## Resistors in Parallel

This is a simulation of the current, voltage, and power relationships in parallel resistors.

The two sliders control R1 and R2, respectively. As you vary R1 and R2, the voltages, currents, and power for both are calculated and displayed dynamically.

When two resistors are connected in parallel, the voltage across both resistors is the same. The battery current is split between the two resistors. The smallest resistance has the largest current through it.

The algebraic expression for the total resistance R of parallel resistors is derived like so:

Ohm's law applies to each resistor separately:

Ohm's law for resistor 1 is E1 = I1 R1
Ohm's law for resistor 2 is E2 = I2 R2
The voltages across each resisor E1 and E2 must be equal because they are both connected directly to the battery:
E1 = E2 = E where E is the battery voltage.
And the sum of the currents through the two resistors must be equal to the battery current, according to Kirchoff's law:
I = I1 + I2
where I is the battery current. Substituting the expression for I1 and I2 from the Ohm's law expressions for each resistor into the previous equation:
I = (E1 / R1) + (E2 / R2)
But E1=E2=E, so
I = (E / R1) + (E / R2)
Factoring out E:
I = E (1 / R1 + 1 / R2)
This can be rearranged to make it closer to the form of Ohm's law.
E = I 1/((1 / R1) + (1 / R2))
You can see that this equation has the form of Ohm's law for the total circuit, where E is the voltage across the two resistors, I is the total current, and therefore 1/((1 / R1) + (1 / R2)) is the total resistance of resistors R1 and R2 in parallel.

### Student Questions for Series and Parallel Resistors

1. Open (double-click on) the "Series" and on the "Parallel" stacks and and inspect the screen layouts. In both of these stacks, the left slider controls R1 and the right slider controls R2. The battery voltage is determined by the number in the little box to the left of the sliders. The simulation uses Ohm's and Joule's Laws to compute the currents, voltages, and power dissipated for each resistor.

2. Vary R1 (by dragging the triangle pointer on the left slider scale) and note the effect on the current, voltage, and power dissipation of R2 in the two circuits. For the series circuit, does changing R1 causes changes in the voltage across and the power in R2? _____ What about for the parallel circuit? _____ .

3. In the parallel circuit, what is the relationship between the total battery current (shown at the top of each circuit next to the little arrow) and the currents through the two resistors?

Is this a general relationship that is always true for all values of resistance and battery voltage?

4. In the series circuit, what is the relationship between the battery voltage (shown in the box to the left of the sliders) and the voltages across the two resistors? Is this a general relationship that is always true for all values of resistance and battery voltage?

5. Sometimes you need a resistor value that you don't have on hand. In such cases, you may be able to create the needed value by using a series or parallel of two available resistors. For example, suppose you needed a 123K resistor. (a) How close can you get to that value by using a series combination of two resistors?

(b) How close can you get to that value by using a parallel combination of two resistors?

6. Note that the resistances R1 and R2 are not really continuously variable, but rather vary in discrete intervals. This is done to acknowledge the fact that commercially available resistors are not manufactured in all possible resistance values but rather in a standard sequence of values spaced about 10 to 20% apart. This, for example, they make 1 KOhms and 1.1 KOhms, but not 1.05 KOhms as a standard value. Also, standard resistors are accurate to only 5% or 10% or their nominal values. (Precision resistors, accurate to 1% or even 0.1%, are available but are costly and are manufactured mostly for "even" values).

Suppose you needed a 98K resistor with an accuracy of 1%, but had only a 100K 1% resistor and a set of standard 10% resistors in all standard values. (a) How you you create the desired resistance using the 100K 1% resistors and one of the standard 10% resistors?

(b) How is it possible to get away with using one 10% resistor, when you need a 1% overall accuracy. How much relative error in the 98K resistance is caused by a 10% error in the 10% resistor?

7. Set R1 to 100 Ohms. Vary R2 and observe the power dissipated in R2 in the series circuit. What happens to the power in R2 when R1< What happens to the power in R2 when R1>>R2? Why?

What happens to the power in R2 when R1=R2? Is this a general observation, valid under all conditions?