Question

How would you connect a pair of equal resistors across a battery in order to get the most power dissipation in the resistors?

Answer 1

Resistors in Series and Parallel

Objects can be connected to a circuit in one of two ways: series or parallel.

When objects are connected in series, any current that flows through object 1 must also flow through object 2. The total voltage drop across the two objects is equal to the voltage drop across the first object plus the voltage drop across the second object. Therefore:

Therefore, when two resistors are connected in series, they act like a single resistor with a resistance,. We call this the equivalent resistance.

When objects are connected in parallel, the voltage across all of the objects must be the same. To see why, consider the circuit shown below:

Since the positive end of the battery is directly connected to the left side of both resistors, and the negative end of the battery is directly connected to the right side of both resistors, the voltage across both resistors is equal to .

In general, the current flowing through the resistors will not be the same. However, the total current flowing through both resistors is equal to:

When two resistors are connected in parallel, the formula for the equivalent resistance can be rewritten as:

Notice that that the equivalent resistance of two resistors connected in parallel is always less than either of the two resistors individually, since:

Also, when identical resistors are connected in parallel, the equivalent resistance is given by:

So, .

Example: A toaster and a iron are connected in parallel to a source. A breaker is connected in series with the power source. Will the breaker throw when both appliances are being used at the same time?

Start by writing the known and unknowns: When resistors are connected in series or in parallel, the current and voltage can be related using the equivalent resistance. Plugging the resistances into the formula for the equivalent resistance gives: Notice that this is less than the resistance of either the iron or the toaster. Plugging the equivalent resistance into Ohm's Law gives: Since is greater than the limit on the breaker, the breaker will throw.