An infinite plane of charge with surface charge density 3.2 μC/m² has a 20-cm-diameter circular hole cut out of it. What is the electric field strength directly over the center of the hole at a distance of 12 cm? Hint: Can you create this charge distribution as a superposition of charge distributions for which you know the electric field?

Verified step by step guidance

1. Recognize that the problem involves a superposition principle. The electric field at the center of the hole can be calculated by considering the infinite plane of charge (with surface charge density \( \sigma = 3.2 \mu C/m^2 \)) and subtracting the contribution of the circular hole (treated as a disk of charge with the same surface charge density).
2. Write the expression for the electric field due to an infinite plane of charge. The electric field produced by an infinite plane of charge is uniform and given by \( E_{plane} = \frac{\sigma}{2\varepsilon_0} \), where \( \varepsilon_0 \) is the permittivity of free space.
3. Determine the electric field due to the circular hole. Treat the hole as a disk of charge with radius \( r = 0.1 \) m (since the diameter is 20 cm). The electric field at a distance \( z = 0.12 \) m above the center of a uniformly charged disk is given by \( E_{disk} = \frac{\sigma}{2\varepsilon_0} \left( 1 - \frac{z}{\sqrt{z^2 + r^2}} \right) \).
4. Apply the superposition principle. The total electric field at the point directly above the center of the hole is \( E_{total} = E_{plane} - E_{disk} \). Substitute the expressions for \( E_{plane} \) and \( E_{disk} \) into this equation.
5. Simplify the expression for \( E_{total} \) by substituting the given values: \( \sigma = 3.2 \mu C/m^2 \), \( z = 0.12 \) m, \( r = 0.1 \) m, and \( \varepsilon_0 = 8.85 \times 10^{-12} \) C^2/(N·m^2). This will yield the electric field strength directly over the center of the hole.

Key Concepts

Electric Field Due to a Plane of Charge

An infinite plane of charge creates a uniform electric field that is perpendicular to its surface. The strength of this electric field (E) is given by the formula E = σ / (2ε₀), where σ is the surface charge density and ε₀ is the permittivity of free space. This concept is crucial for understanding how the electric field behaves in the presence of a charged plane.

Superposition Principle

The superposition principle states that the total electric field created by multiple charge distributions is the vector sum of the electric fields produced by each distribution individually. In this problem, the infinite plane of charge can be thought of as the sum of the field from the full plane and the field from a circular hole, allowing for easier calculation of the resultant field at the specified point.

Electric Field of a Circular Hole

To find the electric field at the center of a circular hole in an infinite plane, one can consider the hole as a negative charge distribution that cancels out part of the field from the plane. The electric field due to a circular disk can be calculated using integration techniques, and its effect can be subtracted from the field of the full plane to determine the net electric field at the center of the hole.