apacitance and capacitors
apacitors apacitors are devices that store energy in an electric field. apacitors are used in many every-day applications Heart defibrillators amera flash units apacitors are an essential part of electronics. apacitors can be micro-sized on computer chips or super-sized for high power circuits such as FM radio transmitters.
Definition of apacitance The definition of capacitance is Q V The units of capacitance are coulombs per volt. The unit of capacitance has been given the name farad (abbreviated F) named after British physicist Michael Faraday (79-867) F V A farad is a very large capacitance Typically we deal with F (0-6 F), nf (0-9 F), or pf (0 - F) 3
I-The parallel-plate capacitor The capacitance of a device depends on the area of the plates and the distance between the plates Where: q V A is the area of one of the plates, d is the separation, e o e 0 is a constant (permittivity of free space), e 0 = 8.85 0 - /N m A d d +Q -Q A A 4
Parallel plates capacitor =F, and the distance between them is d=mm, what is the area of each plate A? q V e o A d A 8.30 m 5
Example: A parallel plate capacitor has plates.00 m in area, separated by a distance of 5.00 mm. A potential difference of 0,000 V is applied across the capacitor. Determine the capacitance the charge on each plate
II-ylindrical apacitor onsider a capacitor constructed of two collinear conducting cylinders of length L. The inner cylinder has radius r = a and the outer cylinder has radius r = b. Both cylinders have charge per unit length with the inner cylinder having positive charge and the outer cylinder having negative charge. We will assume an ideal cylindrical capacitor The electric field points radially from the inner cylinder to the outer cylinder. The electric field is zero outside the collinear cylinders. 7
III-Spherical apacitor onsider a spherical capacitor formed by two concentric conducting spheres with radii r and r Let s assume that the inner sphere has charge +q and the outer sphere has charge q. The electric field is perpendicular to the surface of both spheres and points radially outward 8
ombinations of capacitors It is very often that more than one capacitor is used in an electric circuit We would have to learn how to compute the equivalent capacitance of certain combinations of capacitors 5 3 3 4 9
a. Parallel combination onnecting a battery to the parallel combination of capacitors is equivalent to introducing the same potential difference for both capacitors, A total charge transferred to the system from the battery is the sum of charges of the two capacitors, By definition, a Q V & Q V +Q +Q Thus, eq would be V=V ab -Q -Q Q eq eq V Q Q V V,: V V V b 0
Parallel combination: Analogous formula is true for any number of capacitors, (parallel combination) eq 3... It follows that the equivalent capacitance of a parallel combination of capacitors is greater than any of the individual capacitors
Example: A 3 F capacitor and a 6 F capacitor are connected in parallel across an 8 V battery. Determine the equivalent capacitance and total charge deposited. a +Q +Q V=V ab -Q -Q b
b- Series combination onnecting a battery to the serial combination of capacitors is equivalent to introducing the same charge for both capacitors, A voltage induced in the system from the battery is the sum of potential differences across the individual capacitors, By definition, V Q Thus, & eq V would Q be a V Q V eq eq V Q Q,: Q Q Q V=V ab b c +Q -Q +Q -Q 3
Series combination: Analogous formula is true for any number of capacitors,... eq 3 (series combination) It follows that the equivalent capacitance of a series combination of capacitors is always less than any of the individual capacitance in the combination eq 4
Example: A 3 F capacitor and a 6 F capacitor are connected in series across an 8 V battery. Determine the equivalent capacitance and total charge deposited. a +Q V=V ab c -Q +Q -Q b
Find the equivalent capacitance between points a and b in the combination of capacitors shown in Figure. 6
Energy stored in a charged capacitor onsider a battery connected to a capacitor A battery must do work to move electrons from one plate to the other. The work done to move a small charge dq across a voltage V is: W = V dq. As the charge increases, V increases so the work to bring dq increases. Using calculus we find that the energy (U) stored on a capacitor is given by: V U QV Q V V q q V. 7
The electric field energy in parallel-plate capacitor Find electric field energy density (energy per unit volume) in a parallel-plate capacitor Recall Thus, U V e A d u U / volume energy density 0 volume Ad V Ed e A Ed d 0 = ( ) /( A d ) and so, the energy density is u e 0E - - - - - - - - 8
Example: In the circuit shown V = 48V, = 9F, = 4F and 3 = 8F. (a) determine the equivalent capacitance of the circuit, (b) determine the energy stored in the combination by calculating the energy stored in the equivalent capacitance, V 3 9
apacitors with dielectrics A dielectrics is an insulating material (rubber, glass, etc.) onsider an insolated, charged capacitor Insert a dielectric Q -Q Q -Q q V e o A d V 0 V Notice that the potential difference decreases (k = V 0 /V ) Since charge stayed the same (Q=Q 0 ) capacitance increases k d o V V o d dielectric constant: k = / 0 Dielectric constant is a material property apacitance is multiplied by a factor k when the dielectric fills the region between the plates completely e 0 A d 0
How does an insulating dielectric material reduce electric fields by producing effective surface charge densities? Reorientation of polar molecules - - - - - - - - - - - - - - - - -
Discussion (apacitance and capacitors)
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Example: In the following circuit: = 0 F, = 5.0 F, 3 = 4.0 F. -Find the equivalent capacitance of the whole combination. - If V = 00 volts, what is the charge Q 3 on 3? 3-What is the total energy stored in the circuit? V + - = 0 F = 5.0 F 3 = 4.0 F 3 4
Example: apacitors in Series and Parallel Three capacitors are connected as shown. (a) Find the equivalent capacitance of the 3-capacitor combination. (b) The capacitors, initially uncharged, are connected across a 6.0 V battery. Find the charge and voltage for each capacitor. eq eq 5
Example: Tow series plates capacitor : 6 µf & 4 µf are connected together and charged by voltage 00 V,, Find each charge and Voltage? 6
Example: A parallel plates capacitor, filed with two different dielectrics: K,K. Find the capacitance. A A/ A/ d 7
Example: How many capacity of µf must reach in parallel So they can be charge if 300V voltage difference? 8
Example: A parallel plate capacitor whose plates have an area of m and are separated by a distance of cm. The capacitor is charged to an initial voltage of 3 kv and then disconnected from the charging source. An insulating material is placed between the plates, completely filling the space, resulting in a decrease in the capacitors voltage to kv. Determine: - the original capacitance,. - the dielectric constant of the material. 3- the new capacitance 4- the charge on the capacitor. 5- The stored electric energy before and after inserting dielectric. 9
Example: k 30