Find the equivalent cappacitance between A and B. (a) Three conducting concentic shells of radii r,2r and (b) An isoltated ball-shaped conductor of radius r surrounded by an adjuacent layer of dielectric (K) and outer radius 2r. Two conducting concentic spherical shells having radii r_(1) and r_(2)(r_(2)gtr_(1)), calculate capacity of the system if (i) The inner shell is given a charge and outer is connected to the earth. (ii) The outer shell is given a charge and inner is connected to the earth. (d) Find capacitance of the system (i) Two shells of radii r_(1) and r_(2) having charges are connected by a metallic wire, the separation between shells is large. (ii) Two shells of radii r_(1) and r_(2) carry equal and opposite charges and are separated by a distance d.

Find the equivalent cappacitance between A and B. (a) Three conductin

1.	Find the equivalent cappacitance between A and B. (a) Three conducting concentic shells of radii r,2r and (b) An isoltated ball-shaped conductor of radius r surrounded by an adjuacent layer of dielectric (K) and outer radius 2r. Two conducting concentic spherical shells having radii r_(1) and r_(2)(r_(2)gtr_(1)), calculate capacity of the system if (i) The inner shell is given a charge and outer is connected to the earth. (ii) The outer shell is given a charge and inner is connected to the earth. (d) Find capacitance of the system (i) Two shells of radii r_(1) and r_(2) having charges are connected by a metallic wire, the separation between shells is large. (ii) Two shells of radii r_(1) and r_(2) carry equal and opposite charges and are separated by a distance d.
Answer» SOLUTION :(a) Two capacitors, Capacitor `1,C_(1)=(4pi in_(0)r.2r)/((2r-r))=8piin_(0)r` Capacitor `2,C_(2)=(4pi in_(0)2r.3r)/((3r-2r))=24piin_(0)r` connected a battery between `A` and `B`, and check polarity, `+,-,+,-,`. `C=(C_(1)C_(2))/(C_(1)+C_(2))=+pi in_(0)r` (b) capacitor `1`: `C_(1)=(4pi in_(0)K.r.2r)/(2r-r)=8pi in_(0)Kr` Capacitor `2`: Isolated sphere of radius `2r` and outer radius infinite. `C_(2)=4pi in_(0)2r=8pi in_(0)r` `C_(1)` and `C_(2)` in series, as shown in `(a)` `C-(eq)=(C_(1)C_(2))/(C_(1)+C_(2))=(8pi in_(0)Kr)/(K+1)` (c) (i) Let total charge on outer shell is `Q`' `V_(p)=(1)/(4pi in_(0))((Q)/(r_(2))+(Q')/(r_(2)))=0impliesQ'=-Q` `C=(4pi in_(0)r_(1)r_(2))/(r_(20-r-(1))` (ii) Let charge on inner shell be `Q'`. `V_(p)=(1)/(4piin_(0))((Q')/(r_(1))+(Q)/(r_(2)))=0` `Q'=(-Qr_(1))/(r_(2))` This capacitor is combination of two capacitors (i) capacitor `1,C_(1)=(4piin_(0)r_(1)r_(2))/(r_(2)-r_(1))` (ii) Capacitor `2`, `C_(2)=4pi in_(0)r_(2)` (isolated spherical capacitor) These capacitor are in parallel. the POSITIVE plates of capacitors are at one place. the `-ve` plate of capacitor `1` is connected to ground i.e., its potential is zero. the `-ve` plate of capacitor `2`is at infine i.e., its potential is zero. hence, positive plates at one point and negative plates at another point and hence `C_(1)` and `C_(2)` are in parallel. `C_(eq)=C_(1)+C_(2)=4pi in_(0)r_(2)((r_(1))/(r_(2)-r_(1))+1)=(4piin_(0)r_(2)^(2))/((r_(2)-r_(1)))` (d) (i) `C_(1)=4pi in_(0)r_(1)` `C_(2)=4pi in_(0)r_(2)` `C_(1)` and `C_(2)` are in parallel. theirpositive platesare connected as shown and negative plates are at infine. `C_(eq)=C_(1)+C_(2)=4piin_(0)(r_(1)+r_(2))` (ii) Here, we will find capacitance with the HELP of energy. energy stored in a conductor, `U=(Q^(2))/(2C)` Energy of two point charges sepearated by `r`, `U'=(1)/(4piin_(0)) (Q_(1)Q_(2))/(r)` so potential energy of system `U=U_(1)+U_(2)+U_(12` `=(Q^(2))/(2C_(1))+(Q^(2))/(2C_(2))+(1)/(4piin_(0))((Q)(-Q))/(d)` `=(Q^(2))/(2xx4piin_(0)r_(1))+(Q^(2))/(2xx4piin_(0)r_(2))-(Q^(2))/((4piin_(0)d)` `=(Q^(2))/(8piin_(0))((1)/(r_(1))-(1)/(r_(2))-(2)/(d))` .....(i) For a system, `U=(Q^(2))/(2C_(eq))` Equating (i) and (ii), we get `(1)/(C_(eq))=(1)/(4piin_(0))((1)/(r_(1))+(1)/(r_(2))-(2)/(d))` `C_(eq)=(4piin_(0))/(((1)/(r_(1))+(1)/(r_(2))-(2)/(d)))` If `drarroo,C_(eq)=(4piin_(0))/((1)/(r_(1))+(1)/(r_(2)))` `(1)/(C_(eq))=(1)/(4piin_(0)r_(1))+(1)/(4piin_(0)r_(2))=(1)/(C_(1))+(1)/(C_(2))` i.e., the given system is equivalent to two capacitors `C_(1)=4pi in_(0)r_(1)` and `C_(2)=4piin_(0)r_(2)` in series. . . . .