Equivalent Resistance - AP Physics 1

The new circuit has two resistors in parallel: R2 and the new one attached. To find the equivalent resistance of these two branches, we use the following expression:

In this new equivalent circuit everything is in series, so we can simply add up the resistances:

Now we can use Ohm's law to calculate the total current through the circuit:

We will be working backwards on this problem, using the current to find the resistance. We know the voltage and desired current, so we can calculate the total necessary resistance:

Then we can calculate the equivalent resistance of the two resistors that are in parallel (R2 and our unknown):

Now we can calculate what the resistance between point A and B:

Rearranging for the desired resistance:

We can use the equation for equivalent resistance of parallel resistors to solve this equation:

We know the equivalent resistance, and we know that the resistance of each of the four resistors is equal:

Since we know the power loss and voltage of the circuit, we can calculate the equivalent resistance of the circuit using the following equations:

Substituting Ohm's law into the equation for power, we get:

Rearranging for resistance, we get:

This is the equivalent resistance of the entire circuit. Now we can calculate R4 using the expression for resistors in parallel:

We can use Ohm's law to calculate the equivalent resistance of the circuit:

Now we can use the expression for combining parallel resistors to calculate R1:

We will need to test the values of each answer to find the one that generates an equivalent resistance of .

We know that when condensing parallel resistors, the equivalent resistance will never be larger than the largest single resistance, and will always be smaller than the smallest resistance. Therefore, two of the answer options cen be eliminated immediately.

After we have narrowed our choices down to the other options answers, we just have to test them with the following formula:

We will test the incorrect answer first:

Now for the correct answer:

The goal of this question is to realize that when two resistors are connected in parallel, the equivalent resistance is lower than either of the two original resistors. But when two resistors are connected in series, the equivalent resistance is the sum of the two original resistors. Therefore to minimize our equivalent resistance, we want all the resistance to be in the parallel resistors, leaving for the resistor in series.

Because this circuit is neither purely series or purely parallel, we must simplify it before we solve it. Replace the right branch, which is purely series, with its equivalent resistance:

Now we have a purely parallel circuit, each branch having a resistance of . Apply the parallel formula and solve:

For resistors all in series, the equivalent resistance is equal to the sum of the resistances.

The reciprocal of the equivalent resistance for resistors in parallel is equal to the sum of the reciprocals of the resistances:

In the question stem, we're told that a circuit containing three identical resistors connected in parallel has an equivalent resistance equal to . We are then asked to solve for the resistance of each individual resistor.

To start with, it's important to remember that resistors in parallel add inversely. Thus, the inverse of the equivalent resistance is equal to the sum of the inverse of each individual resistor. Put another way:

Since we know the three resistors we're dealing with are identical, we can assign each of them a value of .

And rearranging, we obtain:

The equivalent resistance of resistors connected in series is the sum of the resistance values of each resistor, or

In our problem,

The equivalent resistance of resistors connected in parallel is given by the following equation,

In our problem,

The equivalent resistance of resistors connected in series is the sum of the resistance values of each resistor, or

In our problem,

The equivalent resistance of resistors connected in parallel is given by the following equation,

In our problem,

Remember the rules of adding resistors. For resistors in series,

and for resistors in parallel,

The idea is to start from the side furthest away from the battery and work back toward it. Notice that and are in parallel. We can add them in parallel so that they have an equivalent resistance ,

This can be calculated, but for 3 resistors we can leave it in equation form until the end. This looks like:

Notice that is now in series with ,

In order to find the equivalent resistance, you must take small steps and slowly work towards finding the total equivalent resistance.You can combine resistors if they are in parallel. The expression to find parallel equivalent resistance is:

You can also combine resistors in series, or one after the other. The expression to find resistors in series is:

To start off, you can combine R2 and R3, since they are both in parallel:

Next, you can combine some of the series resistors together.

After this, you can combine the following resistors in parallel:

Then, combine and in series:

Now get rid of the last parallel by combining and in parallel:

Now finally, add the remaining resistors in series to find the equivalent resistance:

Since the resistors are in parallel to a battery, the voltage drop across each resistor has to be equal to the voltage gain across the battery. Therefore, the voltage drop for any of the resistors will be .

In order to find the equivalent resistance of resistors in parallel, we add the inverses of their values, as shown below

Finally