We use Ohm's law, , to find the current in the circuit.  In Ohm's law  is the voltage in the circuit,  is the current in the circuit and  is the circuit's resistance.

Solving the equation for , we have

.

We use Ohm's law, , to find the current in the circuit.  In Ohm's law  is the voltage in the circuit,  is the current in the circuit and  is the circuit's resistance.

In our problem,

We use Ohm's law, , to find the current in the circuit.  In Ohm's law  is the voltage in the circuit,  is the current in the circuit and  is the circuit's resistance.

Solving Ohm's law for resistance gives us

.

A few rules of circuits that will help here for series circuits are:

Method 1:

The current through  is given and that will be the same current going through  since there are no current junctions. Since the sum of all voltage drops must equal the emf of the battery, the voltage drop across  can be found:

We can use Ohm's law to find the resistance:

Method 2:

Ohm's law:

The resistance values are added to get , so

Solve for :

To answer this question we need to use the definition of Ohm’s law.

where  is voltage,  is current, and  is resistance. As the equation suggests, to determine the effect of voltage on current we need information regarding the resistance. For example, increasing the voltage will increase the current if resistance decreases or if resistance stays the same. On the other hand, increasing the voltage will decrease the current if the resistance increases drastically; therefore, we cannot determine the effect of voltage on current without knowing anything about the resistance (we need to know if resistance increases, decreases, or stays the same). Similarly, we cannot determine the effect of resistance on the current without knowing about the voltage.

Recall that the definition of current is the amount of electrons flowing through a circuit in a given amount of time. We can increase the amount of electrons flowing through a circuit (for a given time) if we increase the current. Using Ohm’s law, we can determine conditions that will increase the current.

Solving for current we get

This means we can get a high current if we increase voltage or decrease resistance. Note that that changing the voltage source from a direct source to an alternating source will generate an alternating current with similar amplitude; it won’t increase the total current of the system and, subsequently, the amount of electrons flowing through the circuit per given time.

To calculate the amount of electrons flowing through this circuit we need to first calculate the current. Use Ohm’s law:

Solve for current:

The question gives us the voltage and resistance; therefore, the current flowing through this circuit is

Recall that current is the amount of electrons flowing through a circuit per unit time.

Solving for the charge of electrons gives us

Recall that an electron contains ; therefore, the number of electrons in this circuit is

This means that  electrons are flowing through this circuit every five seconds.

Using ohm's law, the resistance is determined to be which is calculated to be . Ohm's law is used again to find the current at  with the same resistance.

Recall Ohm's Law:

 where  is the voltage,  is the current, and  is the resistance.

Since the two quantities we are interested in are on the same side of the equation, they are inversely proportional. Hence, if one increases, the other one decreases by the same ratio. Since the current is doubling, the resistance must halve for the circuit to be the same.

To answer this question, we'll need to find an expression for the value of the variable resistor that would make the voltmeter read zero.

First, it's important to realize what situation would result in a reading of zero from the voltmeter. For there to be no reading, that means that there cannot be any voltage difference between the top row of resistors and the bottom row. For this to happen, the voltage drop for the resistors on the left of the voltmeter must be equal, and the same is true for the two resistors to the right of the voltmeter. In other words, both rows of resistors will experience the same voltage decrease as current flows through, thus the difference of voltage drop in the top and bottom row will be identical.

So let's consider the top and bottom resistors on the left side first. In the top left corner, the voltage of the first resistor will be  from Ohm's law. Moreover, the voltage drop of the bottom left resistor will be . These two voltages will need to be equal to one another in order to have the voltmeter read zero.

Now let's take a look at the other resistors on the right. The voltage of the third resistor will be  and the voltage of the variable resistor will be . Just as before, these two resistors will also need to be equal in voltage.

Now that we have the two expressions shown above, we can isolate the term for the variable resistance in terms of the other three resistors to find our answer.