Problem 60 - College Physics

Question

\(\bullet$$\bullet\) In a certain region of space, the electric field \(E\) is uniform; i.e., neither its direction nor its magnitude changes in the region. (a) Use Gauss's law to prove that this region of space must be electrically neutral; that is, there must be no charge in this region. (b) Is the converse true? That is, in a region of space where there is no charge, must \(\vec{E}\) be uniform? Explain.

Step-by-Step Solution

Answer

(a) The region is electrically neutral. (b) No, the field may be non-uniform due to external influences.

Step 1: Understand Gauss's Law

Gauss's law states: \( \oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enclosed}}}{\varepsilon_0} \), where \( \oint \vec{E} \cdot d\vec{A} \) is the electric flux through a closed surface, \( Q_{\text{enclosed}} \) is the net charge enclosed within the surface, and \( \varepsilon_0 \) is the permittivity of free space. In a uniform electric field, the flux through a closed surface may be zero if the net enclosed charge is zero.

Step 2: Check the Field Uniformity Implications

Given that the electric field \(\vec{E}\) is uniform (the same in magnitude and direction everywhere in the region), the net electric flux through a Gaussian surface in any part of this region will be zero unless there is a charge within the surface. If the field has no divergence, it suggests there is no net charge enclosed.

Step 3: Conclusion for Part (a)

For a region with no change in \(\vec{E}\), \( Q_{\text{enclosed}} = 0 \) because the net flux is zero. Therefore, the region must be electrically neutral, confirming no charge is distributed within this space according to Gauss's law.

Step 4: Analyzing the Converse Scenario

Consider if the absence of charge (\( Q_{\text{enclosed}} = 0 \)) implies a uniform \( \vec{E} \). Gauss's law does not guarantee uniformity of \( \vec{E} \), only that the field lines form closed loops or extend to infinity. Complex boundary conditions or sources outside this space can cause a non-uniform field even with no internal charge.

Step 5: Conclusion for Part (b)

The converse is not true; a region without charge doesn't necessarily have a uniform electric field. External influences or variations outside the considered space may result in a non-uniform field even when no internal charge is present.

Key Concepts

Electric Field

An electric field is a region of space around a charged object where other charges experience a force. This concept is fundamental in understanding how charges interact without direct contact. The electric field, represented as \( \vec{E} \), is a vector quantity, meaning it has both a direction and a magnitude.

I'm sure you're curious about what determines the strength and direction of this field. It’s simple! The strength of the electric field is proportional to the amount of charge and inversely proportional to the square of the distance from the charge. The direction is determined by the nature of the charge: fields point away from positive charges and towards negative ones.

One way to visualize electric fields is by using field lines. These lines provide a snapshot of the electric field at any point in space:

• They are drawn closer together where the field is stronger and spread apart where it is weaker.
• Arrows on the lines indicate the direction the field would exert a force on a positive charge.

This visualization aids in understanding concepts like flux and how fields affect charges in space.

Electric Flux

Electric flux, denoted as \( \Phi_E \), is an essential concept in electrostatics that helps us understand how electric fields pass through surfaces. It is defined as the product of the electric field and the area through which the field lines pass. Mathematically, it is expressed as:\[ \Phi_E = \oint \vec{E} \, \cdot \, d\vec{A} \]In simpler terms, it measures how much of the electric field "flows" through a surface. Think of it like the flow of water through a net.

If you have a closed surface, imagine a balloon for instance, the net electric flux is related to the net charge inside the balloon. According to Gauss's law, zero net flux implies that there are no net charges enclosed by the surface, assuming the field is uniform and there are no external influences.

Some key points to help grasp this concept:

• A positive flux indicates that the field is leaving the surface.
• A negative flux means the field is entering the surface.
• If the electric field is uniform and there are no charges inside, the net flux is zero.

This understanding is crucial, especially when considering scenarios involving charged objects and varying field distributions.

Permittivity of Free Space

The permittivity of free space, symbolized by \( \varepsilon_0 \), is a fundamental physical constant that plays a crucial role in electrostatics. It quantifies the ability of the vacuum to allow the formation of electric fields. In essence, it acts as a measure of how much resistance is encountered when forming an electric field in a vacuum.

The value of \( \varepsilon_0 \) is approximately \( 8.854 \times 10^{-12} \, \text{F/m} \) (farads per meter). This might seem like a tiny number, but it has huge implications for electric interactions!

Why is \( \varepsilon_0 \) important? Here are a few reasons:

• It appears in Coulomb's law, determining the force between two charges.
• In Gauss's law, it's used to relate electric flux to the charge enclosed within a surface.
• It sets the scale for the electric field strength that can be sustained in free space.

Understanding \( \varepsilon_0 \) helps provide a deeper insight into how electric fields behave in different environments, and its uniform application makes calculations involving electrostatics consistent worldwide.

Electric Charge

Electric charge is a fundamental property of matter, similar to mass. It causes matter to experience a force when placed in an electric field. The charge can be positive or negative, and objects can either attract or repel each other based on their respective charges.

Basic properties of electric charge include:

• Like charges repel, unlike charges attract.
• It is quantized, meaning charge comes in discrete amounts.
• Charge is conserved; the total charge in an isolated system remains constant.

Charge is typically measured in coulombs (C). In real-world applications, look at subatomic particles: electrons carry a negative charge, while protons are positively charged. Neutrons are neutral, having no charge.

Charge distribution, be it point charges, continuous charge distributions over surfaces, or volumes, can influence the electric field. According to Gauss's law, the distribution of electric charge determines the resulting electric field and, consequently, the electric flux across surfaces.

Uniform Electric Field

A uniform electric field is one where the field lines are parallel, and the strength (magnitude) of the field is the same at every point. This type of field commonly appears in theoretical exercises and experiments due to its simplicity.

In a uniform electric field:

• The field strength doesn't vary with location.
• The field lines are evenly spaced, illustrating constant magnitude.
• It exerts the same force on a charge, regardless of the charge's location.

This concept plays a crucial role when analyzing situations involving capacitors, where a uniform field exists between plates.

The importance of understanding a uniform electric field extends to Gauss's law applications. Such a field results in no net electric flux through a Gaussian surface enclosing no charge. Hence, this signifies that the space inside is electrically neutral, with no internal divergences in the field due to absence of charges.

Remember, while uniform fields are common in simple systems, real-world electric fields can be more complex, often requiring additional considerations for factors like boundary conditions or external influences.