Show how would you join three resistors, each of resistance 9Ω so that the equivalent resistance of the combination is (i) 13Ω (ii) 6Ω?

The possible way of the combination are:

The resistor of 9Ω can combine parallel and one in series.

$$\begin{gathered} \left(\frac{1}{R_1} + \frac{1}{R_2} \right)+ R_3 \\ \left(\frac{1}{9} + \frac{1}{9}\right) + 9 \\ \left(\frac{2}{9}\right) + 9 \\ 4.5 \Omega + 9\Omega = 13.5 \Omega \end{gathered}$$

(ii) 2 resistors connected in series and one in parallel.

R₁ + R₂ =
= 9Ω + 9Ω = 18Ω

Now

18Ω and 9Ω are connected

$\frac{18 x 9}{18 +9} = 6\Omega$

a) Joule's Law of Heating states that the amount of heat produced in a conductor is directly proportional to the square of current passing through it, directly proportional to the resistance of the conductor, directly proportional to the time for which current passed i.e

$$\begin{gathered} H\propto I^2 ... \left(i\right) \\ H\propto R... \left(ii\right) \\ H\propto t .... \left(iii\right) \\ combining \left(i\right), \left(ii\right) and \left(iii\right) \\ H\propto I^{2}Rt Joule \end{gathered}$$

b) Resistance of first lamp = R₁
Resistance of second lamp =R₂

$$\begin{gathered} R = \frac{V^2}{P}, \\ {R}_1 = \frac{220 x 220}{100} = 48 \Omega \\ {R}_2 = \frac{220 x 220}{60} = \frac{2420}{3}\Omega \\ They are connected in parallel \\ \frac{1}{R} = \frac{1}{R_1} +\frac{1}{R_2 } = \frac{1}{484} +\frac{3}{2420} \\ R = \frac{484 x 2420}{3\left(484 + {\displaystyle \frac{2420}{3}}\right)} = \frac{605}{2}\Omega \end{gathered}$$

The resistance of a conductor depends on the following factor:

(i) The resistance of a conductor is directly proportional to length (l) of the conductor $R\propto \mathcal{l}$

(ii) The resistance of a conductor is inversely proportional to the area of the cross-section of the conductor.

$R\propto \frac{I}{A}$

(iii)Resistance also depends on a material of the conductor

$\therefore R= \rho \frac{\mathcal{l}}{A}$

(iv) Resistance and resistivity also depend on temperature.

(b) Metal are good conductors of electricity because they have free electrons to conduct electricity. Whereas glass does not have free electrons hence it is a bad conductor of electricity.

Alloys have a high resistivity in comparison to pure metals. Therefore, alloys are used for making coils for electric heating devices rather than a pure metal.

Resistance (R) = Slope of line =$\frac{1-0.5}{0.2-0.1} = \frac{0.5}{0.1} = 5\Omega$