Physics

We need to calculate the force on the middle charge at due to the other two charges at and . According to Coulomb's Law, the force between two point charges and separated by a distance is given by:

Here, the force between charges separated by 2 nm is 2 pN. We need to find:

• The force between charges at and .
• The force between charges at and .

- The distance between and is 2 nm.

- The distance between and is 4 nm.

1. Since the force for 2 nm separation, for (same distance), the force is 2 pN.

2. For , at 4 nm, using inverse square law:

The force directions for and are opposite (assuming positive direction is towards larger x). Therefore, net force on middle charge:

This calculated net force =

The problem is related to calculating the gravitational potential energy between two bodies using Newton's law of gravitation. The gravitational potential energy between two masses and , separated by a distance , is given by the formula:

where is the gravitational constant.

In this question, both masses and are kg each. Thus, the gravitational potential energy can be calculated as follows:

Substitute the given values into the formula:

Simplify the expression:

This matches the expression given in the first option:

Next, let's evaluate if other options can also be considered:

The second option is:

This option includes an additional term , which represents an added constant energy with dimensions of energy. Such an addition does not affect the validity of the gravitational potential energy formula, as potential energy is only defined up to an arbitrary constant. Therefore, this option is valid as well.

The third option is:

This is incorrect because the potential energy formula involves rather than .

The fourth option is:

This formula neither uses the correct variables nor follows the inverse relationship with the distance, and hence it is incorrect.

In conclusion, the correct representations of gravitational potential energy for the given scenario are:

These options are consistent with the principle that potential energy can be defined up to an arbitrary constant.