Electricity And Magnitism, Resistance

To begin, the resistance of resistors in series is

$R_{series} = \sum\limits_{n} R_{n} = R_{1} + R_{2} + R_{3} + \ldots$

and the resistance of resistors in parallel is

$R_{parallel} = \frac{1}{\sum\limits_{n} \frac{1}{R_{n}}} = \frac{1}{\frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + \ldots}$

It's not too difficult to see that if you keep adding resistors into a series circuit, you are adding positive numbers to a positive sum and increasing the total series resistance. Similarly, if you add resistors into a parallel circuit, you are adding the inverse of a positive number (which is still positive) to the positive denominator of the parallel resistance, decreasing it in value.

The optimal strategy to increase the resistance to a maximum is therefore to start with a single $0.1\Omega$ resistor and continue adding resistors in series. This results in a total resistance of

$R_{series} = \sum\limits_{n} R_{n} = 0.1\Omega + 0.1\Omega + 0.1\Omega + \ldots = 0.1\Omega * 10 = \boxed{1 \Omega}$

Analogously, the optimal strategy to minimize the resistance is to start with a single $0.1 \Omega$ resistor and continue to add resistors in parallel. The total parallel resistance is therefore

$R_{parallel} = \frac{1}{\sum\limits_{n} \frac{1}{R_{n}}} = \frac{1}{\frac{1}{0.1\Omega} + \frac{1}{0.1\Omega} + \frac{1}{0.1\Omega} + \ldots} = \frac{1}{\frac{1}{0.1\Omega} * 10} = \boxed{0.01\Omega}$

Because there is only one option with both of these answers, it therefore must be correct.

Maximum resistance obtained for series connection = n*R= 10 *0.1=1ohm

Minimum resistance obtained for parallel connection = R/n = 0.1/10= 0.01ohm

To obtain the higest possible resistance rom a set of resistors, they need to be connected in series. The effective resistance will be the sum of all resistances constituting the series. R= r1+ r2 + r3 +...r10 or R=r 10 since all r= 0.1 Therefore higest resistance= 0.1 10=1ohm Lowest possibible resistance can be got by connecting the resistors in parallel.The RECIPROCAL of the effective resistance is equal to the sum of RECIPROCALS of the the component resistances. Since all resistances are same in this case the effective resistance can be simply calculated as R/n So lowest resistance = 0.1/10 =0.01 ohm

the largest resistance can be obtained from series connection where Rs= r1+r2+r3+.........+r10 so, Rs=0.1+0.1+0.1+........+.01=1ohm the smallest res can be obtained from parallel connection where 1/Rp = 1/r1 + 1/r2 .............. +1/r10 1/Rp= 1/0.1 +1/0.1 +.............+1/0.1= 100 so Rp= 1/100 = 0.01ohm

For the smallest resistance all the resistors must be connected in Parallel.

Hence,

Minimum Resistance = R/10 = 0.01 ohm.

For the largest resistance all the resistors must be connected in Series.

Hence,

Maximum Resistance = 10R = 1 ohm.

Maximum resistance can be obtained when the resistors are connected in series combination Hence,R(series)=0.1x10=1 ohm And minimum can be obtained by putting all the resistors in parallel hence,R(parallel)=0.1/10=0.01