Heating effect of resistors: For a given applied voltage, four identical heating coils will produce maximum heat when they are connected in which configuration?

Introduction / Context:The heating effect of electric current is governed by Joule's law, which states that the heat generated (H) in a resistor is proportional to V^2 / R for a given voltage V. Therefore, the effective resistance of the connection arrangement determines how much heat is produced.

Given Data / Assumptions:

• Four identical heating coils, each of resistance R.
• Applied voltage is constant.
• Heat generated H ∝ V^2 / R_eq, where R_eq is the equivalent resistance of the network.

Concept / Approach:

To maximize heat, we must minimize the effective resistance. The smallest possible equivalent resistance occurs when all resistors are in parallel. For resistors in parallel, total resistance decreases with the number of branches.

Step-by-Step Solution:

Verification / Alternative check:

Heat ∝ 1/R_eq. For all parallel, R_eq = R/4, so H = V^2 / (R/4) = 4 V^2 / R, the largest among all cases.

Why Other Options Are Wrong:

• All in series: R_eq = 4R ⇒ minimum heat.
• Two parallel pairs in series: R_eq = R ⇒ less heat than parallel case.
• Mixed arrangement: R_eq = 2R/3 ⇒ still higher than R/4.

Common Pitfalls:

• Forgetting that constant voltage is applied (not constant current). Under constant current, series would maximize heat, but here voltage is constant, so parallel gives maximum heat.

Final Answer:

All in parallel