Recall that for sinusoids of form , the period is given by:

For this problem, the fundamental frequency is when , which means that its frequency is:

The correct answer is  and .

This is because at both waves maxima they would add constructively in the form of  in the positive axis.

When they interfere destructively they would subtract in the form of  in the positive axis.

Therefore the answer is  and .

The formula for wavelength depending on the harmonic is as follows:

Where  is the wavelength,  is the length of the pipe and n is the harmonic number. Substituting our values in the equation we obtain:

Therefore the correct answer is  long.

To answer this question, we need to understand the concept of standing waves. A standing wave is a wave that exhibits both nodes and antinodes. A node is a fixed position where there is no displacement from the wave, and an antinode is a point of maximum displacement of the wave.

A standing wave can result from two individual waves traveling in opposite directions and interfering with each other. For instance, when a wave traveling through one medium (such as the air in the pipe) hits an interface that is denser (such as the end of the pipe), the wave will reflect back. However, this reflected wave will be shifted  out of phase. As a result, the reflected wave will interact with the incident wave, and the two will interfere. This interference takes on a specific pattern in which there is destructive interference in the location of nodes, and constructive interference in the location of the antinodes. In such a situation, it doesn't appear that the wave is traveling at all, but rather it looks as if it's standing still, hence the name standing wave.

To solve this question, we need to use the expression for a pipe closed at one end and open at the other.

With this expression, we just need to plug in the values given to us in the question to solve for the length of the pipe.

This is a question where knowing how to effectively sift through a problem statement and choose only the information you need will really help. We are given a bunch of values, but only need to know one thing, which is that the distance between the two charges is doubled.

Coulomb's law is as follows:

We can rewrite this for the initial and final scenarios:

We can divide one equation by the other to set up a ratio:

We know that the final radius is double the intial, which is written as:

Substituting this in we get:

Rerranging for the final force, we get:

Use Coulomb's law.

Plug in known values and solve.

Note that a positive value for electric force corresponds to a repulsive force. This should make sense since the charge on both particles are the same sign (positive).

Use Coulomb's law.

Note that the electric force between two charges of the same sign (both positive or both negative) is a positive value. This indicates a repulsive force.

Recall that Coulomb's law tells us that the magnitude of force between two point charges is given as:

Here,  is force between two particles,  are the charges of each of the two particles, and  is the distance between the two charges. In our case,  and  are identical since each is the charge of a proton which is given as:, and 

Thus, plug in known values and solve.

To determine if the force is attractive or repulsive, we only need to examine the sign of the charges. Since both protons have the same sign for their charge (positive charges) they will repel. 

Use Coulomb's law to find the electric force between the charges:

The electric force between two point charges is given by Coulomb's law:

Now, plug in the given charges (both the same magnitude), the given constant, and the distance between the charges (in meters) to get our answer: