Question

A very long uniform line of charge with charge per unit length = +5.00 μC/m lies along the x-axis, with its midpoint at the origin. A very large uniform sheet of charge is parallel to the xy-plane; the center of the sheet is at z = +0.600 m. The sheet has charge per unit area σ = +8.00 μC/m², and the center of the sheet is at x = 0, y = 0. Point A is on the z-axis at z = +0.300 m, and point B is on the z-axis at z = -0.200 m. What is the potential difference VAB = VA - VB between points A and B? Which point, A or B, is at higher potential?

Answer 1

The potential difference VAB between points A and B is determined by the electric field due to the uniform sheet of charge, as the contribution from the line of charge is constant. The potential at point A is higher than at point B due to its closer proximity to the positively charged sheet.

Step-by-step explanation:

The question is asking for the potential difference between two points A and B located along the z-axis in the presence of a uniform line of charge and a uniform sheet of charge. To calculate this, we need to consider the contributions to the potential from both the line of charge and the sheet of charge separately, as the total potential at a point in space is the sum of potentials due to each charge distribution.

Firstly, the potential due to a very long uniform line of charge at any point is constant for any location along the z-axis because every position is equidistant from the line. As a result, the line of charge makes no contribution to the potential difference between points A and B.

Secondly, the potential due to a uniform sheet of charge is proportional to the distance from the sheet. Since the sheet is parallel to the xy-plane, the potential increases linearly with z. Given the surface charge density σ = +8.00 µC/m², and the separation of point A and B from the sheet center, we can calculate the potential difference.

The electric field E due to the sheet of charge is E = σ / (2ε₀), where ε₀ is the permittivity of free space. The potential difference between two points a distance z apart in a uniform electric field is given by V = E ⋅ z. Plugging in the values:

Calculate the electric field due to the sheet: E = σ / (2ε₀).

Calculate the potential difference: VAB = E ⋅ (zA - zB) = E ⋅ 0.500 m (since point A is at 0.300 m and point B at -0.200 m).

Since the charge on the sheet is positive and point A is closer to the sheet than point B, point A will be at a higher potential than point B. The numbers used in the calculation will confirm the potential difference VAB and also indicate which point is at a higher potential.