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:

Conservation of energy:

Assuming no external work done

Electric potential energy:

Solving for velocity:

Plugging in values:

Force in an electric field:

Using conservation of energy, assuming no external work on system:

Definition of electric potential energy:

Definition of electric potential:

Combining equations:

Assuming final velocity is zero:

 is the distance traveled

Converting  to 

Plugging in values:

Solving for 

Force in an electric field:

Using conservation of energy, assuming no external work on system:

Definition of electric potential energy:

Combining equations:

Assuming initial velocity and final electric potential is zero:

The charge,  will be equal to the electron charge time the number of electrons missing, 

Converting  to  and plugging in values:

In this question, we're given the charge of a particle, the distance that it travels, and the electric field within which this movement occurs. We're asked to find the magnitude of the change in electrical potential energy that this particle undergoes.

We can begin this problem by writing an expression for the electric potential energy.

Since we have the particle's charge, but not its electric potential, we need to find a way to obtain this term. To do this, we can make use of the distance the particle travels, as well as the electric field.

Combining these expressions, we can obtain our answer.

Because there is no charge inside the pyramid, the total flux for the entire shape must be 0. Since the field is vertical, there must be an equal but opposite amount of flux from the base of the pyramid as the faces.

Gauss' Law is 

We have both the field strength and the area, so we just multiply them together. We don't have to worry about cross-products because the field is hitting the base at a 90^o angle.

There is nonzero electric flux going through the cube because it encloses charges, so there's more electric field lines going out than going in.

Gauss' Law:

Because we know the amount of charge enclosed and we know epsilon naught (the permittivity of free space), the area of the cube and the electric field strength is irrelevant; we can just calculate it with the charge.

That gives us the total electric flux. What we want is the flux from a single face. Since there are 6 faces, we can just divide that number by 6 to get our answer. Once we do that, we get .