Problem 11 - University Physics with Modern Physics

Question

22.11. In a certain region of space, the electric field \(\overrightarrow{\boldsymbol{E}}\) is uniform. (a) Use Gauss's law to prove that this region of space must be electrically neutral; that is, the volume charge density \(\rho\) must be zero. (b) Is the converse true? That is, in a region of space where there is no charge, must \(\overrightarrow{\boldsymbol{E}}\) be uniform? Explain.

Step-by-Step Solution

Answer

(a) Uniform \(\overrightarrow{E}\) implies neutral space (\(\rho = 0\)). (b) No charge doesn't ensure uniform \(\overrightarrow{E}\).

Step 1: Understand Gauss's Law

Gauss's Law states that the electric flux \(\Phi_E\) through a closed surface is equal to the charge enclosed \(Q_{\text{enc}}\) divided by the permittivity of free space \(\varepsilon_0\): \[ \Phi_E = \oint \overrightarrow{E} \cdot d\overrightarrow{A} = \frac{Q_{\text{enc}}}{\varepsilon_0} \]This implies that the electric flux is directly proportional to the charge enclosed by the surface.

Step 2: Apply Gauss's Law to the Uniform Electric Field

In a region of space with a uniform electric field \(\overrightarrow{E}\), the flux through a closed Gaussian surface is zero because the electric field is the same entering and exiting each differential area \(d\overrightarrow{A}\). Hence, \[ \oint \overrightarrow{E} \cdot d\overrightarrow{A} = 0 \]From Gauss's Law, if the electric flux is zero, then the enclosed charge \(Q_{\text{enc}}\) must also be zero.

Step 3: Relate Enclosed Charge to Charge Density

The enclosed charge \(Q_{\text{enc}}\) is related to the volume charge density \(\rho\) of the region by: \[ Q_{\text{enc}} = \int \rho \, dV \]If \(Q_{\text{enc}} = 0\), then \(\rho\) must be zero throughout the volume for it to hold over any arbitrary Gaussian surface. This proves that the region must be electrically neutral when the electric field is uniform.

Step 4: Examine the Converse Statement

The converse states that in a region of space where there is no charge, \(\overrightarrow{E}\) must be uniform. Consider an example: space outside a uniformly charged spherical shell has no enclosed charge \(Q_{\text{enc}} = 0\), but the electric field is radial and non-uniform. Thus, while a uniform electric field implies no charge, no charge does not imply a uniform electric field.

Key Concepts

Uniform Electric Field

In physics, an electric field is said to be uniform when it has the same magnitude and direction at every point within a given region. This consistency means that for any point in this space, the force experienced by a test charge is identical, leading to predictable and even electric interactions. Uniform electric fields are often generated between two large, parallel, conducting plates, where charges are distributed evenly.
When we consider Gauss's Law in relation to a uniform electric field, we find that the electric flux through any closed surface is zero. This is because the number of electric field lines entering the surface equals the number of lines exiting. Thus, no net flux passes through.

• The uniformity in the electric field results in balance, ensuring that the region harbors no net charge.
• If the electric flux is zero, by Gauss's Law, the enclosed charge must also be zero.

Therefore, a uniform electric field strongly suggests an electrically neutral region.

Electric Flux

Electric flux represents the quantity of electric field lines penetrating through a given surface. It provides a valuable way to visualize and calculate how electric fields behave over surfaces. Mathematically, it is expressed as the integral of the electric field dot product with a differential area vector over a closed surface: \[ \Phi_E = \oint \overrightarrow{E} \cdot d\overrightarrow{A} \]
Here, electric flux is contingent on both the strength and orientation of the electric field relative to the surface.

• For a uniform electric field, where field lines pass equally everywhere, the flux can easily be computed as the product of the field strength and the perpendicular area.
• In situations with no net enclosed charge, like those under uniform field conditions, the total electric flux becomes zero. This balances the inflow and outflow of field lines through any closed Gaussian surface.

Understanding electric flux is fundamental when applying Gauss's Law to determine charge distributions.

Volume Charge Density

Volume charge density, represented by \( \rho \), is the quantity of charge per unit volume at a point in space, essential for describing how charge is distributed within a body. When \( \rho = 0 \), it indicates a charge-free region. By Gauss's Law, when the total electric flux through a closed surface is zero, the volume charge density within that surface must also be zero, suggesting electrical neutrality.
Let's delve into how this works:

• The equation relating enclosed charge to charge density is \( Q_{\text{enc}} = \int \rho \ dV \).
• When \( Q_{\text{enc}} = 0 \), this integral signifies that \( \rho \) must be zero throughout the volume, meaning no charge is present to affect the electric field.

This absence of charge helps understand why a uniform electric field does not allow a non-zero volume charge density. However, it's imperative to note the reverse isn't necessarily true: a region with zero volume charge density doesn't guarantee a uniform electric field, as this can also describe non-uniform fields outside charged bodies, such as a sphere's exterior.