Question

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=(4piepsilon_0)/(1/a+1/b+-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 1

Hence proved.

Given:

Two spheres have radii a and b and their centers are at a distance d apart.

Capacitance of this system is   C =  $\frac{4 \pi \varepsilon_{o}}{\frac{1}{a}+\frac{1}{b} \pm \frac{2}{d}}$

Show that:

As d approaches infinity the above result reduces to that of two isolated spheres in  series.

Explanation:

As d approaches infinity $\frac{1}{d}$  approaches to zero.

So that Capacitance of this system is  C=   $\frac{4 \pi \varepsilon_{o}}{\frac{1}{a}+\frac{1}{b} }$

                                     C = $\frac{4 \pi \varepsilon_{o}(ab)}{(a+b) }$

When capacitor is series the equivalent capacitance = $\frac{C_{1} C_{2}}{C_{1}+C_{2}}$

Where $C_1$ = capacitance of first sphere = $\varepsilon_{o}$ × $\frac{A}{D}$ = 4$\pi$$\varepsilon_{o}$a

           $C_2$ = capacitance of Second sphere = $\varepsilon_{o}$ × $\frac{A}{D}$ = 4$\pi$$\varepsilon_{o}$b

Equivalent capacitance = $\frac{C_{1} C_{2}}{C_{1}+C_{2}}$ =  (4$\pi$$\varepsilon_{o}$a)(4$\pi$$\varepsilon_{o}$b) / (4$\pi$$\varepsilon_{o}$a)+(4$\pi$$\varepsilon_{o}$b)

                                                    = $\frac{4 \pi \varepsilon_{o}(ab)}{(a+b) }$

So from above equation it is proved that  d approaches infinity the above result reduces to that of two isolated spheres in series.

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