We need to find the voltage, and we have the resistance, so if we can find the current, then we can use Ohm's Law to find the voltage.

The definition of current is amount of charge that flows through a point in  time, so current can be calculated using this equation:

We're told how many electrons have passed through a resistor, and we know how much charge a single electron has. If we convert the number of electrons into total amount of charge, we can divide that number by the amount of time in seconds to find the current.

Now we have the amount of charge in  and the amount of time in seconds, so we divide the two.

Now that we have the current, we can use Ohm's Law to find the voltage.

Therefore, the potential difference is 2.96V.

Write the formula for the potential difference.

Substitute the work and charge. The unit is in volts.

Use the equations for potential electric energy and electric potential:

Substitute.

Plug in known values and solve.

Electric field strength is the slope of the electric potential:

Remember to convert into SI units (meters):

This problem is solved using the work-kinetic energy theory: . In an electric field, work is equal to charge times change in potential: 

Since an electron has a negative charge, decreasing potential increases its kinetic energy, the opposite of what would happen to a proton with its positive charge. Combine these equations:

Plug in known values and solve

In order to solve for electrical potential energy, we'll need to remember the equation for it.

In the above expression,  represents electrical potential energy,  and  represent different point charges, and  represents the distance between their centers. In this example, one of these charges will be the source of the external electric field, while the other charge will be the one that is undergoing a transposition from point A to point B.

Furthermore, we can remember the equation for voltage:

With both these equations in mind, we can combine the two:

This above expression tells us that the electrical potential energy of a system is directly proportional to the voltage change and to the charge of the test charge that is undergoing the voltage change.

Plug in the values and solve for electrical potential energy:

For this question, we're presented with a situation in which a positively charged particle is traveling through an electric field along a defined path. We're then asked to determine the amount of energy we need to input in order to make this process occur.

We'll need to use an equation that relates charge, electric field, and energy, which we can derive as follows.

Next, we need to make sure to use the correct value for distance. Since we know that the electric force is a conservative force, we know that the movement of the particle from A to B is independent of the path taken. In other words, for a conservative force, the only thing that matters is the initial state and the final state. Also, since the electric field is oriented horizontally from right to left, we only care about the movement of the particle along the horizontal direction. Thus, we do not use the  value (distance of path taken), but instead we use the  value (net distance traveled along the electric field). 

Plugging in the values we have, we can obtain our answer:

Converting units:

Use the equation for electric potential energy:

Conservation of energy:

Assuming no external work done

Electric potential energy:

Solving for velocity:

Plugging in values:

Conservation of energy:

Assuming no external work done

Electric potential energy:

Solving for velocity:

Plugging in values: