# Electrical Circuits

## Parallel Circuits


### Section Key Terms


### Resistors in Parallel

In the previous section, we learned that resistors in series are resistors that are connected one after the other. If we instead combine resistors by connecting them next to each other, as shown in , then the resistors are said to be connected in parallel. Resistors are in parallel when both ends of each resistor are connected directly together.
Note that the tops of the resistors are all connected to the same wire, so the voltage at the top of the each resistor is the same. Likewise, the bottoms of the resistors are all connected to the same wire, so the voltage at the bottom of each resistor is the same. This means that the voltage drop across each resistor is the same. In this case, the voltage drop is the voltage rating V of the battery, because the top and bottom wires connect to the positive and negative terminals of the battery, respectively.
Although the voltage drop across each resistor is the same, we cannot say the same for the current running through each resistor. Thus,  
   are not necessarily the same, because the resistors  
   do not necessarily have the same resistance.
Note that the three resistors in  provide three different paths through which the current can flow. This means that the equivalent resistance for these three resistors must be less than the smallest of the three resistors. To understand this, imagine that the smallest resistor is the only path through which the current can flow. Now add on the alternate paths by connecting other resistors in parallel. Because the current has more paths to go through, the overall resistance (i.e., the equivalent resistance) will decrease. Therefore, the equivalent resistance must be less than the smallest resistance of the parallel resistors.
To find the equivalent resistance  
   of the three resistors  
 , we apply Ohm’s law to each resistor. Because the voltage drop across each resistor is V, we obtain
or
We also know from conservation of charge that the three currents  
   must add up to give the current I that goes through the battery. If this were not true, current would have to be mysteriously created or destroyed somewhere in the circuit, which is physically impossible. Thus, we have
Inserting the expressions for  
   into this equation gives
or
This formula is just Ohm’s law, with the factor in parentheses being the equivalent resistance.
Thus, the equivalent resistance for three resistors in parallel is
The same logic works for any number of resistors in parallel, so the general form of the equation that gives the equivalent resistance of N resistors connected in parallel is

### Practice Problems


### Resistors in Parallel and in Series

More complex connections of resistors are sometimes just combinations of series and parallel. Combinations of series and parallel resistors can be reduced to a single equivalent resistance by using the technique illustrated in . Various parts are identified as either series or parallel, reduced to their equivalents, and further reduced until a single resistance is left. The process is more time consuming than difficult.
Let’s work through the four steps in  to reduce the seven resistors to a single equivalent resistor. To avoid distracting algebra, we’ll assume each resistor is 10  
  . In step 1, we reduce the two sets of parallel resistors circled by the blue dashed loop. The upper set has three resistors in parallel and will be reduced to a single equivalent resistor  
  . The lower set has two resistors in parallel and will be reduced to a single equivalent resistor  
  . Using the equation for the equivalent resistance of resistors in parallel, we obtain
These two equivalent resistances are encircled by the red dashed loop following step 1. They are in series, so we can use the equation for the equivalent resistance of resistors in series to reduce them to a single equivalent resistance  
  . This is done in step 2, with the result being
The equivalent resistor  
   appears in the green dashed loop following step 2. This resistor is in parallel with resistor  
 , so the pair can be replaced by the equivalent resistor  
 , which is given by
This is done in step 3. The resistor  
   is in series with the resistor  
 , as shown in the purple dashed loop following step 3. These two resistors are combined in the final step to form the final equivalent resistor  
 , which is
Thus, the entire combination of seven resistors may be replaced by a single resistor with a resistance of about 14.5  
  .
That was a lot of work, and you might be asking why we do it. It’s important for us to know the equivalent resistance of the entire circuit so that we can calculate the current flowing through the circuit. Ohm’s law tells us that the current flowing through a circuit depends on the resistance of the circuit and the voltage across the circuit. But to know the current, we must first know the equivalent resistance.
Here is a general approach to find the equivalent resistor for any arbitrary combination of resistors:
1. Identify a group of resistors that are only in parallel or only in series.
2. For resistors in series, use the equation for the equivalent resistance of resistors in series to reduce them to a single equivalent resistance. For resistors in parallel, use the equation for the equivalent resistance of resistors in parallel to reduce them to a single equivalent resistance.
3. Draw a new circuit diagram with the resistors from step 1 replaced by their equivalent resistor.
4. If more than one resistor remains in the circuit, return to step 1 and repeat. Otherwise, you are finished.

### Practice Problems


### Check Your Understanding


### Section Summary

1. The equivalent resistance of a group of N identical resistors R connected in parallel is R/N.
2. Connecting resistors in parallel provides more paths for the current to go through, so the equivalent resistance is always less than the smallest resistance of the parallel resistors.
3. The same voltage drop occurs across all resistors in parallel; however, the current through each resistor can differ.

### Key Equations


### Concept Items


### Critical Thinking


### Problems


### Test Prep Multiple Choice


### Test Prep Short Answer


### Test Prep Extended Response
