#### Quiz

Q1) A current of 5 ampere flows into a circuit where two identical resistors are attached in parallel. What current would flow through one of the resistors?

A. Both resistors carry 5 A

B. each resistor will carry 2.5 A

C. It is impossible to tell unless you know the size of the resistance of each resistor

D. The resistor would carry less than 5 A but you cant tell how much energy was lost before

Answer) B.

Q2) Four resistors having equal values are wired as a parallel circuit. How does the equivalent resistance of the circuit compare to the resistance of a single resistor?

A. The equivalent resistance is greater than the resistance of any single resistor.

B. The equivalent resistance is the same as the resistance of any single resistor.

C. The equivalent resistance is one-fourth the resistance value of a single resistor.

D. The equivalent resistance is one-half the resistance value of a single resistor.

Answer) C.

A. The equivalent resistance is greater than the resistance of any single resistor.

B. The equivalent resistance is the same as the resistance of any single resistor.

C. The equivalent resistance is one-fourth the resistance value of a single resistor.

D. The equivalent resistance is one-half the resistance value of a single resistor.

Answer) C.

#### Resistors in parallel equivalent resistance Simulation

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**RESISTORS IN PARALLEL**

As discussed above, when a single bulb in a series light set burns out, the entire string of lights goes dark because the circuit is no longer closed.What would happen if there were alternative pathways for the movement of charge, as shown in Figure 1.

A wiring arrangement that provides alternative pathways for the movement of a charge is a parallel arrangement. The bulbs of the decorative light set shown in the schematic diagram in Figure 1 are arranged in parallel with each other.

*Figure 1. These decorative lights are wired in parallel. Notice that in a parallel arrangement there is more than one path for current.*

**Resistors in parallel have the same potential differences across them**

To explore the consequences of arranging resistors in parallel, consider the two bulbs connected to a battery in Figure 13(a). In this arrangement, the left side of each bulb is connected to the positive terminal of the battery, and the right side of each bulb is connected to the negative terminal. Because the sides of each bulb are connected to common points, the potential difference across each bulb is the same. If the common points are the battery’s

terminals, as they are in the figure, the potential difference across each resistor is also equal to the terminal voltage of the battery. The current in each bulb, however, is not always the same.

*Figure 2. (a) This simple parallel circuit with two bulbs connected to a battery can be represented by (b) the schematic diagram shown on the right.*

**The sum of currents in parallel resistors equals the total current**

In Figure 2, when a certain amount of charge leaves the positive terminal and reaches the branch on the left side of the circuit, some of the charge moves through the top bulb and some moves through the bottom bulb. If one of the bulbs has less resistance, more charge moves through that bulb because the bulb offers less opposition to the flow of charges.

Because charge is conserved, the sum of the currents in each bulb equals the current $\ I$ delivered by the battery. This is true for all resistors in parallel.

$\ I = I_{1} + I_{2} + I_{3}.....$

The parallel circuit shown in Figure 2 can be simplified to an equivalent resistance with a method similar to the one used for series circuits. To do this, first show the relationship among the currents.

$\ I = I_{1} + I_{2}$

Then substitute the equivalents for current according to $\Delta V = IR$.

$\frac{\Delta V}{R_{eq}}=\frac{\Delta V_{1}}{R_{1}} + \frac{\Delta V_{2}}{R_{2}}$

Because the potential difference across each bulb in a parallel arrangement equals the terminal voltage ($\Delta V$ = $\Delta V_{1}$ = $\Delta V_{2}$), you can divide each side of the equation by $\Delta V$ to get the following equation.

$\frac{1}{R_{eq}}=\frac{1}{R_{1}} + \frac{1}{R_{2}}$

An extension of this analysis shows that the equivalent resistance of two or more resistors connected in parallel can be calculated using the following equation.

RESISTORS IN PARALLEL$\frac{1}{R_{eq}}=\frac{1}{R_{1}} + \frac{1}{R_{2}}$.....The equivalent resistance of resistors in parallel can be calculated using a reciprocal relationship. |

Notice that this equation does not give the value of the equivalent resistance directly. You must take the reciprocal of your answer to obtain the value of the equivalent resistance.

Because of the reciprocal relationship,

*the equivalent resistance for a parallel arrangement of resistors must always be less than the smallest resistance in the group of resistors.*