a. Two spheres have radii a and b and their centres are at a distance d apart. Show that the capacitance of this system is C=a1+b1±d24πϵ0 provided that d is large compared with a and b. b. Show that as d approaches infinity the above result reduces to that of two islotated spheres inseries.

Solution: a. PD,V=V1+V2 =4πϵ01[(aq−dq)−(b−q+dq) ∴C=Vq−a1+b1−d24πϵ0 If −q is given to first shere and +q to second sphere, then C=a1+b1+d24πϵ0 b. If d→∞, then C=a1+b14πϵ0=a+b(4πϵ0)(ab) in series, C=t=C1+C2C1C2=(4πϵ0a)+(4πϵ0b)(4πϵ0a)(4πϵ0b)=(a+b(4πϵ0)ab)

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a. Two spheres have radii {a} and {b} and their centres are at

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a. Two spheres have radii $a$ and $b$ and their centres are at a distance d apart. Show that the capacitance of this system is
$C = \frac{4 π ϵ _{0}}{\frac{1}{a} + \frac{1}{b} \pm \frac{2}{d}}$
provided that $d$ is large compared with $a$ and $b$ .
b. Show that as $d$ approaches infinity the above result reduces to that of two islotated spheres inseries.

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a. Two spheres have radii

a

and

b

and their centres are at a distance d apart. Show that the capacitance of this system is

C = \frac{4 π ϵ _{0}}{\frac{1}{a} + \frac{1}{b} \pm \frac{2}{d}}

provided that

d

is large compared with

a

and

b

.
b. Show that as

d

approaches infinity the above result reduces to that of two islotated spheres inseries.

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Text solutionVerified

Solution: a.

P D, V = V_{1} + V_{2}

= \frac{1}{4 π ϵ _{0}} [(\frac{q}{a} - \frac{q}{d}) - (\frac{- q}{b} + \frac{q}{d})

∴ C = \frac{q}{V} - \frac{4 π ϵ _{0}}{\frac{1}{a} + \frac{1}{b} - \frac{2}{d}}

- q

is given to first shere and

+ q

to second sphere, then

C = \frac{4 π ϵ _{0}}{\frac{1}{a} + \frac{1}{b} + \frac{2}{d}}

b. If

d \to \infty

, then

C = \frac{4 π ϵ _{0}}{\frac{1}{a} + \frac{1}{b}} = \frac{( 4 π ϵ _{0} ) ( a b )}{a + b}

in series,

C_{\neq = t} = \frac{C _{1} C _{2}}{C _{1} + C _{2}} = \frac{( 4 π ϵ _{0} a ) ( 4 π ϵ _{0} b )}{( 4 π ϵ _{0} a ) + ( 4 π ϵ _{0} b )} = (\frac{( 4 π ϵ _{0} ) a b}{a + b})

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Question Text	a. Two spheres have radii $a$ and $b$ and their centres are at a distance d apart. Show that the capacitance of this system is $C = \frac{4 π ϵ _{0}}{\frac{1}{a} + \frac{1}{b} \pm \frac{2}{d}}$ provided that $d$ is large compared with $a$ and $b$ . b. Show that as $d$ approaches infinity the above result reduces to that of two islotated spheres inseries.
Answer Type	Text solution:1
Upvotes	150

Text solutionVerified

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