Question

Four charges are placed in a three-dimensional space. The charges have magnitudes \(+3 q,-q,+2 q,\) and \(-7 q .\) If a Gaussian surface encloses all the charges, what will be the electric flux through that surface?

Short answer

Answer: The electric flux through the Gaussian surface is \(\frac{-3q}{\varepsilon_0}\).

Step-by-step solution

Determine the net charge enclosed by the Gaussian surface

To find the net charge enclosed by the Gaussian surface, we will sum all the charges within the surface: \((+3q) + (-q) + (+2q) + (-7q)\).

Calculate the net charge

Now we will calculate the total charge: \(+3q - q + 2q - 7q = -3q\).

Apply Gauss's Law

With the net charge enclosed by the Gaussian surface, we can find the electric flux using Gauss's Law: \(\Phi_E = \frac{Q_{enclosed}}{\varepsilon_0}\).

Calculate the electric flux

The net charge is \(-3q\), so the electric flux through the Gaussian surface is: \(\Phi_E = \frac{-3q}{\varepsilon_0}\). Thus, the electric flux through the Gaussian surface is \(\frac{-3q}{\varepsilon_0}\).

Key concepts

Electric Flux

Electric flux is a measure of the amount of electric field passing through a given area. Imagine it like the number of electric field lines crossing a surface. It tells us how the electric field "flows" through a surface, like water flowing through a net. The more field lines crossing perpendicularly, the greater the flux.
Electric flux (\(\Phi_E\)) through a closed surface depends on the net charge enclosed within that surface and is calculated using Gauss's Law. Essentially:

• The electric field contributes more to flux if it is stronger.
• Flux is also greater when the field lines are perpendicular to the surface.
• When field lines run parallel to the surface, they don't contribute to the flux.

Be mindful that electric flux considers both the strength and orientation of the field across a surface.

Gaussian Surface

A Gaussian surface is an imaginary closed surface used in Gauss's Law to simplify the calculation of electric fields. You can think of a Gaussian surface as a balloon you could inflate to completely envelop charges in three-dimensional space.
The purpose of the Gaussian surface is to utilize symmetry in problems, allowing for an easier calculation of electric flux. When applying Gauss's Law, the choice of a Gaussian surface should:

• Completely enclose the charge(s) involved in the calculations.
• Take advantage of any symmetry present, making the mathematics simpler.
• Be chosen such that it is easier to calculate the electric field over it.

Gaussian surfaces align the theory of electric fields with practical calculations, shining a light on how field lines interact with enclosed charges.

Net Charge

Net charge is the total amount of electric charge contained within a specific area or volume. It is the sum of all positive and negative charges in a given space. Think of net charge as the balance between positive and negative charges.
To find the net charge within a Gaussian surface:

• Add the magnitudes of all positive charges.
• Add the magnitudes of all negative charges.
• Subtract the total negative charge from the total positive charge.

In exercises like the original problem, correctly identifying and calculating net charge is crucial. It directly influences the electric flux through the surface.

Three-dimensional Space

Three-dimensional space provides a real-world context in which electric charges are arranged not just on a flat plane, but with depth and spatial separation. This means that charges can be enclosed from all directions.
Understanding this space allows students to visualize how electric fields and Gaussian surfaces operate in all dimensions, not just along a single axis. In practice:

• Charges can be placed in various configurations, which affects calculation.
• Spatial understanding helps when determining symmetry for Gaussian surfaces.
• Three-dimensional visualization improves comprehension of flux pathways.

Recognizing the role of three-dimensional space bridge concepts from theoretical exercises to tangible physical scenarios.