A copper wire of length L and cross-sectional area A carries a current I and dissipates power P as heat. An aluminium wire of the same length L and the same cross-sectional area A is now connected to the same current I. Compared to P, how much power will be dissipated in the aluminium wire?

Introduction / Context:
This question checks your understanding of electric power dissipation in resistive conductors and how it depends on the material of the wire. Copper and aluminium are common conducting materials, but they have different electrical resistivities. When current is fixed, the power lost as heat in a wire depends directly on its resistance. Recognising which material has higher resistivity helps you decide whether the power dissipation increases or decreases when switching from copper to aluminium while keeping dimensions and current the same.

Given Data / Assumptions:
• Copper wire: length L, area A, current I, power dissipated P. • Aluminium wire: same length L and same cross-sectional area A. • The same current I is passed through the aluminium wire. • Resistivity of copper is lower than the resistivity of aluminium (rho_cu < rho_al). • Wires are ohmic and obey P = I^2 * R.

Concept / Approach:
The resistance of a wire is given by R = rho * L / A, where rho is resistivity. For a given length and cross-sectional area, resistance is directly proportional to resistivity. The power dissipated in a resistor carrying a constant current is P = I^2 * R. If the same current I passes through two wires of identical dimensions but different resistivities, then the wire with higher resistivity has higher resistance and therefore dissipates more power for the same current. Aluminium has higher resistivity than copper, so the aluminium wire will have a larger resistance and hence more power dissipation than the copper wire under the same current.

Step-by-Step Solution:
Step 1: Write the resistance of the copper wire: R_cu = rho_cu * L / A. Step 2: Write the resistance of the aluminium wire: R_al = rho_al * L / A. Step 3: Since L and A are the same, R_al / R_cu = rho_al / rho_cu. Step 4: From material properties, rho_al is greater than rho_cu, so R_al is greater than R_cu. Step 5: Power in copper wire: P = I^2 * R_cu. Step 6: Power in aluminium wire: P_al = I^2 * R_al. Step 7: Because R_al > R_cu and I is the same, P_al > P. Step 8: Therefore, the aluminium wire dissipates more power than the copper wire.

Verification / Alternative check:
Take simple numbers for illustration. Suppose that for the given dimensions, the copper wire has resistance 1 ohm and aluminium wire has resistance 1.6 ohm (higher because of higher resistivity). With the same current I = 2 A, the copper wire dissipates P_cu = I^2 * R_cu = 4 * 1 = 4 W. The aluminium wire dissipates P_al = 4 * 1.6 = 6.4 W, which is clearly greater than 4 W. This numerical example confirms the general conclusion that power dissipation increases when replacing copper with aluminium under the same current and dimensions.

Why Other Options Are Wrong:
Option a (Equal to P): Power would be equal only if resistivities were the same or current was adjusted to compensate, which is not the case here. Option b (Less than P): This would require aluminium to have lower resistivity than copper, which is incorrect. Option d (Exactly 2P): There is no information suggesting that the resistance of aluminium is exactly twice that of copper; the actual ratio depends on precise resistivity values. Option e (Zero): Power is zero only when current is zero or resistance is zero, neither of which applies in this situation.

Common Pitfalls:
A frequent mistake is to focus on the fact that aluminium is also a good conductor and assume it behaves the same as copper. While aluminium is indeed a conductor, its resistivity is higher, leading to higher resistance for identical geometry. Another error is to think in terms of P = V^2 / R and forget that here voltage is not specified; the current I is fixed. In constant current scenarios, power scales directly with resistance, while in constant voltage situations, power scales inversely with resistance. Always pay close attention to whether current or voltage is being held constant.

Final Answer:
With the same current and same dimensions, the aluminium wire dissipates greater than P (more power) compared to the copper wire.