Science (086) — AI-generated practice question

(i) Derivation – Parallel Combination

Let three resistors R₁, R₂, R₃ be connected in parallel across points X and Y. The potential difference V across each is the same.

By Ohm's law, currents through each resistor:
$$I_1 = \frac{V}{R_1}, \quad I_2 = \frac{V}{R_2}, \quad I_3 = \frac{V}{R_3}$$

Total current: $I = I_1 + I_2 + I_3$

If $R_p$ is the equivalent resistance:
$$\frac{V}{R_p} = \frac{V}{R_1} + \frac{V}{R_2} + \frac{V}{R_3}$$

$$\boxed{\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}}$$

Practical advantage: If one appliance fails, others continue to work (each branch is independent), unlike in series where failure of one breaks the entire circuit.

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(ii) Calculation

Given: $\rho_{\text{nichrome}} = 100 \times 10^{-6}\ \Omega\text{m}$, $\rho_{\text{copper}} = 1.62 \times 10^{-8}\ \Omega\text{m}$; same length and area.

Since $R = \rho\,\dfrac{l}{A}$, for same $l$ and $A$:

$$\frac{R_{\text{nichrome}}}{R_{\text{copper}}} = \frac{\rho_{\text{nichrome}}}{\rho_{\text{copper}}} = \frac{100 \times 10^{-6}}{1.62 \times 10^{-8}} \approx 6173$$

Preferred material for toaster element: Nichrome, because its resistivity is very high (generates more heat), and it does not oxidise (burn) readily at high temperatures.

Source: Chapter 11, Sections 11.5 and 11.6

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• The derivation must show the step: same V, currents add, divide throughout by V — examiners look for this logic explicitly.
• The practical advantage point is standard and must specifically contrast with series.
• For part (ii), since $l$ and $A$ are identical, resistance ratio equals resistivity ratio directly — show the substitution clearly.
• The reason for choosing nichrome must include two points: high resistivity AND resistance to oxidation at high temperature. Both are in Table 11.2's explanatory text and are commonly tested.