Voltage Across Resistors In Parallel

[Bashyboy]

Homework Statement

Consider the circuit shown in the figure below. (R = 31.0 Ω.)

(a) Find the current in the 31.0 Ω resistor.

(b) Find the potential difference between points a and b.

Homework Equations

The Attempt at a Solution

I am reading the solution of this problem given by the author, and for the most part I understand it, except for this one critical part:

"In diagram (iii), the current above goes through the equivalent resistor $R_{iii}$ to give a voltage drop across this resistor of $ΔV = IR_{iii}$. n diagram (ii), we see that this voltage drop is $ΔV_{ab}$ and is the same across the 10.0-Ω resistor and the 5.00-Ω resistor."

Why is the voltage drop the same across the two resistors in parallel?

 

[CWatters]

Why is the voltage drop the same across the two resistors in parallel?

By definition two resistors in parallel are both connected to the same nodes. In capture.jpg these nodes are a and b.

If you connect two resistors in parallel to an ideal 9V battery they will both have 9V across them.

 

[Bashyboy]

So, it simply follows from definition, and there isn't any other reason?

 

[CWatters]

No you missunderstood what I said. The definition of parallel is that the resistors are connected between the same two nodes. They have the same voltage drop because they are both connected to the same nodes.

Imagine two climbers go up a mountain. They start at the same point at the bottom and reach the same point at top. They might take different routes but both will measure the same change in altitude because they started and finished at the same places.