For resistors aligned in series, the equivalent resistance is the sum of the individual resistances.

Since all the resistors in this problem are equal, we can simplify with multiplication.

The formula for resistance in parallel is:

We are given the values for each individual resistor, allowing us to solve for the total resistance.

Think of resistors as doors, preventing the flow of people (electrons). Imagine the following scenarios: a large group of people are in a room and all try to leave at once. If the five resistors are in series, that's like having all of these people trying to go through all five doors before they can leave. In a circuit, all the electrons in the current mast pass through every resistor.

If the resistors are in parallel, it's like having five separate doors from the room. All of a sudden, the group can leave MUCH faster, encountering less resistance to their flow out of the room. The path of the electrons can split, allowing each particle to pass through only one resistor.

From a formula perspective, the resistors in series are simply summed together to find the equivalent resistance.

In parallel, however, the reciprocals are summed to find the reciprocal equivalent resistance.

Adding whole numbers will always give you a much greater result than adding fractions. For the exact same set of resistors, arrangement in series will have a greater total resistance than arrangement in parallel.

In basic resistors, energy lost due to resistance is converted into heat. In some cases, other conversions also take place (such as generation of light in a lightbulb), but heat is still dissipated along with any alternative conversations. Lightbulbs, batteries, and other types of resistors will become hot as current passes through them.

The formula for resistance is: 

Above,  is length,  is the cross-sectional area of the wire, and  is the resistivity of the material, and is a property of that material. The resistivity is constant for a given material, and thus cannot be changed by altering the dimensions of the wire.

Let’s start with the resistivity equation

The area of a wire is the area of a circle.  So let’s substitute that into our equation

This can be simplified to 

Since we know that both resistors have the same resistivity and the same resistance, we can set these equations equal to each other.

Many things fallout which leaves us with

We know that the second wire is twice the length as the first

So we can substitute this into our equation

The length of the wire drops out of the equation

Now we can solve for the diameter of the longer wire.

Take the square root of both sides

Therefore the ratio of the long to the short wire is

  Or

We will use the resistivity equation to solve for this.  We know

Length = 

Resistance = 

ρ of Tungsten = 

The equation for resistivity is

We can rearrange this equation to solve for 

In this case the area of the wire is the area of a circle which is equal to 

We can rearrange this to get the radius by itself

To find the diameter we need to multiply this value by 2.

At higher temperature, the atoms are moving more rapidly and are arranged in a less orderly way.  Therefore it is expected that these fast moving atoms are more likely to interfere with the flow of electrons.  If there is more interference in the flow of electrons, then there is a higher resistivity.

Resistors are in parallel when the electric current passes through two or more branches or connected parts at the same time before it combines again. After the current leaves the battery it travels to . Then the current splits and travels between the other two resistors before the current coming back together and connecting back to the battery.

Adding resistors in parallel decrease the overall equivalent resistance as they are added using the equation

Since the overall equivalent resistance is decreased, if the voltage is constant the overall current would increase.

Therefore 

This shows the inverse relationship between the two values.