# AP Physics 1 : Resistivity

## Example Questions

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### Example Question #1 : Resistivity

Consider the circuit:

The current flowing through the entire circuit is . What is the value of R3?

Explanation:

We know the voltage and total current of the circuit, so we can calculate an equivalent resistance using Ohm's law:

Now we can use the expression for condensing parallel resistors to calculate R3:

### Example Question #2 : Resistivity

Which of the following factors would decrease the resistance through an electrical cord?

Increasing the length of the cord

Increasing the cross-sectional area of the cord

Decreasing the cross-sectional area of the cord

Increasing the resistivity

Increasing the cross-sectional area of the cord

Explanation:

The equation for resistance is given by .

From this equation, we can see the best way to decrease resistance is by increasing the cross-sectional area, , of the cord. Increasing the length, , of the cord or the resistivity, , will increase the resistance.

### Example Question #3 : Resistivity

An electrician wishes to cut a copper wire  that has no more than  of resistance. The wire has a radius of 0.725mm. Approximately what length of wire has a resistance equal to the maximum  ?

10cm

2.6cm

9.6m

960m

38m

960m

Explanation:

To relate resistance R, resistivity , area A, and length L we use the equation.

Rearranging to isolate the quantity we wish to solve for, L, gives the equation . We must first solve for A using the radius, 0.725mm.

Plugging in our numbers gives the answer, 960m.

### Example Question #4 : Resistivity

What is the current in a circuit with a  resistor followed by a  resistor that are both in parallel with a  resistor? The voltage supplied to the circuit is 5V.

Explanation:

Resistors in serries add according to the formula:

Resistors in parallel add according to the formula:

We can find the total equivalent resistance:

Now we can use Ohm's law to find the current:

Solve:

### Example Question #5 : Resistivity

In the circuit above, your aim is to limit the current to , so you must design a resistor to serve in the place of .

To make this resistor, you have a spool of mystery metal, which has a cross sectional area of  and a resistivity of, . What length of wire should you cut?

Explanation:

First, find out how much total resistance should be in the circuit in order to get the desired current:

Determine what the second resistance should be.

Find the necessary length of wire.

### Example Question #6 : Resistivity

What is the resistance of a  length of round copper wire with a radius of ?

Explanation:

Resistance and resistivity are related as follows:

### Example Question #7 : Resistivity

Two students are performing a lab using lengths of wire as resistors. The two students have wires made of the exact same material, but Student B has a wire the has twice the radius of Student A's wire. If Student B wants his wire to have the same resistance as Student A's wire, how should Student B's wire length compare to Student A's wire?

Student B's wire should be twice as long as Student A's wire

Student B's wire should be four times as long as Student A's wire

Student B's wire should be the same length as Student A's wire

Student B's wire should be  the length of Student A's wire

Student B's wire should be  the length of Student A's wire

Student B's wire should be  the length of Student A's wire

Explanation:

Resistance is proportional to length, and inversely proportional to cross-sectional area. Area depends on the square of the radius:  , so Student B's wire has  times the cross-sectional area of Student A's wire. In order to compensate for the increased area, Student B must make his wire  the length of Student A's wire. This can be shown mathematically using the equation for resistance:

### Example Question #8 : Resistivity

By how much will resistivity change if resistance and length are constant, and cross sectional area is doubled?

The resistivity will double

The resistivity will be halved

The resistivity will not change

The resistivity will double

Explanation:

Recall the formula for resistance  is given by

, where  is the cross sectional area,  is resistivity, and  is length.

Solve for resistivity:

From this, we can tell that resistivity is proportional to cross sectional area by:

Since  is doubled, and resistance  and length  are constant, resistivity  will also be doubled.

### Example Question #9 : Resistivity

Ratio is given by:

Resistivity of first resistor: Resistivity of second resistor

What is the ratio of resistivity of 2 resistors with identical resistances and area, where the first resistor is twice the length of the second resistor?

Explanation:

Resistivity  is given by:

, where  is the resistance,  is the area, and  is the length.

Since both have identical resistances and area, the first resistor will have half the resistivity since it has twice the length. Therefore the resistivity relation is

### Example Question #10 : Resistivity

What is the resistance of a copper rod with resistivity of , diameter of , and length of ?

Explanation:

The equation for resistance is as follows: . Where  is resistivity,  is the length of the wire, and  is the cross section of the wire which can be found using .

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