Electrical apacity Synopsis Electrical apacity i) Electrical capacity of a conuctor is its ability to store electric charge i The potential acuire by a conuctor is irectly proportional to the charge given to it ie, Q ie, Q or Q where the of proportionality is calle the electrical capacity of the conuctor Thus the capacity of a conuctor is efine as the ratio of the charge to the potential iv) Its SI unit is fara v) milli fara ( mf) -3 fara micro fara ( μ F) -6 fara pico fara ( pf) - fara vi) The capacity of a spherical conuctor in fara is given by raius of the conuctor 9 4 πε r, where r v If we imagine Earth to be a uniform soli sphere then the capacity of earth 3 64 4πε R 7μF mf 9 9 Parallel plate apacitor i) onenser (usually, a combination of two conuctors) is a evice by means of which larger amount of charge can be store at a given potential by increasing its electric capacity apacitance of a capacitor or conenser is the ratio of the charge on either of its plates to the potential ifference between them i apacity of a parallel plate conenser without meium between the plates ε A A area of each plate; istance between the plates

Iv) With a meium of ielectric K completely filling the space between the plates ε K A v) The ielectric of a ielectric material is efine as the ratio of the capacity of the parallel plate conenser with the ielectric between the plates to its capacity with air or vacuum between the plates K apacity of the conenser with ielectric meium between plates apacity of the same coneser with air as meium between plates vi) When a ielectric slab of thickness t is introuce between the plates ε A t t k ε A t k v In this case the istance of separation by capacity k t an hence the vi To restore the capacity to original value the istance of separation is to be increase by t k ix) a) If a metal slab of thickness t is introuce between the plates because for metals K is infinity εa t b) If a number of ielectric slabs are inserte between the plates, each parallel to plate surface, then euivalent capacity A t t tn K K Kn If those slabs completely fill up the gap between the plates leaving without any air gap εa t t t n K K Kn x) In a parallel plate capacitor, the electric fiel at the eges is not uniform an that fiel is calle as the fringing fiel

xi) Electric fiel between the plates is uniform electric intensity E σ ε Q Aε Q Here σ is the surface charge ensity on the plates Q/A x Potential ifference between the plates E Q A ε xi Force on each plate Q Q F EQ εae ε A xiv) Energy store per unit volume of the meium 3 ombination of onensers i) When conensers are connecte in series ) All plates have the same charge in magnitue ε ) Potential ifferences between the plates are ifferent 3) : : 3 : : 3 4) Euivalent capacity is then, 3 5) The euivalent capacity is less than the least iniviual capacity 6) Energies of the conensers E : E : E 3 : : 3 7) Total energy of the combinatione E E 3 When conensers are connecte in parallel ) PD across each conenser is same ) harge of each conenser is ifferent Q : Q : Q 3 : : 3 3) Euivalent capacity of the combination 3 4) The euivalent capacity is greater than the greatest iniviual capacity 5) Energies of the conensers E : E : E 3 : : 3 6) Total energy of the combinatione E E 3 i When n ientical conensers each of capacity ) ombine in series, the effective capacity s /n ) ombine in parallel, the effective capacity p nc E 3 3

3) Ratio of the effective capacities s : p : n iv) Mixe group: If there are N capacitors each rate at capacity an voltage, by combining those we can obtain effective capacity rate at an voltage For this n capacitors are connecte in a row an m such rows are connecte in parallel Then n an m n where mn N v) If p an s are the euivalent capacities of two capacitors of capacity an in parallel an series respectively then P p 4PS P P 4 P S an vi) Two capacitors are connecte in parallel to a battery as shown in the figure i) v Two capacitors are connecte in parallel to a battery as shown in the figure i) vi If n ientical capacitors each of capacity are connecte in a suare then a) The resultant capacity between any two ajacent corners A an B 4 3 b) The resultant capacity between any two opposite corners A an ix) If n ientical capacitors each of capacity are connecte in a polygon then a) The resultant capacity between any two ajacent corners 4 b) The resultant capacity between any two opposite corners n n n x) a) If n ientical capacitors are given then they can be connecte in n ifferent ways by taking all the conensers at a time (n > ) b) In n ifferent capacitors are given then they can be connecte in n ifferent ways by taking all the conensers at a time

xi) In a parallel plate capacitor, if n similar plates at eual istance are arrange such that alternate plates are connecte together, the capacitance () of the arrangement is ( n )ε A for air or vacuum an it becomes ielectric meium of ielectric K 4 Energy of capacitor i) The electrostatic energy store in a charge capacitor is eual to Q or Q or ( n )ε AK This energy is store in the uniform electric fiel that is present between the plates of the capacitor 5 ombination of charge capacitors i) If two conensers of capacities an are charge to potentials an respectively an are joine in parallel (ve plate connecte to ve plate), then the common potential Q Q The loss of energy in this process (manifeste as heat) is given by U ( ) ( ) i When two conensers of capacities an charge to potentials an are connecte anti-parallel (ve plate connecte to ve plate) as shown in the figure a) ommon potential b) Loss of energy Q Q ( ) ( ) c) Loss of energy is more in this case compare with previous case in a 3

6 apacitance of spherical conenser a) apacitance of single isolate sphere 4πε R where R is its raius b) In two concentric spheres (outer raius a an inner raius b) i) When the inner is charge an the outer is earthe, then 4πεεr ab a b 4 Kab πε ( a b) When the inner sphere is earthe, then 4πεε ra a b 4πεKa a b 7 Introuction of ielectric in a charge capacitor A ielectric slab (K) is introuce between the plates of the capacitor SNo 3 4 5 Physical uantity apacity harge PD Electric Intensity permanently connecte K time isconnecte 4

8 The istance between the plates of conenser is increase by n times SNo Physical uantity apacity harge 3 PD 4 Electric Intensity 5 Energy store in conenser permanently connecte n time n times n time n times 9 ombination of charge spherical rops isconnecte n times n times Remain n times When n ientical charge small spherical rops are combine to form a big rop Sno a b c e Quantity Raius harge apacity Potential Energy For each For the big charge small rop rop r R n /3 r Q n n /3 n /3 ν ν n 5/3 ν 5