Electrical potential energy is given by the equation .

Electrical potential energy is inversely proportional to the distance between the two charges. If the energy is quadrupled, then  (the distance between the two equal charges) must have decreased proportionally.

For the energy to be quadrupled, the radius must be quartered.

Given the equation and plugging in the values of e and k, we get that

It is important to keep in mind that the charge  is given in the question and must be incorporated into the formula by multiplying each charge by that value.

A positive test charge will naturally move from high potential to low potential. If it is moved in the opposite direction, then the electric field will do work against its motion (negative work). This be seen from the equation for electric field work:

 is the work done by the electric field,  is the charge, and  is the potential difference. If  is positive (the final potential is higher than the initial potential) and  is also positive, then work done by the field is negative.

Relevant equations:

Step 1: Since the work done to assemble the charges equals their potential energy in this arrangment, find the potential energy between each pair of charges. Work is equal to change in potential energy; since the charges start at infinite distance, initially potential energy is equal to zero.

Charges 1 and 2

Charges 1 and 3

Charges 2 and 3

Step 2: Add together all these potential energies to find the total energy of the arrangement.

Since the given voltage is the root mean squared voltage we must multiply the voltage by  to find the maximum voltage.

We determine that the maximum voltage delivered by the outlet is .

Relevant equations:

For the primary coil, we have:

And for the secondary coil:

Plugging these in yields:

To answer this question you need to understand the relationship between electrical conductivity, , and electrical resistivity, :

This means that the electrical conductivity is the reciprocal of the electrical resistivity; therefore, the electrical resistivity of this block is:

Recall the definition of resistivity:

Here,  is the resistance,  is the cross-sectional area, and  is the length of the block. The question gives us resistance and length of the block, and we calculated resistivity; therefore, solving for the area of the block gives us:

The cross-sectional area of the block is .

Of the given answer choices, the only valid conclusion is that the block has a square cross-section with a height and width of  because this square has an area equal to the cross-sectional area of the block (). 

The electrical conductivity is the reciprocal of electrical resistivity; therefore, an increase in electrical resistivity will lead to a decrease in electrical conductivity, and vice versa.

Recall the definition of electrical resistivity:

Here,  is resistance,  is cross-sectional area, and  is the length of the rod. This equation reveals that an increase in resistance and area will increase resistivity, whereas an increase in length will decrease resistivity. Since conductivity is the reciprocal of resistivity, increasing resistance and area will decrease conductivity, whereas increasing the length will increase conductivity.

The question is asking about the rates of charge flow. Recall that the current is defined as the amount of charge flowing through a point in a given time; therefore, we are looking for the amount of current flowing through the two rods. Current flow is higher for a material with higher electrical conductivity. This means that we need to find the rod with the higher electrical conductivity.

Since there are no easy equations to find conductivity, we need to find the resistivity first. Electrical conductivity is the reciprocal of electrical resistivity (measure of the ability of a material to resist current flow); therefore, an increase in resistivity leads to a decrease in conductivity, and vice versa. Resistivity is defined as:

Here,  is the resistance,  is the cross-sectional area, and  is the length of the rod. The question states that the two rods have the same geometrical properties; therefore, the area and the length of the rods are the same. However, the resistance of rod A is higher. This means that the resistivity is higher and conductivity, consequently, is lower for rod A. Since it has a lower conductivity, rod A has a lower charge flow rate. 

Batteries are a series of electrochemical cells that use the electrical energy associated with the movement of electrons to perform other processes. The electrical energy is not itself usable; instead, batteries convert the electrical energy into other forms of energy, such as chemical energy, that can be used to drive a process. The electrical energy is derived from the movement of electrons associated with redox reactions. 

Batteries contain one or more electrochemical cells. Each electrochemical cell houses a cathode and an anode, which facilitate a redox reaction. The cathode is the site of the reduction half-reaction, whereas the anode is the site of the oxidation half-reaction. Reduction is the process by which reactants gain electrons and become more negative; oxidation is the process by which reactants lose electrons and become more positive. As such, electrons are products at an anode and reactants at a cathode. The redox reaction proceeds in the spontaneous reaction direction when the battery is being used, but proceeds in the nonspontaneous reaction direction when the battery is being recharged after use. Recall that nonspontaneous reactions require energy. When you are charging a battery, you are supplying energy in the form of voltage to drive the nonspontaneous reaction. On the other hand, when you are discharging (or using) the battery, the spontaneous redox reaction is supplying energy to drive another process.