They are so many

Electricity and Magnetism Level 2

In the image above, gray dots are capacitors of capacitance $1\mu f$ .

That gray area shows that, there is infinite arrangement of capacitors, such that first circle has 4 capacitors , second one has 8 capacitors, third 16 capacitors and next 32 and so on, as shown.

Find the equivalent capacitance in $\mu f$ between the two terminals $A$ (at the center) and $B$ (at infinity).

The answer is 2.

3 solutions

Baljeet Dhanoa
Jun 14, 2015

We can see that as we go outwards from pt a to b the first 4 capacitors appear to be in parallel then the next 8 appear to be in parallel this is because each circle is an equipotential surface , the system of the first four capacitors is in series with the system of the next 8 and so on till infinity.

Thus , 1/c = (1/4 (1))/ (1-1/2) using sum of infinite gp c= 2

Lu Chee Ket
Dec 14, 2015

~ 4 C ~ 8 C ~ 16 C ~ 32 C ~ ... (in series to infinite terms.):

$\dfrac{1}{\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+...}$ = 2

OR step by step using $C_{series~capacitance} = \dfrac{C_1 C_2}{C_1 + C_2}$ :

4   
8   2.666666667
16  2.285714286
32  2.133333333
64  2.064516129
128 2.031746032
256 2.015748031
512 2.007843137
1024 2.003913894
...

Answer: $\boxed{2}$

Santhosh Roshan
Jun 28, 2015

JUST A SUM OF GEOMETRIC PROGRESSION 4,8,........... AS THE CAPACITORS OR IN PARALLEL CONNECTION .

They are so many

The answer is 2.

3 solutions

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