A 33 Ω half-watt resistor and 330 Ω half-watt resistor are conneted across a 12 V source. Which one(s) will overheat?

Concept

The power across any resistor is given by:

\(P={V^2\over R}=I^2R\)

where, P = Power

V = Supply voltage

I = Current

R = Resistance

Calculation

Since the phrase is "the resistors are connected across the voltage source, I assumed that the resistors are in parallel. So, same voltage. V = 12V

The power dissipated in 33 Ω is:

\(P_{33}={12^2\over 33}=4.36 \space W\)

The power dissipated in 330 Ω is:

\(P_{33}={12^2\over 330}=0.436 \space W\)

The rating of the 33 Ω resistor is half-watt and the power dissipated in it is 4.36 which is more than the rating. Hence, the 33 Ω resistor will overheat.

Additional Information
If we take a series connection of resistors, then:

\(I={12\over 33+330}=0.033A\)

\(P_{33}=(0.033)^2(33)=0.035\space W\) < 0.5 W. It will not overheat.

\(P_{330}=(0.033)^2(330)=0.35\space W\) < 0.5. It will also not overheat.

In case of series connection, neither of the resistor will overheat.