5.1: Flux
- Page ID
- 148074
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5.1 Flux
Learning Objectives
- Relate flux to electric field, area, and angle using trigonometry and vector dot products.
- Represent the area of a surface using vector concepts.
5.1.1 Electric Flux: Field and Area
Model
Explore
- Use the model to fill in the table.
| Ring | Ring Size (big / small) | Electric Field Strength (strong / weak) | Number of Electric Field Lines Through the Ring |
|---|---|---|---|
| 1 | |||
| 2 | |||
| 3 |
Critical Thinking & Concept Invention
Recall that the “number of electric field lines through the ring” represents Electric Flux \((\Phi_E)\).
- Describe the relationship between electric field lines and electric flux.
- Identify two different proportional relationships for Electric Flux:
\[ \Phi_E \propto \]
\[ \Phi_E \propto \]
5.1.2 Electric Flux: Angle
Model
Explore
- Use the model to fill in the table.
| Ring | Angle (\(\theta\)) Between Electric Field and Ring Surface | \(\cos(\theta)\) | Number of Electric Field Lines Through the Ring |
|---|---|---|---|
| 1 | |||
| 2 | |||
| 3 |
Critical Thinking & Concept Invention
- How does electric flux vary with \(\theta\)?
Electric flux increases when... - How does electric flux vary with \(\cos(\theta)\)?
Electric flux increases when... - Identify a proportional relationship for electric flux.
Reminder: \[ \cos(0^\circ)=1 \] and \[ \cos(90^\circ)=0 \]
\[ \Phi \propto \cos(\theta) \qquad \text{or} \qquad \Phi \propto \frac{1}{\cos(\theta)} \]
Summary
The angle between the field and the surface affects the overall flux through that surface. The greatest flux occurs when the field is perpendicular to the surface, which is parallel to the normal vector.
Apply
Find the flux in both rings.
5.1.3 Area Vector
Model
Image Credit: OpenStax University Physics Vol 2 (https://openstax.org/details/books/university-physics-volume-2)
Explore
Compare Scenario 1 (left) and Scenario 2 (right) by identifying which one is Greater, Lesser, or the Same.
Recall that the “number of field lines” through a surface represents the Flux \((\Phi)\).
| \(|\vec{E}|\) | \(\Phi_E\) | Area \(S\) | \(\vec{A}\) | Length of \(\hat{n}\) | |
|---|---|---|---|---|---|
| 1 | |||||
| 2 | |||||
| 3 |
Critical Thinking & Concept Invention
- Why do we use a vector to represent the area of a surface?
- What determines the magnitude and direction of the vector \(\vec{A}\)?
- How can the flux be the same for both surfaces if: \[ \Phi_E=\vec{E}\cdot\vec{A} \]
Summary
We can represent a surface area as a vector that incorporates the magnitude of the area with the direction of its normal vector.
The normal vector \(\hat{n}\) is a unit vector. It has a magnitude of 1 and points directly outward from the surface (at a \(90^\circ\) angle to the surface).
\[ \vec{A}=A\hat{n} \]
The flux calculation is a dot product between the electric field and the area vector for the identified surface. Dot products are also equal to the product of the two magnitudes and a cosine factor of the angle between the two vectors.
The surface can be real or imaginary.
\[ \Phi_E=\vec{E}\cdot\vec{A} \]
\[ \Phi_E=\vec{E}\cdot\hat{n}A \]
\[ \Phi_E=EA\cos\theta \]
Apply
Find the electric flux through a rectangular area \(3\text{ cm}\times2\text{ cm}\) between two parallel plates where there is a constant electric field of \(30\text{ N/C}\) for the following orientations of the area:
- Parallel to the plates.
- Perpendicular to the plates.
- With a normal vector at a \(30^\circ\) angle with the direction of the electric field. Note that this angle can also be given as: \[ 180^\circ+30^\circ \]