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Electrical analogies of mechanical elements such as springs, masses, and dash pots provide the answer. The "deep" connection is simply that the differential equations have the same form. In electric circuit theory, the across variable is voltage while the through variable is current. The analogous quantities in mechanics are force and velocity. Note that ...

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The maximum charge a capacitor stores depends on the voltage $V_0$ you've used to charge it according to the formula: $$Q_0=CV_0$$ However, a real capacitor will only work for voltages up to the breakdown voltage of the dielectric medium in the capacitor. So in reality, for every capacitor there is a maximum possible charge $Q_{max}$ given by: $$... 10 Short answer: this is a textbook example of the limitations of ideal circuit theory. There seems to be a paradox until the underlying premises are examined closely. The fact is that, if we assume ideal capacitors and ideal superconductors, i.e., ideal short circuits, there appears to be unexplained missing energy. What's not being considered is the ... 8 A battery generates a voltage by a chemical reaction. There is a class of chemical reactions called redox reactions that involve the transport of electrons, and you can use the reaction to drive electrons through an external circuit. This is the basis of a battery. The battery will continue to provide power until all the reagents have been used up and the ... 7 The three capacitors are connected in parallel. There are only two nodes in this circuit. A series connection requires at least three. The equivalent capacitance is just the sum of the three capacitances. UPDATE: The circuit can be redrawn such that the parallel connection is manifest. 6 Everything you've probably learned about capacitors, especially including the statement that opposite plates of the capacitor carry opposite charges, applies only to a capacitor in a circuit. If your capacitor is floating, so that the plates are not connected to anything, the charge on the plates is not going to change. If you hook up only one plate to a ... 6 Power lines do cause corona discharges (power line inspection video), which produce some UV light. If the camera is picking up some UV light that might cause a few dots on the picture. If you inspect the pictures closer you will notice that the pattern only appears sometimes and only in two of the cameras of the car (the front and the rear facing camera). ... 6 I don't think these distortions are necessarily caused by the power lines. Going along some way forward on your link, under the power lines and then some more, you get to this image, which shows the distortion in front of the car over an area far from the power lines, when the car is also pretty far from them. EDIT: There's also some distortion inside ... 5 In general, no. Specific gravity refers to the density of a substance - how heavy a liter of it is. Dielectric constant refers to the response of the substance to an electric field, which depends on the chemistry of the substance. They are not physically related in any obvious way. It is possible to find two liquids with very similar densities and ... 5 As the two charged bodies attract, they have unlike charges. So, Assuming your two charged bodies as conductors and charged equally, the system may be considered as a Capacitor. If you place a dielectric like glass of some Relative permittivity \epsilon_r (3.7 to 10) which fills the empty space between the bodies, then the capacitance would be ... 5 Crazy Buddy's answer and related comments have made the point that you could indeed use a capacitor to charge a battery, but the amount of energy stored in capacitors is generally less than in batteries so it wouldn't charge the battery very much. However there is a new generation of capacitors called ultracapacitors that are being developed with electric ... 5 Every system likes to decrease its electrostatic energy. The charges on the plates are almost in stable equilibrium. The charges on the opposite plates attract them, and the charges on the same plate repel them with almost the same force. However, a capacitor has fringe fields: These may be negligible when calculating the field inside a capacitor, but ... 5 There is no such thing as inherent capacitance of a object. Capacitance is a property between two conductors. Your question therefore makes no sense. Capacitance of a typical human to where? In the case of touch screens, the sensors are measuring capacitance back to their own ground. How well a human touching a sensor couples back to that ground has a ... 4 Important notice: My previous result was a little bit incorrect. I found the factor 1/2 by comparison with the textbook V.V. Batygin, I.N. Toptygin «Problems in electrodynamics». Let's denote the radius of the inner sphere S_1 as a, the radius of the outer sphere S_2 as b and the displacement as c, so that c\ll a,b. We choose the origin of the ... 4 Yes, however you will change the capacitance of the capacitor based on the dielectric properties of the glass you are using and the material you are replacing (which might be air or even vacuum). The general force between the two plates can be calculated as:$$F = \dfrac{1}{2}\dfrac{Q^2}{Cd}$$Where Q is the charge, C is the capacitance and d is the ... 4 First of all, thanks for this question because it made me think about relativity which was always fun! It's true that E'=\frac{1}{\gamma} E. You say that relativity states that the energy should increase by a factor of \gamma. This is certainly true for a massive particle whose energy is \gamma mc^2, but why would you expect this to hold for the ... 4 I think the simplest setting you can see this is in the reactance of a capacitor when subjected to an alternating voltage source: If you subject a capacitor of capacitance C to a voltage v(t)=V_0\cos(\omega t), then the current i(t) leading to its plates, which have charge q(t), will obey$$i(t)=\frac{dq}{dt}=C\frac{dv}{dt}.$$This is best ... 4 First, note that the light bulb is essentially just a glorified resistor. As current flows through the filament, Joule heating causes the filament to get hot and emit light. When one places a capacitor in a circuit containing a light bulb and a battery, the capacitor will initially charge up, and as this charging up is happening, there will be a nonzero ... 4 The analogy of a system of water pipes to represent an electrical circuit is well established and actually very useful, especially at elementary schools. Pressure = voltage and rate of water flow = current. Switches can be modelled as valves and resistors as restrictions in the pipe. Anyhow, the hydraulic analogy of a capacitor can be though of as a rubber ... 4 There are 3 main reasons for using a capacitor. First it stores the energy, so it can deliver a pulse of energy that is far larger than the battery can. Remember it may take several seconds of battery energy to fully charge the flash capacitor. Then the capacitor releases all that in less than a millisecond (10^{-3}s) or even just a few microseconds, so ... 4 Here is my understanding: When you increase the distance between electrodes - capacitance drops, but stored charge remains the same, as electrons have nowhere to go. Same charge in lower capacitance means higher voltage potential. Without that part of stored energy would just vanish :-) 4 Toby, I agree that this is really counter intuitive and I was also quite surprised as well when I first saw this very demonstration. I am an undergraduate TA and this is how I explained it in my lab section. I hope this helps. I see two parts to a full explanation: (1) Why is the electric field constant and (2) why does the potential difference (or voltage) ... 4 This is true, however the bending is not much until one comes closer to the edge so it is usually neglected or too small to depict. Here's what I get when I simulate the system: This is the same system as a vector plot: 4 Well, you have think about the definition of capacitance, as dmckee pointed out in his comment. For two conductors both charged with charge Q and at a potential difference V, capacitance is$$ C = \frac{Q}{V} $$So capacitance is a proportionality constant between charge on two conductor and the potential difference. Now, if you consider two parallel ... 4 Suppose you have two conductors kept at voltages V_1 and V_2 and they have charges Q_1 and Q_2 respectively. Then, one has the relation$$ Q_1 = C_{11}\ V_1 + C_{12}\ V_2\quad \textrm{and}\quad Q_2 = C_{12}\ V_1 + C_{22} \ V_2 \ , $$which defines a (symmetric) Capacitance matrix that is determined by the geometries of the two conductors. This ... 3 A good analogy for an electrical circuit is to model it by a system of pipes with water flowing through them. The water represents the electrons and the voltage represents the pressure in the pipes. In this model the capacitors would be modelled by header tanks i.e. the pressure can pump water up into the tanks until the pressure from the height of the water ... 3 Current does flow through the circuit but only for a short time, as a transient effect, while the charge builds up on the plates, and after that current flow stops. You can think of electrons in the wire to the left as being attracted to the positive terminal on the battery, so they move towards it. However, because the circuit is open, soon enough the ... 3 Imagine taking a disc of some conductor and putting it in an electric field (with the plane of the disc at right angles to the field lines). The free electrons in the conductor will move in response to the field, so the face at the positive side gets a net negative charge and the other face gets a net positive charge. You get a transient current through the ... 3 The charge will redistribute as follows:$$\frac{+Q}{2}\begin{vmatrix} \frac{+Q}{2} & \frac{-Q}{2} \\ \text{charged plate} & \text{free plate} \end{vmatrix}\frac{+Q}{2}  Note that the charges on the inner and outer sides of each plate are distinct. In a capacitor (even one not in a circuit), it is mandatory that the opposing faces have ...

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Two things stand out to my eye: 1) In your volume integral, when you switched to spherical coordinates, you put in a $4 \pi r \ dr$ where there should be a $4 \pi r^2 dr$. 2) Your evaluation of the 1-D integral is not quite right. I think these two are enough to explain the discrepancy.

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