Question

What would be the consequence of setting the potential at \(+100 \mathrm{~V}\) at infinity, rather than taking it to be zero there? a) Nothing; the field and the potential would have the same values at every finite point. b) The electric potential would become infinite at every finite point, and the electric field could not be defined. c) The electric potential everywhere would be \(100 \mathrm{~V}\) higher, and the electric field would be the same. d) It would depend on the situation. For example, the potential due to a positive point charge would drop off more slowly with distance, so the magnitude of the electric field would be less.

Short answer

Answer: The most plausible consequence would be that the electric potential everywhere would be 100 V higher, and the electric field would remain the same.

Step-by-step solution

Understanding electric potential at infinity

According to the convention, the reference point for the electric potential is taken to be at infinity, and its value is set to zero. This choice is arbitrary and helps simplify calculations. The idea behind this exercise is to evaluate whether the choice of a different value for the electric potential at infinity changes anything related to electric potential and field at finite distances.

Evaluating option a

A common notion is that the potential at infinity and how it is defined do not affect the potential difference between two points or the electric field at any point in the finite region. Therefore, one might think that there would be no difference in the values of the electric field and potential at every finite point, but we need to further analyze other options to make a conclusion.

Evaluating option b

Let's consider the case where the electric potential becomes infinite at every finite point. This would happen if we divide by a distance approaching zero. However, we are not changing the charge distribution or the distance between the charges. Thus, this option seems implausible because there's no reason why changing the reference value of the potential at infinity would result in the electric potential becoming infinite in every finite point and making the electric field undefined.

Evaluating option c

According to this option, increasing the potential at infinity by a constant value (+100 V) would result in an increase of the same constant value at every other point in space. This means the electric potential difference between any two points in space would remain the same, and since the electric field is related to the potential difference, the electric field would also be unchanged. This seems like a plausible explanation, but let's evaluate the last option before making a final conclusion.

Evaluating option d

This option suggests that the consequences depend on the specific situation. However, as stated earlier, how the electric potential is defined at infinity should not affect potential difference or the electric field at any point in the finite region as they are dependent upon the charge distribution and the distances involved. Therefore, it is unlikely that the answer would depend on the specific situation.

Final conclusion

After evaluating all four options, the most plausible consequence of setting the potential at \(+100 \mathrm{~V}\) at infinity rather than taking it to be zero there would be that the electric potential everywhere would be \(100 \mathrm{~V}\) higher (option c), and the electric field would be the same.