We are designing an electric space heater to operate from 120 V. Two heating elements with resis

step 1 Here it is given minimum power delivered between the two register is 240 by 120 volts supply whe

We are designing an electric space heater to operate from...

We are designing an electric space heater to operate from $120 \mathrm{V}$. Two heating elements with resistances $R_{1}$ and $R_{2}$ are to be used that can be operated in parallel, separately, or in series The highest power is to be $1280 \mathrm{W}$, and the lowest power is to be $240 \mathrm{W}$. What values are needed for $R_{1}$ and $R_{2} ?$ What intermediate power settings are available?

Key Concepts

Ohm’s Law

Ohm’s Law is the fundamental relationship between voltage, current, and resistance in an electrical circuit. It states that the voltage across a resistor is equal to the product of the current through it and its resistance (V = IR). In designing a heater, this law allows us to determine how much current will be drawn by a resistor when a fixed voltage is applied, and it underpins the calculations for power dissipation in different circuit configurations.

Electrical Power Calculation

Electrical power in a resistive element is calculated using the formula P = V²/R when a constant voltage is applied. This relation shows that for a given applied voltage, the power delivered—and hence the heating effect—depends inversely on the resistance. Choosing proper resistor values is therefore critical when the design requires specific power outputs (like a maximum of 1280 W and a minimum of 240 W) under different wiring configurations.

Series and Parallel Resistive Networks

Connecting resistors in series or parallel changes their effective resistance and, consequently, the power they dissipate. In series, resistances add together (R_total = R1 + R2), leading to a decrease in power for a constant applied voltage. In parallel, the effective resistance is given by R_parallel = (R1 R2)/(R1 + R2), which is lower than either individual resistor and results in a higher power draw. Understanding these configurations is essential for designing a heater with multiple operating modes.

Design Constraints in Resistive Heating

When designing electrical heating systems, engineers must choose resistor values that meet safety and performance specifications over a range of configurations. Constraints such as a maximum power rating (to avoid overheating) and a minimum power output (to ensure effective heating) dictate the permissible resistor values. The interplay between individual resistor values and their series or parallel combinations is critical to achieving the desired range of power outputs.

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