Question

Suppose you have a large spherical balloon and you are able to measure the component \(E_{n}\) of the electric field normal to its surface. If you \(\operatorname{sum} E_{\mathrm{n}} d A\) over the whole surface area of the balloon and obtain a magnitude of \(10.0 \mathrm{~N} \mathrm{~m}^{2} / \mathrm{C}\), what is the electric charge enclosed by the balloon?

Short answer

Answer: The electric charge enclosed by the balloon is approximately \(8.854 \times 10^{-11} \mathrm{~C}\).

Step-by-step solution

Determine the electric flux from the given information

The electric flux (Φ) through the surface of the balloon can be found by summing the normal component of the electric field (\(E_{n}\)) times the differential area element (dA) over the entire surface. Given, \(\sum E_{n} dA = 10.0 \mathrm{~N} \mathrm{~m}^{2} / \mathrm{C}\). So, the electric flux, Φ = \(10.0 \mathrm{~N} \mathrm{~m}^{2} / \mathrm{C}\).

Use Gauss's Law to find the electric charge enclosed

According to Gauss's Law, the electric flux through a closed surface is equal to the enclosed charge (Q) divided by the permittivity of free space (ε₀): Φ = \(\frac{Q}{\varepsilon_0}\) We know the value of Φ and the permittivity of free space is a constant, ε₀ = \(8.854 \times 10^{-12} \mathrm{~C}^{2} \mathrm{~N}^{-1} \mathrm{~m}^{-2}\).

Calculate the enclosed electric charge

Now, we can solve for the electric charge enclosed (Q) by multiplying the electric flux by the permittivity of free space: Q = Φ × ε₀ Q = \(10.0 \mathrm{~N} \mathrm{~m}^{2} / \mathrm{C}\) × \(8.854 \times 10^{-12} \mathrm{~C}^{2} \mathrm{~N}^{-1} \mathrm{~m}^{-2}\)

Simplify and find the electric charge

Multiply the given values to find the electric charge enclosed by the balloon: Q = \(10.0 \times 8.854 \times 10^{-12} \mathrm{~C}\) Q ≈ \(8.854 \times 10^{-11} \mathrm{~C}\) So, the electric charge enclosed by the balloon is approximately \(8.854 \times 10^{-11} \mathrm{~C}\).

Key concepts

Electric Flux

When learning about electrostatics, it's essential to understand the concept of electric flux. Imagine a net catching fish in a river—the amount of fish caught represents the quantity of water that flows through the net. Similarly, electric flux quantifies the number of electric field lines passing through a given area. It is represented mathematically as \[ \Phi = \sum E_n dA \[ where \( \Phi \) is the electric flux, \( E_n \) is the component of the electric field perpendicular to the surface, and \( dA \) is the differential area element.
In layman's terms, if the field is stronger or the area is larger, more field lines pass through, resulting in a higher flux. This concept is crucial when dealing with Gauss's Law, which relates the flux through a closed surface to the charge enclosed within that surface.

Electric Charge

The universe is built on the foundation of electric charge, an intrinsic property of particles such as protons and electrons. Charge exists in two types—positive and negative. It's the 'glue' that holds atoms together and the 'spark' in our electronics.

Electric charge is measured in coulombs (C), and its effect in space is realized through the electric field it generates. The larger the charge, the stronger the influence it exerts on other charges in its vicinity. In our balloon example, the charge enclosed by the surface determines the electric flux we measure outside.

Electric Field

Think of the electric field as the region of influence around charged particles. Just as gravity's invisible hand tugs at masses, electric fields act upon electric charges. Mathematically, it is represented as force per unit charge: \[ E = \frac{F}{q} \[ where \( E \) is the electric field, \( F \) is the force experienced by a small test charge, and \( q \) is the magnitude of that test charge. A high electric field implies a more potent force a charge would experience in that field. This concept helps us understand how charges interact with each other and is central to explaining phenomena in circuits and electrostatics.

Permittivity of Free Space

The permittivity of free space, denoted by \( \varepsilon_0 \), sets the stage for electromagnetic interactions in a vacuum. It's a fundamental constant that describes how easily electric field lines can permeate the vacuum of space. The lower the permittivity, the 'easier' it is for an electric field to exist.

Its value, \(8.854 \times 10^{-12} \mathrm{~C}^{2} \mathrm{~N}^{-1} \mathrm{~m}^{-2}\), may look complex, but it's as crucial as the speed of light in forming the fabric of our electromagnetic universe. Permittivity isn't just a vacuum affair; different materials have their own specific permittivities which affect how fields behave inside them. When using Gauss's Law, \( \varepsilon_0 \) is the factor that relates electric flux to the enclosed charge, bridging the gap between abstract field lines and tangible charge.