Facts $Q=5.0\cdot 10^{-9} \text{ C}$$P$$R=50 \text{ cm}$$$V = \frac{1}{4\pi\epsilon_0}\frac{q}{r},$$$q$$r$$P$$P$$P$$0 \text{ V}$$P$\begin{align*} V &= \frac{1}{4\pi\epsilon… text/html 2018-05-17T15:16:36+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week2_moleoelectrons&rev=1526570196&do=diff Return to Electric Charge Page Example: Find the total charge for a mole of electrons How much total charge (in coulombs) is in one mole of electrons? Facts * The Avogadro constant is $N_A = 6.022 \cdot 10^{23} \text{ mol}^{-1}$. This is easy to look up, which is what we did. * Note: When we write the unit as $\text{ mol}^{-1}$, we mean particles per mole. We could also write this unit as $mol^{-1}=\frac{1}{mol}$$e = -1.602\cdot10^{-19} \text{ C}$$Q$$N$$e$$Q=N\cdot e$$e$$N$\begin{align*… text/html 2018-05-17T16:49:33+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week2_potential_plots&rev=1526575773&do=diff Return to Electric Potential Plotting Electric Potential with Wolfram Alpha To make the 3D plots in Wolfram Alpha, first you need to create a free notebook with the Wolfram Cloud. Starting here, click on the Wolfram Programming Lab for beginners. If you scroll down to the very bottom of the page, there is a link in blue on the left that says text/html 2021-01-26T21:21:20+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week3_balloon_wall&rev=1611696080&do=diff Return to Electric Force Example: Balloon Stuck to a Wall When you rub part of a rubber balloon against wool (or your hair), electrons will leave the wool, which is slightly conductive, and go onto the balloon. The rubber on the balloon is much less conductive (rubber is more of an insulator than wool), and the electrons will not readily leave the balloon. As a result, the balloon becomes negatively charged. Imagine you bring the negatively charged balloon up to a wall, and it sticks (This is … text/html 2021-05-19T15:01:35+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week3_particle_in_field&rev=1621436495&do=diff Return to Electric Potential Energy Example: Particle Acceleration through an Electric Field Suppose you have a particle with a mass $m$, charge $Q$, initially at rest in an electric field $\vec{E}=E_0\hat{x}$. The electric field extends for a distance $L$ in the $+\hat{x}$-direction before dropping off abruptly to 0. So, the magnitude of the electric field is exactly $E_0$$0$$Q>0$$Q<0$$Q=0$$m$$Q$$L$$i$$f$\begin{align*} \Delta U &= -\int_i^f\vec{F}\bullet d\vec{r} &&&&&& (1) \\ \Delta U &= q\D… text/html 2017-08-28T15:54:36+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week3_ploting_potential&rev=1503935676&do=diff Plotting Potential for Multiple Charges text/html 2018-05-29T14:28:09+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week3_plotting_potential&rev=1527604089&do=diff Return to Superposition Plotting Potential for Multiple Charges To make the 3D plots in Wolfram Alpha, first you need to create a free notebook with the Wolfram Cloud just like we did before. Starting here, click on the Wolfram Programming Lab for beginners. If you scroll down to the very bottom of the page, there is a link in blue on the left that says text/html 2021-05-19T15:08:34+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week3_spaceship_asteroid&rev=1621436914&do=diff Return to Electric Potential Energy Example: Preventing an Asteroid Collision Suppose your friend is vacationing in Italy, and she has lent you her spaceship for the weekend. You have gathered together a group of friends and you are currently cruising through the heavens together and having a great time. You are surrounded by nothingness in all directions. Suddenly, the radar starts beeping ferociously. The ship is on a collision course with an asteroid. You are not too worried about survival … text/html 2021-05-19T14:46:03+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week3_superposition_three_points&rev=1621435563&do=diff Return to superposition Example: Superposition with Three Point Charges Suppose we have a distribution of point charges in a plane near a point $P$. There are three point charges: Charge 1 with charge $-Q$, a distance $2R$ to the left of $P$; Charge 2 with charge $Q$, a distance $R$ above $P$; and Charge 3 with charge $Q$$2R$$P$$P$$-Q$$2R$$P$$Q$$R$$P$$Q$$2R$$P$$$\vec E = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{r}.$$$$V = \frac{1}{4\pi\epsilon_0}\frac{q}{r}.$$$\vec{E}_{tot}=\vec{E}_{1}+\vec{E… text/html 2021-07-22T18:21:57+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week4_charge_cylinder&rev=1626978117&do=diff Example: Electric Field from a Cylindrical Shell of Charge Note: Super Challenge Problem!! -- This is a beyond the scope of this class (so you won't be expected to solve this kind of problem), but it is a cool example of how to expand from lines to areas of charge if you are interested $R$$L$$Q$$R$$P$$z$$z = 0$$z$$P$$z$$Q$$L$$R$$xy$$z$$$\vec{E}=\frac{1}{4\pi\epsilon_0}\frac{Qz}{(R^2+z^2)^{3/2}}\hat{z}$$$P$$\text{d}Q$$\text{d}Q$$\text{d}Q$$\vec{r}$$\text{d}\vec{E}$$P$$\text{d}Q$$\text{d}Q$$\text… text/html 2021-05-25T14:38:07+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week4_charge_ring&rev=1621953487&do=diff Return to line of charge Example: Electric Field from a Ring of Charge Suppose we have a ring with radius $R$ that has a uniform charge distribution with total charge $Q$. What is the electric field at a point $P$, which is a distance $z$ from the center of the ring, along a line perpendicular to the plane of the ring? What happens to the electric field if $z = 0$$P$$P$$z$$Q$$P$$\text{d}Q$$\vec{r}$$$\vec{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^3}\vec{r}$$$P$$\text{d}l$$\lambda$$\text{d}Q$$\tex… text/html 2018-06-12T18:49:32+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week4_tilted_segment&rev=1528829372&do=diff Return to $dQ$ A Tilted Segment of Charge Suppose we have a segment of uniformly distributed charge stretching from the point $\langle 0,0,0 \rangle$ to $\langle 1 \text{ m}, 1 \text{ m}, 0 \rangle$, which has total charge $Q$. We also have a point $P=\langle 2 \text{ m},0,0 \rangle$. Define a convenient $\text{d}Q$ for the segment, and $\vec{r}$ between a point on the segment to the point $P$$\text{d}Q$$\text{d}Q$$\vec{r}$$\langle 0,0,0 \rangle$$\langle 1 \text{ m}, 1 \text{ m}, 0 \rangle$$Q$… text/html 2021-05-25T14:28:10+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week4_two_segments&rev=1621952890&do=diff Return to $dQ$ Example: Two Segments of Charge Suppose we have two segments of uniformly distributed charge, one with total charge $+Q$, the other with $-Q$. The two segments each have length $L$, and lie crossed at their endpoints in the $xy$-plane. The segment with charge $+Q$ lies along the $y$$-Q$$x$$\vec{E}_P$$P$$\vec{r}_P=r_x\hat{x}+r_y\hat{y}$$y$$0$$L$$Q$$x$$0$$L$$-Q$$P$$\vec{r}_P=r_x\hat{x}+r_y\hat{y}$$$\vec{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^3}\vec{r}$$$P$$$\vec{E}_P = \vec{E}_{+… text/html 2017-09-22T15:57:03+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week5_flux_cube_plane&rev=1506095823&do=diff FIXME - this is more or a homework problem for them. So I'm not sure we want to use this one... Example: Flux through a Cube on a Charged Plane Suppose you have a plane of charge with a uniform surface charge density of $\sigma=-4\mu\text{C/m}^2$. What is the electric flux through a cube with side-length $l=0.5 \text{ m}$$\vec{E} = \frac{\sigma}{2\epsilon_0}(\pm\hat{z})$$\pm\hat{z}$$l=0.5 \text{ m}$$\sigma=-4\mu\text{C/m}^2$$\Phi_e$$$\Phi_e=\int\vec{E}\bullet \text{d}\vec{A}$$$$\vec{E} = \frac… text/html 2021-06-04T00:54:54+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week5_flux_cylinder_line&rev=1622768094&do=diff Return to Enclosed Charge notes Example: Flux through a Cylinder on a Line of Charge Suppose you have a line of charge with a uniform linear charge density of $\lambda=15\mu\text{C/m}$. What is the electric flux through a cylinder with radius $R=0.5 \text{ m}$, and length $l=3 \text{ m}$ that is placed so that its axis is aligned with the line of charge? Feel free to use the electric field due to an infinite uniform line of charge: $\vec{E} = \frac{\lambda}{2\pi r\epsilon_0}\hat{r}$$\hat{r}$$r… text/html 2018-07-24T14:52:04+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week5_flux_cylinder&rev=1532443924&do=diff Return to Electric Flux and Area Vectors notes Example: Flux through a Closed Cylinder A constant electric field $\vec{E}$ is directed along the $x$-axis. If we imagine a cylindrical surface with radius $R$ and height $h$ is situated in the field so that the bases of the cylinder are parallel to the $xz$-plane. What is the electric flux through the cylinder?$R$$h$$x$$\Phi_{\text{cylinder}}$$\vec{A}$$\text{d}\vec{A}$$\vec{E}$$$\Phi=\vec{E}\bullet \vec{A}$$$\text{d}\vec{A}$$\text{d}\vec{A}$$y$$x… text/html 2018-07-24T15:02:30+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week5_flux_dipole&rev=1532444550&do=diff Return to Electric Flux through Curved Surfaces notes Example: Flux from a Dipole Suppose you have a two charges, one with value $5 \mu\text{C}$, the other with value $-5 \mu\text{C}$. There are at separate locations, a distance $1 \text{ m}$ apart, and they can be modeled as a dipole. What is the flux through a cylinder with radius $4 \text{ m}$ and length $16 \text{ m}$$q=5 \mu\text{C}$$-q=-5 \mu\text{C}$$1 \text{ m}$$4 \text{ m}$$16 \text{ m}$$\Phi_e$$\text{d}\vec{A}$$\vec{E}$$\left(\vec{E}… text/html 2021-06-04T00:41:13+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week5_flux_tilted_surface&rev=1622767273&do=diff Return to Electric Flux and Area Vectors notes Example: Flux through a Tilted Surface Suppose you have a uniform electric field $\vec{E} = 8\text{ V/m } \hat{x}$. There is a tilted rectangular surface with dimensions $3\text{ m}$ (perpendicular to the field), and $5\text{ m}$ (at an angle of $\theta=30^\circ$ to field). What is the electric flux through the surface?$\vec{E} = 8\text{ V/m } \hat{x}$$3\text{ m}$$3\text{ m}$$30^\circ$$\Phi_e$$\vec{A}$$$\Phi=\vec{E}\bullet \vec{A}$$$z$$+z$$\vec{E}… text/html 2021-06-04T00:47:50+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week5_flux_two_radii&rev=1622767670&do=diff Return to Electric Flux through Curved Surfaces notes Example: Flux through Two Spherical Shells Suppose you have a point charge with value $1 \mu\text{C}$. What are the fluxes through two spherical shells centered at the point charge, one with radius $3 \text{ cm}$ and the other with radius $6 \text{ cm}$? Facts * The point charge has charge $q=1 \mu\text{C}$$3 \text{ cm}$$6 \text{ cm}$$\Phi_e$$\text{d}\vec{A}$$\vec{A}$$$\Phi_e=\int\vec{E}\bullet \text{d}\vec{A}$$$$\vec{E}=\frac{1}{4\pi\e… text/html 2021-06-07T14:02:12+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week5_gauss_ball&rev=1623074532&do=diff Return to Putting Gauss's Law Together Example: Gauss' Law Application -- A Ball of Charge Suppose you have an insulating ball that has been charged somehow. The charging was very thorough and it seems like the ball is pretty much uniformly charged. It has a charge $Q$ and a radius $R$. What is the electric field at a distance $r$$r<R$$r>R$$Q$$R$$\Phi_e$$\vec{E}(\vec{r})$$$\Phi_e=\int\vec{E}\bullet \text{d}\vec{A}$$$$\Phi_e=\frac{Q_{\text{enclosed}}}{\epsilon_0}$$$Q>0$$+$$r$$$\vec{E}(\vec{r})=… text/html 2021-06-08T00:41:02+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week6_charges_circuit&rev=1623112862&do=diff Return to Surface Charges around a Circuit Example: Charge Distribution on the Bends of a Circuit On the circuit shown below, draw how you would expect charge to distribute on the surface of the wire near the bends in the circuit. Facts * Electric field is constant in the wire - created by the surface charges on the wire, not the battery.$+$$-$ text/html 2021-06-08T00:49:28+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week6_drift_speed&rev=1623113368&do=diff Return to current in wires Example: Drift Speed in Different Types of Wires Suppose you have a two wires. Each has a current of $5 \text{ A}$. One is made of copper (Cu) and has radius $0.5 \text{ mm}$. The other is made of zinc (Zn) and has radius $0.1 \text{ mm}$. What are the drift speeds of electrons in each wire? You may want to consult the table below.$I=5 \text{ A}$$r = 0.5 \text{ mm}$$I=5 \text{ A}$$r = 0.1 \text{ mm}$$q=-1.6\cdot 10^{-19} \text{ C}$$n_{\text{Cu}}=8.47\cdot 10^{22} \te… text/html 2021-06-08T00:51:16+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week6_node_rule&rev=1623113476&do=diff Return to current in wires Example: Application of Node Rule Suppose you have the circuit below. You are given a few values: $I_1=8 \text{ A}$, $I_2=3 \text{ A}$, and $I_3=4 \text{ A}$. Determine all other currents in the circuit, using the Current Node Rule. Draw the direction of the current as well. Facts * $I_1=8 \text{ A}$, $I_2=3 \text{ A}$, and $I_3=4 \text{ A}$. * $I_1$, $I_2$, and $I_3$ are directed as pictured.$I_{in}=I_{out}$$A$$I_1$$I_2$$I_{A\rightarrow B}$$I_{in}=I_{out}$$I_1… text/html 2021-06-14T23:50:07+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week7_charging_capacitor&rev=1623714607&do=diff Return to Charging and Discharging Capacitors Looking at a Capacitor as it's Charging Suppose you have a parallel plate capacitor that is disconnected from any power source and is discharged. At time $t=0$, the capacitor is connected to a battery. Sketch the graphs of current ($I$) in the wire, charge ($Q$$\Delta V$$t=0$$I$$Q$$\Delta V$$t=0$$t=0$$t=0$$t=0$$t=0$$I = \frac{\text{d}Q}{\text{d}t}$$$E_{plate} = \frac{Q_{plate}}{2\epsilon_0 A_{plate}}$$$Q^2$$Q^5$$$\Delta V = E\cdot \Delta d$$$Q$$$C … text/html 2021-06-15T13:52:53+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week7_cylindrical_capacitor&rev=1623765173&do=diff Return to Capacitors in Circuit Finding the Capacitance of a Cylindrical Capacitor Find the capacitance of a cylindrical capacitor. The structure of the capacitor is a cylindrical shell inside another cylindrical shell. The two shells become oppositely charged when the capacitor is connected to a power source. The length of the cylinders is $L$$a$$b$$a<b$$L$$a$$b$$$C=\frac{Q}{\Delta V},$$$Q$$\Delta V$$Q$$\Delta V$$\Delta V$$Q$$-Q$$$\Delta V=V_f-V_i=-\int_{r_i}^{r_f} \vec{E}\bullet \text{d}\vec… text/html 2021-06-15T01:02:40+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week7_energy_plate_capacitor&rev=1623718960&do=diff Return to Capacitors in Circuit Energy Stored in a Parallel Plate Capacitor Suppose you have a parallel plate capacitor with a capacitance of $16 \text{ mF}$. You connect it to a 15-Volt battery and leave it to charge. After a while, you suddenly double the area of the plates and wait for another while. What is the energy stored in the capacitor? What would be the energy stored if you had disconnected the capacitor before doubling the area?$C = 16 \text{ mF}$$\Delta V_{\text{battery}}=15 \text… text/html 2018-06-19T14:54:47+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week7_ohms_law&rev=1529420087&do=diff Return to resistors and conductivity Example: Application of Ohm's Law Suppose you have a simple circuit that contains only a 9-Volt battery and a resistor of $120 \Omega$. What is the current in the wire? Facts * $\Delta V = 9\text{ V}$ * $R = 120 \Omega$ Lacking * Current Approximations & Assumptions * The wire has very very small resistance when compared to the 120 $\Omega$$\Delta V = IR$$R = 120\Omega$$\Delta V = 9 \text{ V}$$$I = \frac{\Delta V}{R} = 75 \text{ mA}$$ text/html 2018-06-19T14:54:28+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week7_resistance_wire&rev=1529420068&do=diff Return to resistors and conductivity Resistance of a Wire Suppose you have a wire whose resistance is 60 m$\Omega$. The wire has a length of 2 cm, and has a cross-sectional area of 1 mm$^2$. What would be the resistance if you increase the length of the wire to 6 cm (keeping the area the same)? What would be the resistance if you increase the cross-sectional area to 3 mm$^2$$L = 2 \text{ cm}$$A = 1 \text{ mm}^2$$R = 60 \text{ m}\Omega$$L_{new} = 6 \text{ cm}$$A_{new} = 3 \text{ mm}^2$$$R = \fr… text/html 2021-06-14T23:40:25+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week7_wire_dimensions&rev=1623714025&do=diff Return to Energy in Circuits Example: Changing the Dimensions of a Wire Suppose you have a simple circuit whose wire changes in thickness. The wire is 8 meters long. The first 2 meters of the wire are 3 mm thick. The next 2 meters are 1 mm thick. The last 4 meters are 3 mm thick. The wire is connected to a 12-Volt battery and current is allowed to flow. You use an ammeter and a voltmeter to find that the current through the first 2 meters of wire is $I_1 = 5 \text{ A}$$\Delta V_1 = 1 \text{ V}… text/html 2018-06-26T14:45:32+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week8_cap_parallel&rev=1530024332&do=diff Return to Capacitors in Parallel notes Connecting Already-Charged Capacitors Suppose you have the following setup of already-charged capacitors. The positive plates are all on the top half of the circuit. Capacitors are labeled 1 through 3 for convenience of reference, and the sign of the charge on the plates is indicated. You know that $Q_1 = Q_2 = Q_3 = 1 \text{ mC}$$\Delta V_1 = \Delta V_2 = \Delta V_3 = 20 \text{ V}$$Q_1 = Q_2 = Q_3 = 1 \text{ mC}$$\Delta V_1 = \Delta V_2 = \Delta V_3 = 20… text/html 2021-06-29T00:08:32+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week8_cap_series&rev=1624925312&do=diff Return to capacitors in series notes Example: Capacitors in Series Suppose you have the following circuit. Capacitors are labeled 1 and 2 for convenience of reference. You know that the circuit contains a 12-Volt battery, $Q_1=4.5 \mu\text{C}$, and $C_2=0.5 \mu\text{F}$. What is the capacitance of Capacitor 1? What happens to the charge on $k = 3$$\Delta V_{\text{bat}} = 12\text{ V}$$Q_1 = 4.5 \mu\text{C}$$C_2 = 0.5 \mu\text{F}$$k = 3$$C_1$$$C=\frac{Q}{\Delta V}$$$$\frac{1}{C_{\text{equiv}}}=\… text/html 2021-07-05T21:15:21+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week8_charge_discharge_caps_resistors&rev=1625519721&do=diff Return to Larger Combinations of Circuit Elements notes Challenge Problem: Charging Capacitors through Resistors Suppose you have a the setup shown below. All capacitors have the same capacitance $C$, and all resistors have the same resistance $R$. Capacitor 1 is charged to $Q$ and $-Q$ on its plates, so that the voltage across it is $\Delta V$$t=0$$t=0$$Q$$-Q$$t=0$$\Delta V$$C_1 = C_2 = C_3 = C_4 = C$$R_1 = R_2 = R_3 = R_4 = R$$t=0$$t=0$$t=0$$0$$t=0$$R_{\text{equiv, 3,4}}=R_3+R_4=2R$$$\frac{1… text/html 2021-06-28T23:51:32+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week8_resistors_parallel&rev=1624924292&do=diff Return to Resistors in Parallel Notes Example: Resistors in Series and in Parallel Suppose you have the following circuit. Resistors are labeled 1 through 4 and nodes in the circuit are labeled A, B, and C for convenience of reference. You know that the circuit contains a 12-Volt battery, $I_1 = 50 \text{ mA}$$R_1=80 \Omega$$R_3=300 \Omega$$R_4=500 \Omega$$\Delta V_4=5 \text{ V}$$\Delta V_1$$\Delta V_2$$\Delta V_3$$R_2$$\Delta V_{\text{bat}} = 12\text{ V}$$I_1 = 50 \text{ mA}$$R_1=80 \Omega$$R… text/html 2021-07-05T21:45:49+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week8_resistors_series&rev=1625521549&do=diff Return to Resistors in Series Notes Example: Resistors in Series Suppose you have the following circuit. Resistors are labeled 1 through 3 for convenience of reference. You know that the circuit contains a 12-Volt battery, and $R_1=10 \Omega$, $\Delta V_3=6 \text{ V}$, and the power dissipated through Resistor 1 is $P_1 = 0.1 \text{ W}$. What is the resistance of and power dissipated through Resistor 2?$R_1=10 \Omega$$\Delta V_3 = 6\text{ V}$$\Delta V_{bat} = 12\text{ V}$$P_1 = 0.1 \text{ W}$$… text/html 2021-07-22T18:28:37+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week8_wheatstone&rev=1626978517&do=diff Return to Larger Combinations of Resistors and Capacitors notes The Wheatstone Bridge Suppose you have the following circuit -- it is similar to a well known circuit called a Wheatstone bridge. Resistors are labeled 1 through 4 for convenience of reference, and the fifth element is a light bulb, which also has some resistance. If any current at all flows through the light bulb, it will glow. You know $R_1 = 150 \Omega$$R_2=60 \Omega$$R_3$$R_3=250 \Omega$$R_4$$R_3=500 \Omega$$\Delta V_{\text{ba… text/html 2017-10-20T02:13:27+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week9_current_segment&rev=1508465607&do=diff Magnetic Field from a Current Segment You may have read about how to find the magnetic field from a very long wire of current. Now, what is the magnetic field from a single segment? Suppose we have the configuration shown below. Your observation point is at the origin, and the segment of current $I$$\langle -L, 0, 0 \rangle$$\langle 0, -L, 0 \rangle$$I$$\langle -L, 0, 0 \rangle$$\langle 0, -L, 0 \rangle$$\vec{B}$$$\vec{B}= \int \frac{\mu_0}{4 \pi}\frac{I \cdot d\vec{l}\times \vec{r}}{r^3}$$$\te… text/html 2021-07-05T21:58:52+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week9_detecting_b&rev=1625522332&do=diff Return to Moving Charges make Magnetic Fields notes Magnetic Field near a Moving Charge You are a collector of magnetic field detectors. A fellow detector collector is trying to trim down her collection, and so it's your job to see if an old detector is still working properly, in which case it's yours! Today, you are a magnetic field detector collector, inspector, and hopefully a selector. First, you run a test in which a charged particle ($q = 15 \text{ nC}$$2 \text{ m/s}$$\vec{v} = 2 \text{ … text/html 2021-07-05T21:55:28+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week9_earth_field&rev=1625522128&do=diff Return to Permanent Magnet notes Using the Earth's Magnetic Field for Measurements You have spotted an unidentified flying object! Naturally, you wish to find its charge. You have a compass, a good sense of direction, and keen eyesight. You notice that it is flying due south on a course that will pass directly overhead, and it is $15 \text{ m}$$200 \text{ m/s}$$B_{\text{earth}} = 32 \mu\text{T}$$\vec{B}_{\text{earth}} = 32 \mu\text{T } \hat{y}$$h = 15 \text{ m}$$\vec{v} = -200 \text{ m/s } \ha… text/html 2021-07-07T17:52:07+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week10_current_ring&rev=1625680327&do=diff Challenge Example: Magnetic Field from a Ring of Current Suppose you have a circular ring, in which which there is a current $I$. The radius of the ring is $R$. The current produces a magnetic field. What is the magnetic field at the center of the ring? Facts $I$$R$$\vec{B}$$xy$$$\vec{B} = \int \frac{\mu_0}{4 \pi}\frac{I \cdot d\vec{l}\times \vec{r}}{r^3}$$$\text{d}\vec{l}$$\vec{r}$$\text{d}\vec{l}$$\vec{r}$$\text{d}\vec{l}$$\hat{\phi}$$\hat{\phi}$$\hat{\phi} = -\sin(\phi) \hat{x} + \cos(\phi)… text/html 2021-07-07T18:01:18+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week10_current_segment&rev=1625680878&do=diff Return to Currents make Magnetic Fields notes Magnetic Field from a Current Segment The notes outline how to find the magnetic field from a very long wire of current. Now, what is the magnetic field from a single segment? Suppose we have the configuration shown below. Your observation point is at the origin, and the segment of current $I$ runs in a straight line from $\langle -L, 0, 0 \rangle$$\langle 0, -L, 0 \rangle$$I$$\langle -L, 0, 0 \rangle$$\langle 0, -L, 0 \rangle$$\vec{B}$$$\vec{B}= \… text/html 2017-11-02T13:32:48+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week10_force_on_charge&rev=1509629568&do=diff Magnetic Force on Moving Charge Suppose you have a moving charge ($q=1.5 \text{ mC}$) in a magnetic field ($\vec{B} = 0.4 \text{ mT } \hat{y}$). The charge has a speed of $10 \text{ m/s}$. What is the magnetic force on the charge if its motion is in the $+x$-direction? The $-y$-direction? Facts * The charge is $q=1.5 \text{ mC}$. * There is an external magnetic field $\vec{B} = 0.4 \text{ mT } \hat{y}$$\vec{v} = 10 \text{ m/s } \hat{x}$$\vec{v} = -10 \text{ m/s } \hat{y}$$\vec{F}_B$$$\vec… text/html 2021-07-07T15:42:47+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week10_helix&rev=1625672567&do=diff Return to Path of a Charge through a Magnetic Field notes Helical Motion in a Magnetic Field Suppose you have a moving charge $q=20 \text{ mC}$ in a magnetic field $\vec{B} = 15 \text{ mT } \hat{y}$. The charge has a velocity of $\vec{v} = (3\hat{x} + 2\hat{y}) \text{ m/s}$, and a mass of $m = 1 \text{ g}$. What does the motion of the charge look like? Facts * There is a charge $q = 20 \text{ mC}$. * The charge has velocity $\vec{v} = (3\hat{x} + 2\hat{y}) \text{ m/s}$. * $m = 1 \text{ … text/html 2018-07-03T14:01:34+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week10_radius_motion_b_field&rev=1530626494&do=diff Return to Path of a Charge through a Magnetic Field notes Radius of Circular Motion in a Magnetic Field Suppose you have a moving charge $q>0$ in a magnetic field $\vec{B} = -B \hat{z}$. The charge has a velocity of $\vec{v} = v\hat{x}$, and a mass $m$. What does the motion of the charge look like? What if the charge enters the field from a region with $0$$q$$v\hat{x}$$m$$\vec{B} = -B \hat{z}$$0$$B$$$\vec{F}= q \vec{v} \times \vec{B}$$$$\vec{F}= q \vec{v} \times \vec{B}$$$\vec{v}$$\hat{x}$$\ve… text/html 2018-08-09T18:08:31+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week12_changing_shape&rev=1533838111&do=diff Return to Changing Magnetic Flux notes Flux Through a Changing, Rotating Shape Suppose you have a magnetic field directed in the $-\hat{z}$-direction, into the page. There is a flexible, circular loop situated on the page, in the $xy$-plane. You stretch it out in the $\pm x$-direction like a rubber band to change its area. Then you rotate it $90^\text{o}$$xy$$60^\text{o}$$yz$$$\Phi_B = \int \vec{B} \bullet d\vec{A}$$$d\vec{A}$$$\int \vec{B} \bullet d\vec{A} = \int BdA\cos\theta$$$B$$\theta$$dA… text/html 2018-08-09T18:46:46+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week12_decreasing_flux&rev=1533840406&do=diff Return to The Curly Electric Field and Induced Current notes Decreasing Flux Say we have a bar that is sliding down a pair of connected conductive rails (so current is free to flow through the loop created by the bar and rails), which is sitting in a magnetic field that points into the page. If the bar slides along the rails to decrease the area of the loop, what happens?$$\Phi_B = \int \vec{B} \bullet \text{d}\vec{A}$$$\text{d}\vec{A}$$$\int \vec{B} \bullet \text{d}\vec{A} = \int B\text{d}A\c… text/html 2018-08-09T18:08:09+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week12_flux_examples&rev=1533838089&do=diff Return to Changing Magnetic Flux notes Review of Flux through a Loop Suppose you have a magnetic field $\vec{B} = 0.6 \text{ mT } \hat{x}$. Three identical square loops with side lengths $L = 0.5 \text{ m}$ are situated as shown below. The perspective shows a side view of the square loops, so they appear very thin even though they are squares when viewed face on.$\theta_1 = 0$$\theta_2 = 90^\text{o}$$\theta_3 = 42^\text{o}$$\vec{B} = 0.6 \text{ mT } \hat{x}$$L = 0.5 \text{ m}$$$\Phi_B = \int \… text/html 2021-07-13T12:16:27+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week12_force_between_wires&rev=1626178587&do=diff Return to Magnetic Force on a Current Carrying Wire notes Magnetic Force between Two Current-Carrying Wires Two parallel wires have currents in opposite directions, $I_1$ and $I_2$. They are situated a distance $R$ from one another. What is the force per length $L$ of one wire on the other? Facts * $I_1$ and $I_2$ exist in opposite directions.$R$$\vec{F}_{1 \rightarrow 2 \text{, L}}$$$\vec{B}=\frac{\mu_0 I}{2 \pi r} \hat{z}$$$$\text{d}\vec{F}= I \text{d}\vec{l} \times \vec{B}$$$$ d\vec{F}_… text/html 2021-07-13T12:33:45+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week12_force_loop_magnetic_field&rev=1626179625&do=diff Return to Magnetic Force on a Current Carrying Wire notes Force on a Loop of Current in a Magnetic Field Suppose you have a square loop (side length $L$) of current $I$ situated in a uniform magnetic field $\vec{B}$ so that the magnetic field is parallel to two sides of the loop. What is the magnetic force on the loop of current?$L$$B$$I$$$\left| \vec{F} \right|=IBL\sin\theta$$$\theta$\[ \left| \vec{F} \right| = \begin{cases} IBL\sin \pi = 0 & \text{top} \\ … text/html 2018-04-11T19:36:29+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week12_moving_coils_flux&rev=1523475389&do=diff Flux Through Moving Coils Suppose you have two conducting rings centered on the same axis so that they face one another. One of the rings has a current in it, which produces a magnetic field nearby. The rings are moved towards one another. What happens?$$\Phi_B = \int \vec{B} \bullet \text{d}\vec{A}$$ text/html 2018-08-09T19:19:32+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week14_ac_graph&rev=1533842372&do=diff Return to Changing Flux from an Alternating Current notes Analyzing an Alternating Current Graph Suppose you are given the following graph of current over time. You can see that the first peak is at the point where $t=0.01\text{ s}$, and $I=0.3\text{ A}$. The graph is shown below. Find the amplitude, period, and frequency of the current, and give an equation that describes the alternating current.$(t=0.01\text{ s}, I=0.3\text{ A})$$t=0$$I=0$$I=0$$0.3\text{ A}$$0.04\text{ s}$$$f = \frac{1}{\tau… text/html 2021-07-22T13:51:37+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week14_b_field_capacitor&rev=1626961897&do=diff Return to Changing Electric Fields notes Magnetic Field from a Charging Capacitor Suppose you have a parallel plate capacitor that is charging with a current $I=3 \text{ A}$. The plates are circular, with radius $R=10 \text{ m}$ and a distance $d=1 \text{ cm}$ apart. What is the magnetic field in the plane parallel to but in between the plates?$R=10 \text{ m}$$d=1 \text{ cm}$$I=3 \text{ A}$$I$$$\vec{E} = \frac{Q/A}{\epsilon_0} \hat{x}$$$Q$$A$$\hat{x}$$$\int \vec{B}\bullet \text{d}\vec{l} = \mu… text/html 2021-07-13T13:26:15+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week14_changing_current_rectangle&rev=1626182775&do=diff Return to Changing Magnetic Fields with Time notes Changing Current Induces Voltage in Rectangular Loop Suppose you have an increasing current through a long wire, $I(t) = I_0 \frac{t}{t_0}$. Next to this wire, there is a rectangular loop of width $w$ and height $h$. The side of the rectangle is aligned parallel to the wire so that the rectangle is a distance $d$$I(t) = I_0 \frac{t}{t_0}$$w$$h$$h$$d$$V_{ind}$$I_{ind}$$$B = \frac{\mu_0 I}{2 \pi r}$$$$\Phi_B = \int \vec{B} \bullet \text{d}\vec{A… text/html 2021-07-22T13:56:13+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=184_notes:examples:week14_step_down_transformer&rev=1626962173&do=diff Return to Changing Flux from an Alternating Current notes Designing a Step-down Transformer Recall the discussion on voltage transformers. We designed a step-up transformer in the notes, which is used to convert small voltages from a generator into high voltages, which get carried long distances to residential areas. High-voltage power lines are dangerous, though, because the potential difference between the power lines and the ground is so enormous. Before the lines enter a residential area, …