Capacitos and Capacitance Capacitos ae devices that can stoe a chage Q at some voltage V. The geate the capacitance, the moe chage that can be stoed. The equation fo capacitance, C, is vey simple: C Q V C V Q V [ F] 1F 1 faad Even an isolated conducting sphee has capacitance. We just take the equation fo the electic potential of a point chage (which also applies to conducting sphees, and find the above atio: Michael Faaday R Q kq Q V C R V R k
Find the capacitance of Old Spaky C k 0.0m 1. 10 F 9 9.00 10 N m / C. pf A capacitance of ~ 0 pf is not vey big. If this Van de Gaaf geneato could each 1,000,000 V, the total chage on the dome would be only 0 μc. This is one eason this device is safe. The othe is that high voltage causes chage to tavel along the suface of objects (such as ou bodies).
It is moe pactical to build a conducto out of two conductos, chaging them equal and opposite, as an electic dipole. With the ight geomety, this will ceate a contained electic field, and the possibility of much lage capacitance than can be achieved with a sphee of equivalent size. The geneic capacito
Easily calculable capacito #1: paallel plates Hee s the example we ve been efeing to fo some time: the paallel plate capacito. We assume that the plates ae so lage compaed to thei sepaation d that we can ignoe the finge field, and that all the electic field is inside. Then, as we found ealie, V Ed. And, since all the chage is on the inne sufaces of the conductos, E σ /ε 0 C Q V σa Ed σa ε d σ 0 ε 0 What happens to C if d is made vey small??? d A This is a simple esult. Notice that C depends only on ε 0 and the dimensions of the capacito. We will see the same thing with all othe capacitos.
Calculable capacito #: concentic sphees This calculation is also easy. We use the equation fo the electic potential of a sphee of a given adius and chage. We apply it once to the inne sphee, and again to the oute, to find the voltage diffeence between the two. (By Gauss s Law, the only electic field is between the two sphees.) ΔV V a V b kq a kq b kq b a b a Then, the capacitance of a spheical capacito (in vacuum) is: C Q ΔV 1 4πε a k b a b a a b 0 b
Calculable capacito #3: coaxial infinite cylindes We ve done most of the wok aleady fo this, by finding the diffeence in electic potential at two diffeent adii in a cylindically symmetic geomety. We simply use that ealie esult fo ΔV, and find C fom the basic equation. Δ a b k V ln λ Notice that to calculate a Q fo the capacitance equation, we need to choose a length, L, so that Q λl: Δ a b a b k L k L V Q C ln λ ln λ Capacitance pe unit length (in vacuum). a b a b k L C ln ln 1 0 πε
Popula design fo ealy capacitos the Leyden Ja In 1745, the Leyden Ja (o Leyden Bottle) was invented by Ewald Jügen von Kleist (1700-1748). The glass inceases the capacitance damatically, as we shall undestand soon A Leyden Battey. What would we call this configuation today? Again, we ll be discussing this futhe Glass
Induction machine invented by James Wimshust c. 1880. The two disks, with naow tin-foil stips glued aound the im, ae otated in opposite diections by a system of pulleys and belts. At each side thee is a conducto, teminating in metal bushes that ub against the tin-foil sectos. The chage is induced in two jaw collectos and stoed in a pai of Leyden jas connected to two sliding electodes. This machine was highly popula and is still used today fo teaching puposes. It woked well even in damp weathe and did not evese polaity. Apat fom laboatoy demonstations, it was used fo medical teatment and as a high-voltage souce fo the fist X-ay tubes. Ealy chaging machines
By ~ 1905. Lage machines!
Moden capacitos: many sizes, shapes, and types http://en.wikipedia.og/wiki/capacito http://www.spakmuseum.com/radios.htm http://www.uoguelph.ca/~antoon/gadgets/caps/caps.html Capacito types: (most common types ae in ed). Metal film: Made fom polyme foil with a laye of metal deposited on suface. They have good quality and stability, and ae suitable fo time cicuits and high fequencies. Mica: Simila to metal film. Often high voltage. Suitable fo high fequencies. Pape: Used fo high voltages. Glass: Used fo high voltages. Stable tempeatue coefficient in a wide ange of tempeatues. Ceamic: Chips of alteing layes of metal and ceamic. Vey common, they find use in low-pecision coupling and filteing applications. Good fo high fequencies. Electolytic: Polaized. Simila to metal film in constuction, but the electodes ae made of aluminum etched fo much highe suface aea, and the dielectic is soaked with liquid electolyte. Can achieve high capacities. Tantalum: Like electolytic. Polaized. Bette pefomance at highe fequencies. Can toleate low tempeatues. Supecapacitos: Made fom cabon aeogel, cabon nanotubes, o highly poous electode mateials. Extemely high capacity. Moe discussion to come on the subject of dielectics
Capacitos connected in paallel Conside stating with unchaged capacitos, then connecting a battey acoss the teminals. What is the total chage in this cicuit? Why do the two capacitos have the same voltage, in equilibium? Use C Q/V Solve fo the total capacitance of this equivalent cicuit, C eq, in tems of C 1 and C. Deive the equation
Capacitos connected in seies Conside stating with unchaged capacitos, then placing a chage +Q on capacito C 1. Why do the two capacitos have the same chage, in equilibium? What is the elationship among the voltages? Use C Q/V Again, solve fo the total capacitance of this equivalent cicuit, C eq, in tems of C 1 and C. Deive the equation
Equations fo capacitos connected in paallel and in seies Paallel: C C C 1 + +... Seies: 1 C 1 C 1 + 1 C +...
Finding the equivalent capacitance of a complex cicuit by successive combination of its elements, using the equations we just deived. These cicuits ae puely capacitive. No esistos, etc. Do
Potential enegy stoed in a capacito We will calculate the wok equied to stat with a dischaged capacito and chage it to a total chage Q. Imagine we ae taking positive chage fom the lowe plate in incements dq and moving them though the voltage diffeence V ceated by the electic field. The wok equied fo this move is qv. With each dq that is moved, V inceases, so that the wok fo each dq ises as the capacito chages. We ae descibing the integation at ight, with the final answe being the potential enegy stoed in the capacito. U V +Q A + + + + dq d E _ -Q q C 1 C dw Vdq dq qdq This is the potential enegy in any chaged capacito since the deivation is geneal! Fo the case of paallel plates: ε C 0A d U Q 0 Q d ε A 0 + Q C
Geneal expessions fo potential enegy in a capacito, and enegy density of the electic field. Thee ae two othe foms of the geneal expession fo the potential enegy in a capacito, deivable fom the expession on the last page by using V Q/C to eliminate one of the thee vaiables, V, Q, o C, fom the equation: Q CV U C QV Whee is the enegy stoed? Amazingly, in the electic field itself! Fo the paallel plate capacito, which has a constant electic field, it is easy to calculate the enegy in this field, stating fom the second fom above: U 1 CV 1 ε 0A ( Ed) d 1 ε 0E ( Ad) The facto (Ad) is the volume of this capacito. If we divide by this facto, the esult will be the enegy density, a geneal fomula that applies (point by point) to all electic fields in a vacuum: u 1 ε 0E
Do Chaging one capacito fom anothe: final conditions, and enegy.
Why have we been saying in a vacuum? If we fill the volume of any capacito with a dielectic mateial, we will see the following: C 0 C Fo the same chage Q, the voltage on the capacito in vacuum, V 0, will be geate than the voltage, V, on the same capacito filled with dielectic. Since C Q/V, the capacitance has been inceased by V 0 /V. Vey useful, but how does this happen?
What s a dielectic mateial? It s an insulato. And, the popety of a dielectic that causes the incease of capacitance is its polaizability. Thee ae two classes of dielectics, with diffeent polaization mechanisms: Pola molecules Non-pola molecules Patial alignment with E E induces polaization in each atom o molecule In both cases, each dipole has an inteio electic field that points opposite to the applied field E. So, the total electic field is educed. Then, since V in a capacito is the integal of E with distance, V is educed.
The dielectic constant, K, is the facto by which C inceases: C KC 0 Dielectic constants fo vaious mateials: Moe dielectic constants, plus some beakdown voltages :
Dielectics in capacitos: binging all the physics togethe Fist, conside a block of dielectic in an extenal field E. Essentially, the block is still net neutal inside, but the polaization has induced two suface chage densities, +/ σ i, with the negative suface chage on the incoming face, and the positive on the outgoing. Using the paallel plate capacito as an example, putting this dielectic in the gap educes the electic field as follows: K C Q / V V0 V0 / d C0 Q / V0 V V / d E E The suface chages detemine E 0 and E: σ σ σ i E0 E ε 0 ε 0 Put these into the top equation to find σ : ι σ σ ε 0 i 1 σ K ε 0 1 σ i σ 1 K 0 E E 0 K (K > 1) E 0 E If K is vey lage, σ i is appoximatelyσ!
Dielectics in capacitos, continued Recall that ε 0 is called the pemittivity of the vacuum. We can imagine that the polaization of a dielectic changes this facto. So we define a pemittivity fo the mateial, ε Kε 0 that takes the polaization into account. Then we can modify the vacuum equations to apply to cases with dielectics, simply by witing ε in place of ε 0. Showing how this woks fo a paallel plate capacito: C ε 0 A KC0 K Kε 0 d A d εa d C εa d U 1 0 ) V CV 1 ( KC 1 K ε 0E ( Ad) 1 εe ( Ad) u 1 ε E We have illustated this simple eplacement ε 0 ε fo a paallel plate capacito, but it applied to all capacitos, and to all equations involving pemittivity.
Using Gauss s Law with dielectics Φ E A KE da Q encl fee ε 0 We wee actually using this law when we calculated the electic field in the dielectic based on the sum of suface chages, and found the equation below. E σ σ i ε 0
Enegy in a capacito with and without dielectic: It depends on what we hold constant! Constant Q: chage and disconnect. K Q U befoe C Q U afte KC +Q -Q K>1 U is smalle with the dielectic is inside the capacito. Constant V: leave connected to battey. K V U befoe 1 CV U afte 1 KCV K>1 U is lage with the dielectic is inside the capacito. Ae thee foces on these slabs as they ae being inseted?
Discuss Examples with dielectics
Discuss Moe examples with dielectics
Examples of seies/paallel capacito cicuits
Discuss Example with a switch: Seies? Paallel? Neithe?
Bewae: these ae not simply paallel/seies connections!