(a) What is the resistance of a \(1.00 \times {10^2}\;\Omega \), a \(2.50\;{\rm{k}}\Omega ,\)and a \({\bf{4}}{\bf{.00}}\;{\bf{k\Omega }}\) resistor connected in series? (b) In parallel?

The values of the three resistances given, are-

\(\begin{align}{}{R_1} = 100\;\Omega \\{R_2} = 2500\;\Omega \\{R_3} = 4000\;\Omega \end{align}\)

Resistance in series: equivalent resistance in series combination can be obtained by adding the resistance of all the resistors.

Resistance in parallel: in parallel combination, the reciprocal of equivalent resistance can be calculated by adding the reciprocal of all the resistances.

(a)

When three resistors are connected in series,

\({R_{eq}} = {R_1} + {R_2} + {R_3}\)

\(\begin{align}{}{R_{eq.}} &= 100\;\Omega + 2500\;\Omega + 4000\;\Omega \\ &= 6600\;\Omega \\ &= 6.6\;{\rm{k}}\Omega \end{align}\)

The equivalent resistance in series combination is given as- \(6.6\;{\rm{k\Omega }}\)

(b

when the resistors are connected in parallel,

\(\begin{align}{}\frac{1}{{{R_{eq}}}} &= \frac{1}{{{R_1}}} + \frac{1}{{{R_2}}} + \frac{1}{{{R_3}}}\\ &= \frac{1}{{100\;\Omega \;}} + \frac{1}{{2500\;\Omega }} + \frac{1}{{4000\;\Omega }}\\ &= \left( {\frac{{200 + 8 + 5}}{{20000}}} \right)\;{\Omega ^{ - 1}}\\ &= \frac{{213}}{{20000}}\;\;{\Omega ^{ - 1}}\end{align}\)

Further,

\(\begin{align}{}{R_{eq}} &= \frac{{20000}}{{213}}\;\Omega \\ &= 93.9\;\Omega \end{align}\)

The equivalent resistance in parallel combination is given as- \(93.9\;\Omega \).